Álgebra geométrica

Algebraic structure designed for geometry

En matemáticas , un álgebra geométrica (también conocida como álgebra de Clifford ) es una extensión del álgebra elemental para trabajar con objetos geométricos como los vectores . El álgebra geométrica se construye a partir de dos operaciones fundamentales, la suma y el producto geométrico. La multiplicación de vectores da como resultado objetos de dimensiones superiores llamados multivectores . En comparación con otros formalismos para manipular objetos geométricos, el álgebra geométrica es notable por admitir la división vectorial (aunque generalmente no para todos los elementos) y la suma de objetos de diferentes dimensiones.

El producto geométrico fue mencionado brevemente por primera vez por Hermann Grassmann , ^[1] quien estaba principalmente interesado en desarrollar el álgebra exterior estrechamente relacionada . En 1878, William Kingdon Clifford amplió en gran medida el trabajo de Grassmann para formar lo que ahora se suele llamar álgebras de Clifford en su honor (aunque el propio Clifford eligió llamarlas "álgebras geométricas"). Clifford definió el álgebra de Clifford y su producto como una unificación del álgebra de Grassmann y el álgebra de cuaterniones de Hamilton . Agregar el dual del producto exterior de Grassmann (el "encuentro") permite el uso del álgebra de Grassmann-Cayley , y una versión conforme de esta última junto con un álgebra conforme de Clifford produce un álgebra geométrica conforme (CGA) que proporciona un marco para las geometrías clásicas . ^[2] En la práctica, estas y varias operaciones derivadas permiten una correspondencia de elementos, subespacios y operaciones del álgebra con interpretaciones geométricas. Durante varias décadas, las álgebras geométricas fueron ignoradas, eclipsadas en gran medida por el cálculo vectorial , recién desarrollado para describir el electromagnetismo. El término "álgebra geométrica" ​​fue popularizado nuevamente en la década de 1960 por David Hestenes , quien defendió su importancia para la física relativista. ^[3]

Los escalares y vectores tienen su interpretación habitual y forman subespacios distintos de un álgebra geométrica. Los bivectores proporcionan una representación más natural de las cantidades pseudovectoriales del cálculo vectorial 3D que se derivan como un producto vectorial , como el área orientada, el ángulo de rotación orientado, el par, el momento angular y el campo magnético . Un trivector puede representar un volumen orientado, etc. Se puede utilizar un elemento llamado cuchilla para representar un subespacio y proyecciones ortogonales sobre ese subespacio. Las rotaciones y las reflexiones se representan como elementos. A diferencia de un álgebra vectorial, un álgebra geométrica acomoda naturalmente cualquier número de dimensiones y cualquier forma cuadrática como en la relatividad .

Ejemplos de álgebras geométricas aplicadas en física incluyen el álgebra del espacio-tiempo (y el álgebra menos común del espacio físico ) y el álgebra geométrica conforme . El cálculo geométrico , una extensión del AG que incorpora diferenciación e integración , se puede utilizar para formular otras teorías como el análisis complejo y la geometría diferencial , por ejemplo, utilizando el álgebra de Clifford en lugar de formas diferenciales . El álgebra geométrica ha sido defendida, sobre todo por David Hestenes ^[4] y Chris Doran ^[5] , como el marco matemático preferido para la física . Los defensores afirman que proporciona descripciones compactas e intuitivas en muchas áreas, incluidas la mecánica clásica y cuántica , la teoría electromagnética y la relatividad . ^[6] El AG también ha encontrado uso como herramienta computacional en gráficos de computadora ^[7] y robótica .

Definición y notación

Hay varias formas diferentes de definir un álgebra geométrica. El enfoque original de Hestenes era axiomático, ^[8] "lleno de significado geométrico" y equivalente al álgebra universal ^{[a] de Clifford. [9]} Dado un espacio vectorial de dimensión finita ⁠ ⁠ ${\displaystyle V}$ sobre un cuerpo ⁠ ⁠ ${\displaystyle F}$ con una forma bilineal simétrica (el producto interno , ^[b] por ejemplo, la métrica euclidiana o lorentziana ) ⁠ ⁠ ${\displaystyle g:V\times V\to F}$ , el álgebra geométrica del espacio cuadrático ⁠ ⁠ ${\displaystyle (V,g)}$ es el álgebra de Clifford ⁠ ⁠ ${\displaystyle \operatorname {Cl} (V,g)}$ , un elemento del cual se llama multivector. El álgebra de Clifford se define comúnmente como un álgebra cociente del álgebra tensorial , aunque esta definición es abstracta, por lo que la siguiente definición se presenta sin requerir álgebra abstracta .

Definición

Un álgebra asociativa unitaria con una ${\displaystyle \operatorname {Cl} (V,g)}$ forma bilineal simétrica no degenerada es el álgebra de ${\displaystyle g:V\times V\to F}$Clifford del espacio cuadrático si ${\displaystyle (V,g)}$ [ ^10]

• contiene ⁠ ⁠ ${\displaystyle F}$ y ⁠ ⁠ ${\displaystyle V}$ como subespacios distintos
• ⁠ ⁠ ${\displaystyle a^{2}=g(a,a)1}$ para ⁠ ⁠ ${\displaystyle a\in V}$
• ⁠ ⁠ ${\displaystyle V}$ genera ⁠ ⁠ ${\displaystyle \operatorname {Cl} (V,g)}$ como un álgebra
• ⁠ ⁠ ${\displaystyle \operatorname {Cl} (V,g)}$ no es generado por ningún subespacio propio de ⁠ ⁠ ${\displaystyle V}$ .

Para cubrir formas bilineales simétricas degeneradas, se debe modificar la última condición. ^[c] Se puede demostrar que estas condiciones caracterizan de manera única al producto geométrico.

En el resto de este artículo, solo se considerará el caso real , ⁠ ⁠ . La notación ${\displaystyle F=\mathbb {R} }$⁠ ⁠ ${\displaystyle {\mathcal {G}}(p,q)}$ (respectivamente ⁠ ⁠ ${\displaystyle {\mathcal {G}}(p,q,r)}$ ) se utilizará para denotar un álgebra geométrica para la cual la forma bilineal ⁠ ⁠ ${\displaystyle g}$ tiene la signatura ⁠ ⁠ ${\displaystyle (p,q)}$ (respectivamente ⁠ ⁠ ${\displaystyle (p,q,r)}$ ).

El producto en el álgebra se llama producto geométrico , y el producto en el álgebra exterior contenida se llama producto exterior (frecuentemente llamado producto cuña o producto externo ^[d] ). Es estándar denotar estos respectivamente por yuxtaposición (es decir, suprimiendo cualquier símbolo de multiplicación explícito) y el símbolo ⁠ ⁠ ${\displaystyle \wedge }$ .

La definición anterior del álgebra geométrica es todavía algo abstracta, por lo que resumiremos aquí las propiedades del producto geométrico. Para multivectores : ${\displaystyle A,B,C\in {\mathcal {G}}(p,q)}$

• ⁠ ⁠ ${\displaystyle AB\in {\mathcal {G}}(p,q)}$ ( cierre )
• ⁠ ⁠ ${\displaystyle 1A=A1=A}$ , donde ⁠ ⁠ ${\displaystyle 1}$ es el elemento identidad (existencia de un elemento identidad )
• ⁠ ⁠ ${\displaystyle A(BC)=(AB)C}$ ( asociatividad )
• ⁠ ⁠ ${\displaystyle A(B+C)=AB+AC}$ y ⁠ ⁠ ${\displaystyle (B+C)A=BA+CA}$ ( distributividad )
• ⁠ ⁠ ${\displaystyle a^{2}=g(a,a)1}$ para ⁠ ⁠ ${\displaystyle a\in V}$ .

El producto exterior tiene las mismas propiedades, excepto que la última propiedad anterior se reemplaza por ⁠ ⁠ ${\displaystyle a\wedge a=0}$ por ⁠ ⁠ ${\displaystyle a\in V}$ .

Nótese que en la última propiedad anterior, el número real ⁠ ⁠ ${\displaystyle g(a,a)}$ no necesita ser no negativo si ⁠ ⁠ ${\displaystyle g}$ no es positivo-definido. Una propiedad importante del producto geométrico es la existencia de elementos que tienen un inverso multiplicativo. Para un vector ⁠ ⁠ ${\displaystyle a}$ , si entonces existe y es igual a ⁠ ⁠ . Un elemento distinto de cero del álgebra no necesariamente tiene un inverso multiplicativo. Por ejemplo, si es un vector en tal que ⁠ ⁠ , el elemento es tanto un elemento idempotente no trivial como un divisor de cero distinto de cero , y por lo tanto no tiene inverso. ^[e] ${\displaystyle a^{2}\neq 0}$ ${\displaystyle a^{-1}}$ ${\displaystyle g(a,a)^{-1}a}$ ${\displaystyle u}$ ${\displaystyle V}$ ${\displaystyle u^{2}=1}$ ${\displaystyle \textstyle {\frac {1}{2}}(1+u)}$

Es habitual identificar y con sus imágenes bajo las incrustaciones naturales y ⁠ ⁠ . En este artículo, se asume esta identificación. En todo momento, los términos escalar y vector se refieren a elementos de y respectivamente (y de sus imágenes bajo esta incrustación). ${\displaystyle \mathbb {R} }$ ${\displaystyle V}$ ${\displaystyle \mathbb {R} \to {\mathcal {G}}(p,q)}$ ${\displaystyle V\to {\mathcal {G}}(p,q)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle V}$

Producto geométrico

Dados dos vectores y ⁠ ⁠ , si el producto geométrico es ^[13] anticonmutativo; son perpendiculares (arriba) porque ⁠ ⁠ , si es conmutativo; son paralelos (abajo) porque ⁠ ⁠ . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle ab}$ ${\displaystyle a\cdot b=0}$ ${\displaystyle a\wedge b=0}$

Interpretación geométrica de elementos de grado en un álgebra exterior real para (punto con signo), ( segmento de línea dirigido o vector), (elemento de plano orientado), (volumen orientado). El producto exterior de vectores se puede visualizar como cualquier forma dimensional (por ejemplo , - paralelepípedo, - elipsoide ) ; con magnitud ( hipervolumen ) y orientación definidas por la de su límite dimensional y en qué lado está el interior. ^[14] [ ^15] ${\displaystyle n}$ ${\displaystyle n=0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle (n-1)}$

Para los vectores ⁠ ⁠ ${\displaystyle a}$ y ⁠ ⁠ ${\displaystyle b}$ , podemos escribir el producto geométrico de dos vectores cualesquiera ⁠ ⁠ ${\displaystyle a}$ y ⁠ ⁠ ${\displaystyle b}$ como la suma de un producto simétrico y un producto antisimétrico:

${\displaystyle ab={\frac {1}{2}}(ab+ba)+{\frac {1}{2}}(ab-ba).}$

Así podemos definir el producto interno de los vectores como

${\displaystyle a\cdot b:=g(a,b),}$

de modo que el producto simétrico se puede escribir como

${\displaystyle {\frac {1}{2}}(ab+ba)={\frac {1}{2}}\left((a+b)^{2}-a^{2}-b^{2}\right)=a\cdot b.}$

Por el contrario, ⁠ ⁠ ${\displaystyle g}$ está completamente determinado por el álgebra. La parte antisimétrica es el producto exterior de los dos vectores, el producto del álgebra exterior contenida :

${\displaystyle a\wedge b:={\frac {1}{2}}(ab-ba)=-(b\wedge a).}$

Luego por simple adición:

${\displaystyle ab=a\cdot b+a\wedge b}$la forma no generalizada o vectorial del producto geométrico.

Los productos internos y externos están asociados con conceptos familiares del álgebra vectorial estándar. Geométricamente, y son paralelos si su producto geométrico es igual a su producto interno, mientras que y son perpendiculares si su producto geométrico es igual a su producto externo. En un álgebra geométrica para la cual el cuadrado de cualquier vector distinto de cero es positivo, el producto interno de dos vectores puede identificarse con el producto escalar del álgebra vectorial estándar. El producto externo de dos vectores puede identificarse con el área con signo encerrada por un paralelogramo cuyos lados son los vectores. El producto vectorial de dos vectores en dimensiones con forma cuadrática definida positiva está estrechamente relacionado con su producto externo. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle 3}$

La mayoría de los casos de álgebras geométricas de interés tienen una forma cuadrática no degenerada. Si la forma cuadrática es completamente degenerada , el producto interno de dos vectores cualesquiera siempre es cero y, en ese caso, el álgebra geométrica es simplemente un álgebra exterior. A menos que se indique lo contrario, este artículo tratará únicamente álgebras geométricas no degeneradas.

El producto exterior se extiende naturalmente como un operador binario bilineal asociativo entre dos elementos cualesquiera del álgebra, satisfaciendo las identidades

${\displaystyle {\begin{aligned}1\wedge a_{i}&=a_{i}\wedge 1=a_{i}\\a_{1}\wedge a_{2}\wedge \cdots \wedge a_{r}&={\frac {1}{r!}}\sum _{\sigma \in {\mathfrak {S}}_{r}}\operatorname {sgn} (\sigma )a_{\sigma (1)}a_{\sigma (2)}\cdots a_{\sigma (r)},\end{aligned}}}$

donde la suma es sobre todas las permutaciones de los índices, con el signo de la permutación , y son vectores (no elementos generales del álgebra). Como cada elemento del álgebra puede expresarse como la suma de productos de esta forma, esto define el producto exterior para cada par de elementos del álgebra. De la definición se deduce que el producto exterior forma un álgebra alternada . ${\displaystyle \operatorname {sgn} (\sigma )}$ ${\displaystyle a_{i}}$

La ecuación de estructura equivalente para el álgebra de Clifford es ^{[16] [17]}

${\displaystyle a_{1}a_{2}a_{3}\dots a_{n}=\sum _{i=0}^{[{\frac {n}{2}}]}\sum _{\mu \in {}{\mathcal {C}}}(-1)^{k}\operatorname {Pf} (a_{\mu _{1}}\cdot a_{\mu _{2}},\dots ,a_{\mu _{2i-1}}\cdot a_{\mu _{2i}})a_{\mu _{2i+1}}\land \dots \land a_{\mu _{n}}}$

donde es el Pfaffian de ⁠ ⁠ y proporciona combinaciones , ⁠ ⁠ , de ⁠ ⁠ índices divididos en ⁠ ⁠ y ⁠ ⁠ partes y ⁠ ⁠ es la paridad de la combinación . ${\displaystyle \operatorname {Pf} (A)}$ ${\displaystyle A}$ ${\textstyle {\mathcal {C}}={\binom {n}{2i}}}$ ${\displaystyle \mu }$ ${\displaystyle n}$ ${\displaystyle 2i}$ ${\displaystyle n-2i}$ ${\displaystyle k}$

El Pfaffian proporciona una métrica para el álgebra exterior y, como señaló Claude Chevalley, el álgebra de Clifford se reduce al álgebra exterior con una forma cuadrática cero. ^[18] El papel que juega el Pfaffian se puede entender desde un punto de vista geométrico desarrollando el álgebra de Clifford a partir de los símplices . ^[19] Esta derivación proporciona una mejor conexión entre el triángulo de Pascal y los símplices porque proporciona una interpretación de la primera columna de unos.

Cuchillas, calidades y bases

Un multivector que es el producto exterior de vectores linealmente independientes se llama cuchilla , y se dice que es de grado ⁠ ⁠ . ^[f] Un multivector que es la suma de cuchillas de grado se llama multivector (homogéneo) de grado ⁠ ⁠ . A partir de los axiomas, con clausura, todo multivector del álgebra geométrica es una suma de cuchillas. ${\displaystyle r}$ ${\displaystyle r}$ ${\displaystyle r}$ ${\displaystyle r}$

Consideremos un conjunto de vectores linealmente independientes que abarcan un subespacio ⁠ ⁠ -dimensional del espacio vectorial. Con ellos, podemos definir una matriz simétrica real (de la misma manera que una matriz de Gram ). ${\displaystyle r}$ ${\displaystyle \{a_{1},\ldots ,a_{r}\}}$ ${\displaystyle r}$

${\displaystyle [\mathbf {A} ]_{ij}=a_{i}\cdot a_{j}}$

Por el teorema espectral , se puede diagonalizar a una matriz diagonal mediante una matriz ortogonal mediante ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {D} }$ ${\displaystyle \mathbf {O} }$

${\displaystyle \sum _{k,l}[\mathbf {O} ]_{ik}[\mathbf {A} ]_{kl}[\mathbf {O} ^{\mathrm {T} }]_{lj}=\sum _{k,l}[\mathbf {O} ]_{ik}[\mathbf {O} ]_{jl}[\mathbf {A} ]_{kl}=[\mathbf {D} ]_{ij}}$

Definir un nuevo conjunto de vectores , conocidos ${\displaystyle \{e_{1},\ldots ,e_{r}\}}$ como vectores base ortogonales, que serán aquellos transformados por la matriz ortogonal:

${\displaystyle e_{i}=\sum _{j}[\mathbf {O} ]_{ij}a_{j}}$

Como las transformaciones ortogonales conservan los productos internos, se deduce que y, por lo tanto, son perpendiculares. En otras palabras, el producto geométrico de dos vectores distintos está completamente especificado por su producto exterior o, de manera más general, ${\displaystyle e_{i}\cdot e_{j}=[\mathbf {D} ]_{ij}}$ ${\displaystyle \{e_{1},\ldots ,e_{r}\}}$ ${\displaystyle e_{i}\neq e_{j}}$

${\displaystyle {\begin{array}{rl}e_{1}e_{2}\cdots e_{r}&=e_{1}\wedge e_{2}\wedge \cdots \wedge e_{r}\\&=\left(\sum _{j}[\mathbf {O} ]_{1j}a_{j}\right)\wedge \left(\sum _{j}[\mathbf {O} ]_{2j}a_{j}\right)\wedge \cdots \wedge \left(\sum _{j}[\mathbf {O} ]_{rj}a_{j}\right)\\&=(\det \mathbf {O} )a_{1}\wedge a_{2}\wedge \cdots \wedge a_{r}\end{array}}}$

Por lo tanto, cada hoja de grado puede escribirse como el producto exterior de vectores. De manera más general, si se permite un álgebra geométrica degenerada, entonces la matriz ortogonal se reemplaza por una matriz de bloques que es ortogonal en el bloque no degenerado, y la matriz diagonal tiene entradas de valor cero a lo largo de las dimensiones degeneradas. Si los nuevos vectores del subespacio no degenerado se normalizan de acuerdo con ${\displaystyle r}$ ${\displaystyle r}$

${\displaystyle {\widehat {e_{i}}}={\frac {1}{\sqrt {|e_{i}\cdot e_{i}|}}}e_{i},}$

entonces estos vectores normalizados deben elevar al cuadrado a o ⁠ ⁠ . Por la ley de inercia de Sylvester , el número total de ⁠ ⁠ y el número total de ⁠ ⁠ a lo largo de la matriz diagonal es invariante. Por extensión, el número total de estos vectores que elevan al cuadrado a y el número total que elevan al cuadrado a es invariante. (El número total de vectores base que elevan al cuadrado a cero también es invariante, y puede ser distinto de cero si se permite el caso degenerado). Denotamos esta álgebra ⁠ ⁠ . Por ejemplo, modela el espacio euclidiano tridimensional , el espaciotiempo relativista y un álgebra geométrica conforme de un espacio tridimensional. ${\displaystyle +1}$ ${\displaystyle -1}$ ${\displaystyle +1}$ ${\displaystyle -1}$ ${\displaystyle p}$ ${\displaystyle +1}$ ${\displaystyle q}$ ${\displaystyle -1}$ ${\displaystyle {\mathcal {G}}(p,q)}$ ${\displaystyle {\mathcal {G}}(3,0)}$ ${\displaystyle {\mathcal {G}}(1,3)}$ ${\displaystyle {\mathcal {G}}(4,1)}$

El conjunto de todos los productos posibles de vectores base ortogonales con índices en orden creciente, incluido el producto vacío, forma una base para toda el álgebra geométrica (un análogo del teorema PBW ). Por ejemplo, la siguiente es una base para el álgebra geométrica ⁠ ⁠ : ${\displaystyle n}$ ${\displaystyle 1}$ ${\displaystyle {\mathcal {G}}(3,0)}$

${\displaystyle \{1,e_{1},e_{2},e_{3},e_{1}e_{2},e_{2}e_{3},e_{3}e_{1},e_{1}e_{2}e_{3}\}}$

Una base formada de esta manera se denomina base estándar para el álgebra geométrica, y cualquier otra base ortogonal para producirá otra base estándar. Cada base estándar consta de elementos. Cada multivector del álgebra geométrica se puede expresar como una combinación lineal de los elementos de la base estándar. Si los elementos de la base estándar son un conjunto de índices, entonces el producto geométrico de dos multivectores cualesquiera es ${\displaystyle V}$ ${\displaystyle 2^{n}}$ ${\displaystyle \{B_{i}\mid i\in S\}}$ ${\displaystyle S}$

${\displaystyle \left(\sum _{i}\alpha _{i}B_{i}\right)\left(\sum _{j}\beta _{j}B_{j}\right)=\sum _{i,j}\alpha _{i}\beta _{j}B_{i}B_{j}.}$

La terminología " -vector" se utiliza a menudo para describir multivectores que contienen elementos de un solo grado. En el espacio de dimensiones superiores, algunos de estos multivectores no son cuchillas (no se pueden factorizar en el producto exterior de vectores). A modo de ejemplo, en no se puede factorizar ; sin embargo, normalmente, estos elementos del álgebra no se prestan a la interpretación geométrica como objetos, aunque pueden representar cantidades geométricas como rotaciones. Solo los - , - , - y -vectores son siempre cuchillas en el -espacio . ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle e_{1}\wedge e_{2}+e_{3}\wedge e_{4}}$ ${\displaystyle {\mathcal {G}}(4,0)}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle (n-1)}$ ${\displaystyle n}$ ${\displaystyle n}$

Versor

Un ⁠ ⁠ ${\displaystyle k}$ -versor es un multivector que puede expresarse como el producto geométrico de vectores invertibles. ^{[g] [21]} Los cuaterniones unitarios (originalmente llamados versores por Hamilton) pueden identificarse con rotores en el espacio 3D de la misma manera que los rotores 2D reales subsumen números complejos; para los detalles, consulte a Dorst. ^[22] ${\displaystyle k}$

Algunos autores utilizan el término "producto versor" para referirse al caso frecuente en el que un operando está "emparedado" entre operadores. Las descripciones de rotaciones y reflexiones, incluidos sus morfismos externos, son ejemplos de este tipo de emparedado. Estos morfismos externos tienen una forma algebraica particularmente simple. ^[h] En concreto, una aplicación de vectores de la forma

${\displaystyle V\to V:a\mapsto RaR^{-1}}$se extiende al morfismo externo ${\displaystyle {\mathcal {G}}(V)\to {\mathcal {G}}(V):A\mapsto RAR^{-1}.}$

Dado que tanto los operadores como el operando son versores, existe la posibilidad de ejemplos alternativos, como rotar un rotor o reflejar un espinor, siempre que se pueda atribuir algún significado geométrico o físico a dichas operaciones.

Por el teorema de Cartan-Dieudonné tenemos que toda isometría puede darse como reflexiones en hiperplanos y como las reflexiones compuestas proporcionan rotaciones entonces tenemos que las transformaciones ortogonales son versores.

En términos de grupo, para un ⁠ ⁠ ${\displaystyle {\mathcal {G}}(p,q)}$ real, no degenerado , habiendo identificado el grupo como el grupo de todos los elementos invertibles de ⁠ ⁠ , Lundholm da una prueba de que el "grupo versor" (el conjunto de versores invertibles) es igual al grupo de Lipschitz ( también conocido como grupo de Clifford, aunque Lundholm desaprueba este uso). ^[23] ${\displaystyle {\mathcal {G}}^{\times }}$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle \{v_{1}v_{2}\cdots v_{k}\in {\mathcal {G}}\mid v_{i}\in V^{\times }\}}$ ${\displaystyle \Gamma }$

Subgrupos del grupo Lipschitz

Denotamos la involución de grado como ⁠ ⁠ ${\displaystyle {\widehat {S}}}$ y la reversión como ⁠ ⁠ ${\displaystyle {\widetilde {S}}}$ .

Aunque el grupo de Lipschitz (definido como ⁠ ⁠ ${\displaystyle \{S\in {\mathcal {G}}^{\times }\mid {\widehat {S}}VS^{-1}\subseteq V\}}$ ) y el grupo versor (definido como ⁠ ⁠ ${\displaystyle \textstyle \{\prod _{i=0}^{k}v_{i}\mid v_{i}\in V^{\times },k\in \mathbb {N} \}}$ ) tienen definiciones divergentes, son el mismo grupo. Lundholm define los subgrupos ⁠ ⁠ ${\displaystyle \operatorname {Pin} }$ , ⁠ ⁠ ${\displaystyle \operatorname {Spin} }$ y ⁠ ⁠ ${\displaystyle \operatorname {Spin} ^{+}}$ del grupo de Lipschitz. ^[24]

Múltiples análisis de espinores utilizan GA como representación. ^[25]

Proyección de calificaciones

Se puede establecer una estructura de espacio vectorial graduado en un álgebra geométrica ${\displaystyle \mathbb {Z} }$ mediante el uso del producto exterior que es naturalmente inducido por el producto geométrico.

Dado que el producto geométrico y el producto exterior son iguales en los vectores ortogonales, esta clasificación se puede construir convenientemente utilizando una base ortogonal . ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$

Los elementos del álgebra geométrica que son múltiplos escalares de son de grado y se llaman escalares . Los elementos que están en el lapso de son de grado ⁠ ⁠ y son los vectores ordinarios. Los elementos en el lapso de son de grado y son los bivectores. Esta terminología continúa hasta el último grado de ⁠ ⁠ -vectores. Alternativamente, ⁠ ⁠ -vectores se llaman pseudoescalares , ⁠ ⁠ -vectores se llaman pseudovectores, etc. Muchos de los elementos del álgebra no se califican mediante este esquema ya que son sumas de elementos de diferente grado. Se dice que dichos elementos son de grado mixto . La calificación de multivectores es independiente de la base elegida originalmente. ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$ ${\displaystyle 1}$ ${\displaystyle \{e_{i}e_{j}\mid 1\leq i<j\leq n\}}$ ${\displaystyle 2}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle (n-1)}$

Esta es una gradación como espacio vectorial, pero no como álgebra. Debido a que el producto de una ⁠ ⁠ ${\displaystyle r}$ -cuchilla y una ⁠ ⁠ ${\displaystyle s}$ -cuchilla está contenido en el espacio de ⁠ ⁠ -cuchillas , el álgebra geométrica es un álgebra filtrada . ${\displaystyle 0}$ ${\displaystyle r+s}$

Un multivector se puede descomponer con el operador de proyección de grado ⁠ ⁠ , que genera la porción de grado de ⁠ ⁠ . Como resultado: ${\displaystyle A}$ ${\displaystyle \langle A\rangle _{r}}$ ${\displaystyle r}$ ${\displaystyle A}$

${\displaystyle A=\sum _{r=0}^{n}\langle A\rangle _{r}}$

A modo de ejemplo, el producto geométrico de dos vectores ya que y y ⁠ ⁠ , para otros que y ⁠ ⁠ . ${\displaystyle ab=a\cdot b+a\wedge b=\langle ab\rangle _{0}+\langle ab\rangle _{2}}$ ${\displaystyle \langle ab\rangle _{0}=a\cdot b}$ ${\displaystyle \langle ab\rangle _{2}=a\wedge b}$ ${\displaystyle \langle ab\rangle _{i}=0}$ ${\displaystyle i}$ ${\displaystyle 0}$ ${\displaystyle 2}$

Un multivector también puede descomponerse en componentes par e impar, que pueden expresarse respectivamente como la suma de los componentes de grado par y de grado impar anteriores: ${\displaystyle A}$

${\displaystyle A^{[0]}=\langle A\rangle _{0}+\langle A\rangle _{2}+\langle A\rangle _{4}+\cdots }$

${\displaystyle A^{[1]}=\langle A\rangle _{1}+\langle A\rangle _{3}+\langle A\rangle _{5}+\cdots }$

Este es el resultado de olvidar la estructura de un espacio vectorial -graduado a un ${\displaystyle \mathrm {Z} }$ espacio vectorial -graduado . El producto geométrico respeta esta gradación más gruesa . Por lo tanto , además de ser un espacio vectorial -graduado , el álgebra geométrica es un álgebra -graduada , también conocida como superálgebra . ${\displaystyle \mathrm {Z} _{2}}$ ${\displaystyle \mathrm {Z} _{2}}$ ${\displaystyle \mathrm {Z} _{2}}$

Restringiéndonos a la parte par, el producto de dos elementos pares también es par. Esto significa que los multivectores pares definen una subálgebra par . La subálgebra par de un álgebra geométrica de ⁠ ⁠ dimensiones es ${\displaystyle n}$isomorfa (sin preservar ni la filtración ni la gradación) a un álgebra geométrica completa de dimensiones. Algunos ejemplos incluyen y ⁠ ⁠ . ${\displaystyle (n-1)}$ ${\displaystyle {\mathcal {G}}^{[0]}(2,0)\cong {\mathcal {G}}(0,1)}$ ${\displaystyle {\mathcal {G}}^{[0]}(1,3)\cong {\mathcal {G}}(3,0)}$

Representación de subespacios

El álgebra geométrica representa los subespacios de como aspas, y por eso coexisten en la misma álgebra con vectores de ⁠ ⁠ . Un subespacio ⁠ ⁠ -dimensional de se representa tomando una base ortogonal y usando el producto geométrico para formar la aspa ⁠ ⁠ . Hay múltiples aspas que representan ⁠ ⁠ ; todas las que representan son múltiplos escalares de ⁠ ⁠ . Estas aspas se pueden separar en dos conjuntos: múltiplos positivos de y múltiplos negativos de ⁠ ⁠ . Se dice que los múltiplos positivos de tienen la misma orientación que ⁠ ⁠ , y los múltiplos negativos la orientación opuesta . ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle k}$ ${\displaystyle W}$ ${\displaystyle V}$ ${\displaystyle \{b_{1},b_{2},\ldots ,b_{k}\}}$ ${\displaystyle D=b_{1}b_{2}\cdots b_{k}}$ ${\displaystyle W}$ ${\displaystyle W}$ ${\displaystyle D}$ ${\displaystyle D}$ ${\displaystyle D}$ ${\displaystyle D}$ ${\displaystyle D}$

Las hojas son importantes ya que las operaciones geométricas como proyecciones, rotaciones y reflexiones dependen de la factorabilidad a través del producto exterior que (la clase restringida de) -hojas proporciona ${\displaystyle n}$ pero que (la clase generalizada de) multivectores de grado no proporciona ${\displaystyle n}$ cuando ⁠ ⁠ ${\displaystyle n\geq 4}$ .

Unidades pseudoescalares

Los pseudoescalares unitarios son cuchillas que desempeñan papeles importantes en GA. Un pseudoescalar unitario para un subespacio no degenerado de es una cuchilla que es el producto de los miembros de una base ortonormal para ⁠ ⁠ . Se puede demostrar que si y son ambos pseudoescalares unitarios para ⁠ ⁠ , entonces y ⁠ ⁠ . Si uno no elige una base ortonormal para ⁠ ⁠ , entonces la incrustación de Plücker da un vector en el álgebra exterior pero solo hasta la escala. Usando el isomorfismo del espacio vectorial entre el álgebra geométrica y el álgebra exterior, esto da la clase de equivalencia de para todo ⁠ ⁠ . La ortonormalidad elimina esta ambigüedad excepto por los signos anteriores. ${\displaystyle W}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle I}$ ${\displaystyle I'}$ ${\displaystyle W}$ ${\displaystyle I=\pm I'}$ ${\displaystyle I^{2}=\pm 1}$ ${\displaystyle W}$ ${\displaystyle \alpha I}$ ${\displaystyle \alpha \neq 0}$

Supongamos que se forma el álgebra geométrica con el conocido producto interno definido positivo en . Dado un plano (subespacio bidimensional) de  ⁠ ⁠ , se puede encontrar una base ortonormal que abarca el plano y, por lo tanto, encontrar un pseudoescalar unitario que represente este plano. El producto geométrico de dos vectores cualesquiera en el espacio de y se encuentra en ⁠ ⁠ , es decir, es la suma de un ⁠ ⁠ -vector y un ⁠ ⁠ -vector. ${\displaystyle {\mathcal {G}}(n,0)}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \{b_{1},b_{2}\}}$ ${\displaystyle I=b_{1}b_{2}}$ ${\displaystyle b_{1}}$ ${\displaystyle b_{2}}$ ${\displaystyle \{\alpha _{0}+\alpha _{1}I\mid \alpha _{i}\in \mathbb {R} \}}$ ${\displaystyle 0}$ ${\displaystyle 2}$

Por las propiedades del producto geométrico, ⁠ ⁠ ${\displaystyle I^{2}=b_{1}b_{2}b_{1}b_{2}=-b_{1}b_{2}b_{2}b_{1}=-1}$ . La semejanza con la unidad imaginaria no es incidental: el subespacio es ⁠ ⁠ -álgebra isomorfa a los números complejos . De esta manera, una copia de los números complejos está incrustada en el álgebra geométrica para cada subespacio bidimensional de en el que la forma cuadrática está definida. ${\displaystyle \{\alpha _{0}+\alpha _{1}I\mid \alpha _{i}\in \mathbb {R} \}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle V}$

A veces es posible identificar la presencia de una unidad imaginaria en una ecuación física. Dichas unidades surgen de una de las muchas cantidades del álgebra real que se elevan al cuadrado a ⁠ ⁠ ${\displaystyle -1}$ , y tienen importancia geométrica debido a las propiedades del álgebra y la interacción de sus diversos subespacios.

En ⁠ ⁠ ${\displaystyle {\mathcal {G}}(3,0)}$ , se produce otro caso familiar. Dada una base estándar que consiste en vectores ortonormales de ⁠ ⁠ , el conjunto de todos los ⁠ ⁠ -vectores está abarcado por ${\displaystyle e_{i}}$ ${\displaystyle V}$ ${\displaystyle 2}$

${\displaystyle \{e_{3}e_{2},e_{1}e_{3},e_{2}e_{1}\}.}$

Al etiquetarlos como ⁠ ⁠ ${\displaystyle i}$ , y (desviándonos momentáneamente de nuestra convención de mayúsculas), el subespacio generado por ⁠ ⁠ -vectores y ⁠ ⁠ -vectores es exactamente ⁠ ⁠ . Se ve que este conjunto es la subálgebra par de ⁠ ⁠ , y además es isomorfo como ⁠ ⁠ -álgebra a los cuaterniones , otro sistema algebraico importante. ${\displaystyle j}$ ${\displaystyle k}$ ${\displaystyle 0}$ ${\displaystyle 2}$ ${\displaystyle \{\alpha _{0}+i\alpha _{1}+j\alpha _{2}+k\alpha _{3}\mid \alpha _{i}\in \mathbb {R} \}}$ ${\displaystyle {\mathcal {G}}(3,0)}$ ${\displaystyle \mathbb {R} }$

Ampliaciones de los productos interiores y exteriores.

Es una práctica común extender el producto exterior de los vectores a toda el álgebra. Esto se puede hacer mediante el uso del operador de proyección de grados mencionado anteriormente:

${\displaystyle C\wedge D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{r+s}}$     (el producto exterior )

Esta generalización es coherente con la definición anterior que implica antisimetrización. Otra generalización relacionada con el producto exterior es el producto conmutador:

${\displaystyle C\times D:={\tfrac {1}{2}}(CD-DC)}$     (el producto del conmutador )

El producto regresivo es el dual del producto exterior (que corresponde respectivamente a "encuentro" y "unión" en este contexto). ^[i] La especificación dual de elementos permite, para las láminas ⁠ ⁠ ${\displaystyle C}$ y ⁠ ⁠ ${\displaystyle D}$ , la intersección (o encuentro) donde se debe tomar la dualidad en relación con una lámina que contiene tanto a ⁠ ⁠ ${\displaystyle C}$ como a ⁠ ⁠ ${\displaystyle D}$ (la lámina más pequeña de este tipo es la unión). ^[27]

${\displaystyle C\vee D:=((CI^{-1})\wedge (DI^{-1}))I}$

con la unidad pseudoescalar ${\displaystyle I}$ del álgebra. El producto regresivo, como el producto exterior, es asociativo. ^[28]

El producto interno de vectores también se puede generalizar, pero de más de una forma no equivalente. El artículo (Dorst 2002) ofrece un tratamiento completo de varios productos internos diferentes desarrollados para álgebras geométricas y sus interrelaciones, y la notación se toma de allí. Muchos autores utilizan el mismo símbolo que para el producto interno de vectores para su extensión elegida (por ejemplo, Hestenes y Perwass). No ha surgido una notación consistente.

Entre estas diferentes generalizaciones del producto interno sobre vectores se encuentran:

${\displaystyle C\;\rfloor \;D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{s-r}}$   (la contracción de la izquierda )

${\displaystyle C\;\lfloor \;D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{r-s}}$   (la contracción correcta )

${\displaystyle C*D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{0}}$   (el producto escalar )

${\displaystyle C\bullet D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{|s-r|}}$   (el producto "punto (gordo)") ^[j]

Dorst (2002) argumenta que es preferible utilizar contracciones en lugar del producto interno de Hestenes; son algebraicamente más regulares y tienen interpretaciones geométricas más claras. Varias identidades que incorporan las contracciones son válidas sin restricción de sus entradas. Por ejemplo,

${\displaystyle C\;\rfloor \;D=(C\wedge (DI^{-1}))I}$

${\displaystyle C\;\lfloor \;D=I((I^{-1}C)\wedge D)}$

${\displaystyle (A\wedge B)*C=A*(B\;\rfloor \;C)}$

${\displaystyle C*(B\wedge A)=(C\;\lfloor \;B)*A}$

${\displaystyle A\;\rfloor \;(B\;\rfloor \;C)=(A\wedge B)\;\rfloor \;C}$

${\displaystyle (A\;\rfloor \;B)\;\lfloor \;C=A\;\rfloor \;(B\;\lfloor \;C).}$

Benefits of using the left contraction as an extension of the inner product on vectors include that the identity ${\displaystyle ab=a\cdot b+a\wedge b}$ is extended to ${\displaystyle aB=a\;\rfloor \;B+a\wedge B}$ for any vector ${\displaystyle a}$ and multivector ⁠ ${\displaystyle B}$⁠, and that the projection operation ${\displaystyle {\mathcal {P}}_{b}(a)=(a\cdot b^{-1})b}$ is extended to ${\displaystyle {\mathcal {P}}_{B}(A)=(A\;\rfloor \;B^{-1})\;\rfloor \;B}$ for any blade ${\displaystyle B}$ and any multivector ${\displaystyle A}$ (with a minor modification to accommodate null ⁠ ${\displaystyle B}$⁠, given below).

Dual basis

Let ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$ be a basis of ⁠ ${\displaystyle V}$⁠, i.e. a set of ${\displaystyle n}$ linearly independent vectors that span the ⁠ ${\displaystyle n}$⁠-dimensional vector space ⁠ ${\displaystyle V}$⁠. The basis that is dual to ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$ is the set of elements of the dual vector space ${\displaystyle V^{*}}$ that forms a biorthogonal system with this basis, thus being the elements denoted ${\displaystyle \{e^{1},\ldots ,e^{n}\}}$ satisfying

${\displaystyle e^{i}\cdot e_{j}=\delta ^{i}{}_{j},}$

where ${\displaystyle \delta }$ is the Kronecker delta.

Given a nondegenerate quadratic form on ⁠ ${\displaystyle V}$⁠, ${\displaystyle V^{*}}$ becomes naturally identified with ⁠ ${\displaystyle V}$⁠, and the dual basis may be regarded as elements of ⁠ ${\displaystyle V}$⁠, but are not in general the same set as the original basis.

Given further a GA of ⁠ ${\displaystyle V}$⁠, let

${\displaystyle I=e_{1}\wedge \cdots \wedge e_{n}}$

be the pseudoscalar (which does not necessarily square to ⁠ ${\displaystyle \pm 1}$⁠) formed from the basis ⁠ ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$⁠. The dual basis vectors may be constructed as

${\displaystyle e^{i}=(-1)^{i-1}(e_{1}\wedge \cdots \wedge {\check {e}}_{i}\wedge \cdots \wedge e_{n})I^{-1},}$

where the ${\displaystyle {\check {e}}_{i}}$ denotes that the ⁠ ${\displaystyle i}$⁠th basis vector is omitted from the product.

A dual basis is also known as a reciprocal basis or reciprocal frame.

A major usage of a dual basis is to separate vectors into components. Given a vector ⁠ ${\displaystyle a}$⁠, scalar components ${\displaystyle a^{i}}$ can be defined as

${\displaystyle a^{i}=a\cdot e^{i}\ ,}$

in terms of which ${\displaystyle a}$ can be separated into vector components as

${\displaystyle a=\sum _{i}a^{i}e_{i}\ .}$

We can also define scalar components ${\displaystyle a_{i}}$ as

${\displaystyle a_{i}=a\cdot e_{i}\ ,}$

in terms of which ${\displaystyle a}$ can be separated into vector components in terms of the dual basis as

${\displaystyle a=\sum _{i}a_{i}e^{i}\ .}$

A dual basis as defined above for the vector subspace of a geometric algebra can be extended to cover the entire algebra.^[29] For compactness, we'll use a single capital letter to represent an ordered set of vector indices. I.e., writing

${\displaystyle J=(j_{1},\dots ,j_{n})\ ,}$

where ⁠ ${\displaystyle j_{1}<j_{2}<\dots <j_{n}}$⁠, we can write a basis blade as

${\displaystyle e_{J}=e_{j_{1}}\wedge e_{j_{2}}\wedge \cdots \wedge e_{j_{n}}\ .}$

The corresponding reciprocal blade has the indices in opposite order:

${\displaystyle e^{J}=e^{j_{n}}\wedge \cdots \wedge e^{j_{2}}\wedge e^{j_{1}}\ .}$

Similar to the case above with vectors, it can be shown that

${\displaystyle e^{J}*e_{K}=\delta _{K}^{J}\ ,}$

where ${\displaystyle *}$ is the scalar product.

With ${\displaystyle A}$ a multivector, we can define scalar components as^[30]

${\displaystyle A^{ij\cdots k}=(e^{k}\wedge \cdots \wedge e^{j}\wedge e^{i})*A\ ,}$

in terms of which ${\displaystyle A}$ can be separated into component blades as

${\displaystyle A=\sum _{i<j<\cdots <k}A^{ij\cdots k}e_{i}\wedge e_{j}\wedge \cdots \wedge e_{k}\ .}$

We can alternatively define scalar components

${\displaystyle A_{ij\cdots k}=(e_{k}\wedge \cdots \wedge e_{j}\wedge e_{i})*A\ ,}$

in terms of which ${\displaystyle A}$ can be separated into component blades as

${\displaystyle A=\sum _{i<j<\cdots <k}A_{ij\cdots k}e^{i}\wedge e^{j}\wedge \cdots \wedge e^{k}\ .}$

Linear functions

Although a versor is easier to work with because it can be directly represented in the algebra as a multivector, versors are a subgroup of linear functions on multivectors, which can still be used when necessary. The geometric algebra of an ⁠ ${\displaystyle n}$⁠-dimensional vector space is spanned by a basis of ${\displaystyle 2^{n}}$ elements. If a multivector is represented by a ${\displaystyle 2^{n}\times 1}$ real column matrix of coefficients of a basis of the algebra, then all linear transformations of the multivector can be expressed as the matrix multiplication by a ${\displaystyle 2^{n}\times 2^{n}}$ real matrix. However, such a general linear transformation allows arbitrary exchanges among grades, such as a "rotation" of a scalar into a vector, which has no evident geometric interpretation.

A general linear transformation from vectors to vectors is of interest. With the natural restriction to preserving the induced exterior algebra, the outermorphism of the linear transformation is the unique^[k] extension of the versor. If ${\displaystyle f}$ is a linear function that maps vectors to vectors, then its outermorphism is the function that obeys the rule

${\displaystyle {\underline {\mathsf {f}}}(a_{1}\wedge a_{2}\wedge \cdots \wedge a_{r})=f(a_{1})\wedge f(a_{2})\wedge \cdots \wedge f(a_{r})}$

for a blade, extended to the whole algebra through linearity.

Modeling geometries

Although a lot of attention has been placed on CGA, it is to be noted that GA is not just one algebra, it is one of a family of algebras with the same essential structure.^[31]

Vector space model

The even subalgebra of ${\displaystyle {\mathcal {G}}(2,0)}$ is isomorphic to the complex numbers, as may be seen by writing a vector ${\displaystyle P}$ in terms of its components in an orthonormal basis and left multiplying by the basis vector ⁠ ${\displaystyle e_{1}}$⁠, yielding

${\displaystyle Z=e_{1}P=e_{1}(xe_{1}+ye_{2})=x(1)+y(e_{1}e_{2}),}$

where we identify ${\displaystyle i\mapsto e_{1}e_{2}}$ since

${\displaystyle (e_{1}e_{2})^{2}=e_{1}e_{2}e_{1}e_{2}=-e_{1}e_{1}e_{2}e_{2}=-1.}$

Similarly, the even subalgebra of ${\displaystyle {\mathcal {G}}(3,0)}$ with basis ${\displaystyle \{1,e_{2}e_{3},e_{3}e_{1},e_{1}e_{2}\}}$ is isomorphic to the quaternions as may be seen by identifying ⁠ ${\displaystyle i\mapsto -e_{2}e_{3}}$⁠, ${\displaystyle j\mapsto -e_{3}e_{1}}$ and ⁠ ${\displaystyle k\mapsto -e_{1}e_{2}}$⁠.

Every associative algebra has a matrix representation; replacing the three Cartesian basis vectors by the Pauli matrices gives a representation of ⁠ ${\displaystyle {\mathcal {G}}(3,0)}$⁠:

${\displaystyle {\begin{aligned}e_{1}=\sigma _{1}=\sigma _{x}&={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\\e_{2}=\sigma _{2}=\sigma _{y}&={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\\e_{3}=\sigma _{3}=\sigma _{z}&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\,.\end{aligned}}}$

Dotting the "Pauli vector" (a dyad):

${\displaystyle \sigma =\sigma _{1}e_{1}+\sigma _{2}e_{2}+\sigma _{3}e_{3}}$ with arbitrary vectors ${\displaystyle a}$ and ${\displaystyle b}$ and multiplying through gives:

${\displaystyle (\sigma \cdot a)(\sigma \cdot b)=a\cdot b+a\wedge b}$ (Equivalently, by inspection, ⁠ ${\displaystyle a\cdot b+i\sigma \cdot (a\times b)}$⁠)

Spacetime model

In physics, the main applications are the geometric algebra of Minkowski 3+1 spacetime, ⁠ ${\displaystyle {\mathcal {G}}(1,3)}$⁠, called spacetime algebra (STA),^[3] or less commonly, ⁠ ${\displaystyle {\mathcal {G}}(3,0)}$⁠, interpreted the algebra of physical space (APS).

While in STA, points of spacetime are represented simply by vectors, in APS, points of ⁠ ${\displaystyle (3+1)}$⁠-dimensional spacetime are instead represented by paravectors, a three-dimensional vector (space) plus a one-dimensional scalar (time).

In spacetime algebra the electromagnetic field tensor has a bivector representation ⁠ ${\displaystyle {F}=({E}+ic{B})\gamma _{0}}$⁠.^[32] Here, the ${\displaystyle i=\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}}$ is the unit pseudoscalar (or four-dimensional volume element), ${\displaystyle \gamma _{0}}$ is the unit vector in time direction, and ${\displaystyle E}$ and ${\displaystyle B}$ are the classic electric and magnetic field vectors (with a zero time component). Using the four-current ⁠ ${\displaystyle {J}}$⁠, Maxwell's equations then become

In geometric calculus, juxtaposition of vectors such as in ${\displaystyle DF}$ indicate the geometric product and can be decomposed into parts as ⁠ ${\displaystyle DF=D~\rfloor ~F+D\wedge F}$⁠. Here ${\displaystyle D}$ is the covector derivative in any spacetime and reduces to ${\displaystyle \nabla }$ in flat spacetime. Where ${\displaystyle \bigtriangledown }$ plays a role in Minkowski ⁠ ${\displaystyle 4}$⁠-spacetime which is synonymous to the role of ${\displaystyle \nabla }$ in Euclidean ⁠ ${\displaystyle 3}$⁠-space and is related to the d'Alembertian by ⁠ ${\displaystyle \Box =\bigtriangledown ^{2}}$⁠. Indeed, given an observer represented by a future pointing timelike vector ${\displaystyle \gamma _{0}}$ we have

${\displaystyle \gamma _{0}\cdot \bigtriangledown ={\frac {1}{c}}{\frac {\partial }{\partial t}}}$

${\displaystyle \gamma _{0}\wedge \bigtriangledown =\nabla }$

Boosts in this Lorentzian metric space have the same expression ${\displaystyle e^{\beta }}$ as rotation in Euclidean space, where ${\displaystyle {\beta }}$ is the bivector generated by the time and the space directions involved, whereas in the Euclidean case it is the bivector generated by the two space directions, strengthening the "analogy" to almost identity.

The Dirac matrices are a representation of ⁠ ${\displaystyle {\mathcal {G}}(1,3)}$⁠, showing the equivalence with matrix representations used by physicists.

Homogeneous models

Homogeneous models generally refer to a projective representation in which the elements of the one-dimensional subspaces of a vector space represent points of a geometry.

In a geometric algebra of a space of ${\displaystyle n}$ dimensions, the rotors represent a set of transformations with ${\displaystyle n(n-1)/2}$ degrees of freedom, corresponding to rotations – for example, ${\displaystyle 3}$ when ${\displaystyle n=3}$ and ${\displaystyle 6}$ when ⁠ ${\displaystyle n=4}$⁠. Geometric algebra is often used to model a projective space, i.e. as a homogeneous model: a point, line, plane, etc. is represented by an equivalence class of elements of the algebra that differ by an invertible scalar factor.

The rotors in a space of dimension ${\displaystyle n+1}$ have $n(n-1)/2+n$ degrees of freedom, the same as the number of degrees of freedom in the rotations and translations combined for an ⁠ ${\displaystyle n}$⁠-dimensional space.

This is the case in Projective Geometric Algebra (PGA), which is used^[33][34][35] to represent Euclidean isometries in Euclidean geometry (thereby covering the large majority of engineering applications of geometry). In this model, a degenerate dimension is added to the three Euclidean dimensions to form the algebra ⁠ ${\displaystyle {\mathcal {G}}(3,0,1)}$⁠. With a suitable identification of subspaces to represent points, lines and planes, the versors of this algebra represent all proper Euclidean isometries, which are always screw motions in 3-dimensional space, along with all improper Euclidean isometries, which includes reflections, rotoreflections, transflections, and point reflections.

PGA combines ${\displaystyle {\mathcal {G}}(3,0,1)}$ with a complement operator to obtain join, meet, distance, and angle formulas.^[36] In effect, the complement switches basis vectors that are present and absent in the expression of each term of the algebraic representation. For example, in the PGA or 3-dimensional space, the complement of the line ${\displaystyle {\boldsymbol {e}}_{12}}$ is the line ⁠ ${\displaystyle {\boldsymbol {e}}_{03}}$⁠, because ${\displaystyle {\boldsymbol {e}}_{0}}$ and ${\displaystyle {\boldsymbol {e}}_{3}}$ are basis elements that are not contained in ${\displaystyle {\boldsymbol {e}}_{12}}$ but are contained in ⁠ ${\displaystyle {\boldsymbol {e}}_{03}}$⁠. In the PGA of 2-dimensional space, the complement of ${\displaystyle {\boldsymbol {e}}_{12}}$ is ⁠ ${\displaystyle {\boldsymbol {e}}_{0}}$⁠, since there is no ${\displaystyle {\boldsymbol {e}}_{3}}$ element.

PGA is a widely used system that combines geometric algebra with homogeneous representations in geometry, but there exist several other such systems. The conformal model discussed below is homogeneous, as is "Conic Geometric Algebra",^[37] and see Plane-based geometric algebra for discussion of homogeneous models of elliptic and hyperbolic geometry compared with the Euclidean geometry derived from PGA.

Conformal model

Working within GA, Euclidean space ${\displaystyle \mathbb {E} ^{3}}$ (along with a conformal point at infinity) is embedded projectively in the CGA ${\displaystyle {\mathcal {G}}(4,1)}$ via the identification of Euclidean points with 1D subspaces in the 4D null cone of the 5D CGA vector subspace. This allows all conformal transformations to be performed as rotations and reflections and is covariant, extending incidence relations of projective geometry to rounds objects such as circles and spheres.

Specifically, we add orthogonal basis vectors ${\displaystyle e_{+}}$ and ${\displaystyle e_{-}}$ such that ${\displaystyle e_{+}^{2}=+1}$ and ${\displaystyle e_{-}^{2}=-1}$ to the basis of the vector space that generates ${\displaystyle {\mathcal {G}}(3,0)}$ and identify null vectors

${\displaystyle n_{\text{o}}={\tfrac {1}{2}}(e_{-}-e_{+})}$ as the point at the origin and

${\displaystyle n_{\infty }=e_{-}+e_{+}}$ as a conformal point at infinity (see Compactification), giving

${\displaystyle n_{\infty }\cdot n_{\text{o}}=-1.}$

(Some authors set ${\displaystyle e_{4}=n_{\text{o}}}$ and ⁠ ${\displaystyle e_{5}=n_{\infty }}$⁠.^[36]) This procedure has some similarities to the procedure for working with homogeneous coordinates in projective geometry, and in this case allows the modeling of Euclidean transformations of ${\displaystyle \mathbb {R} ^{3}}$ as orthogonal transformations of a subset of ⁠ ${\displaystyle \mathbf {R} ^{4,1}}$⁠.

A fast changing and fluid area of GA, CGA is also being investigated for applications to relativistic physics.

Table of models

Note in this list that ⁠ ${\displaystyle p}$⁠ and ⁠ ${\displaystyle q}$⁠ can be swapped and the same name applies; for example, with relatively little change occurring, see sign convention. For example, ${\displaystyle {\mathcal {G}}(3,1,0)}$ and ${\displaystyle {\mathcal {G}}(1,3,0)}$ are both referred to as Spacetime Algebra.^[38]

Geometric interpretation in the vector space model

Projection and rejection

In 3D space, a bivector ${\displaystyle a\land b}$ defines a 2D plane subspace (light blue, extends infinitely in indicated directions). Any vector ${\displaystyle c}$ in 3D space can be decomposed into its projection ${\displaystyle c_{\Vert }}$ onto a plane and its rejection ${\displaystyle c_{\perp }}$ from this plane.

For any vector ${\displaystyle a}$ and any invertible vector ⁠ ${\displaystyle m}$⁠,

${\displaystyle a=amm^{-1}=(a\cdot m+a\wedge m)m^{-1}=a_{\|m}+a_{\perp m},}$

where the projection of ${\displaystyle a}$ onto ${\displaystyle m}$ (or the parallel part) is

${\displaystyle a_{\|m}=(a\cdot m)m^{-1}}$

and the rejection of ${\displaystyle a}$ from ${\displaystyle m}$ (or the orthogonal part) is

${\displaystyle a_{\perp m}=a-a_{\|m}=(a\wedge m)m^{-1}.}$

Using the concept of a ⁠ ${\displaystyle k}$⁠-blade ⁠ ${\displaystyle B}$⁠ as representing a subspace of ⁠ ${\displaystyle V}$⁠ and every multivector ultimately being expressed in terms of vectors, this generalizes to projection of a general multivector onto any invertible ⁠ ${\displaystyle k}$⁠-blade ⁠ ${\displaystyle B}$⁠ as^[l]

${\displaystyle {\mathcal {P}}_{B}(A)=(A\;\rfloor \;B)\;\rfloor \;B^{-1},}$

with the rejection being defined as

${\displaystyle {\mathcal {P}}_{B}^{\perp }(A)=A-{\mathcal {P}}_{B}(A).}$

The projection and rejection generalize to null blades ${\displaystyle B}$ by replacing the inverse ${\displaystyle B^{-1}}$ with the pseudoinverse ${\displaystyle B^{+}}$ with respect to the contractive product.^[m] The outcome of the projection coincides in both cases for non-null blades.^[45][46] For null blades ⁠ ${\displaystyle B}$⁠, the definition of the projection given here with the first contraction rather than the second being onto the pseudoinverse should be used,^[n] as only then is the result necessarily in the subspace represented by ⁠ ${\displaystyle B}$⁠.^[45]The projection generalizes through linearity to general multivectors ⁠ ${\displaystyle A}$⁠.^[o] The projection is not linear in ⁠ ${\displaystyle B}$⁠ and does not generalize to objects ⁠ ${\displaystyle B}$⁠ that are not blades.

Reflection

Simple reflections in a hyperplane are readily expressed in the algebra through conjugation with a single vector. These serve to generate the group of general rotoreflections and rotations.

Reflection of vector ${\displaystyle c}$ along a vector ⁠ ${\displaystyle m}$⁠. Only the component of ${\displaystyle c}$ parallel to ${\displaystyle m}$ is negated.

The reflection ${\displaystyle c'}$ of a vector ${\displaystyle c}$ along a vector ⁠ ${\displaystyle m}$⁠, or equivalently in the hyperplane orthogonal to ⁠ ${\displaystyle m}$⁠, is the same as negating the component of a vector parallel to ⁠ ${\displaystyle m}$⁠. The result of the reflection will be

${\displaystyle c'={-c_{\|m}+c_{\perp m}}={-(c\cdot m)m^{-1}+(c\wedge m)m^{-1}}={(-m\cdot c-m\wedge c)m^{-1}}=-mcm^{-1}}$

This is not the most general operation that may be regarded as a reflection when the dimension ⁠ ${\displaystyle n\geq 4}$⁠. A general reflection may be expressed as the composite of any odd number of single-axis reflections. Thus, a general reflection ${\displaystyle a'}$ of a vector ${\displaystyle a}$ may be written

${\displaystyle a\mapsto a'=-MaM^{-1},}$

where

${\displaystyle M=pq\cdots r}$ and ${\displaystyle M^{-1}=(pq\cdots r)^{-1}=r^{-1}\cdots q^{-1}p^{-1}.}$

If we define the reflection along a non-null vector ${\displaystyle m}$ of the product of vectors as the reflection of every vector in the product along the same vector, we get for any product of an odd number of vectors that, by way of example,

${\displaystyle (abc)'=a'b'c'=(-mam^{-1})(-mbm^{-1})(-mcm^{-1})=-ma(m^{-1}m)b(m^{-1}m)cm^{-1}=-mabcm^{-1}\,}$

and for the product of an even number of vectors that

${\displaystyle (abcd)'=a'b'c'd'=(-mam^{-1})(-mbm^{-1})(-mcm^{-1})(-mdm^{-1})=mabcdm^{-1}.}$

Using the concept of every multivector ultimately being expressed in terms of vectors, the reflection of a general multivector ${\displaystyle A}$ using any reflection versor ${\displaystyle M}$ may be written

${\displaystyle A\mapsto M\alpha (A)M^{-1},}$

where ${\displaystyle \alpha }$ is the automorphism of reflection through the origin of the vector space (⁠ ${\displaystyle v\mapsto -v}$⁠) extended through linearity to the whole algebra.

Rotations

A rotor that rotates vectors in a plane rotates vectors through angle ⁠ ${\displaystyle \theta }$⁠, that is ${\displaystyle x\mapsto R_{\theta }x{\widetilde {R}}_{\theta }}$ is a rotation of ${\displaystyle x}$ through angle ⁠ ${\displaystyle \theta }$⁠. The angle between ${\displaystyle u}$ and ${\displaystyle v}$ is ⁠ ${\displaystyle \theta /2}$⁠. Similar interpretations are valid for a general multivector ${\displaystyle X}$ instead of the vector ⁠ ${\displaystyle x}$⁠.^[13]

If we have a product of vectors ${\displaystyle R=a_{1}a_{2}\cdots a_{r}}$ then we denote the reverse as

${\displaystyle {\widetilde {R}}=a_{r}\cdots a_{2}a_{1}.}$

As an example, assume that ${\displaystyle R=ab}$ we get

${\displaystyle R{\widetilde {R}}=abba=ab^{2}a=a^{2}b^{2}=ba^{2}b=baab={\widetilde {R}}R.}$

Scaling ${\displaystyle R}$ so that ${\displaystyle R{\widetilde {R}}=1}$ then

${\displaystyle (Rv{\widetilde {R}})^{2}=Rv^{2}{\widetilde {R}}=v^{2}R{\widetilde {R}}=v^{2}}$

so ${\displaystyle Rv{\widetilde {R}}}$ leaves the length of ${\displaystyle v}$ unchanged. We can also show that

${\displaystyle (Rv_{1}{\widetilde {R}})\cdot (Rv_{2}{\widetilde {R}})=v_{1}\cdot v_{2}}$

so the transformation ${\displaystyle Rv{\widetilde {R}}}$ preserves both length and angle. It therefore can be identified as a rotation or rotoreflection; ${\displaystyle R}$ is called a rotor if it is a proper rotation (as it is if it can be expressed as a product of an even number of vectors) and is an instance of what is known in GA as a versor.

There is a general method for rotating a vector involving the formation of a multivector of the form ${\displaystyle R=e^{-B\theta /2}}$ that produces a rotation ${\displaystyle \theta }$ in the plane and with the orientation defined by a ⁠ ${\displaystyle 2}$⁠-blade ⁠ ${\displaystyle B}$⁠.

Rotors are a generalization of quaternions to ⁠ ${\displaystyle n}$⁠-dimensional spaces.

Examples and applications

Hypervolume of a parallelotope spanned by vectors

For vectors ⁠ ${\displaystyle a}$⁠ and ⁠ ${\displaystyle b}$⁠ spanning a parallelogram we have

${\displaystyle a\wedge b=((a\wedge b)b^{-1})b=a_{\perp b}b}$

with the result that ⁠ ${\displaystyle a\wedge b}$⁠ is linear in the product of the "altitude" and the "base" of the parallelogram, that is, its area.

Similar interpretations are true for any number of vectors spanning an ⁠ ${\displaystyle n}$⁠-dimensional parallelotope; the exterior product of vectors ⁠ ${\displaystyle a_{1},a_{2},\ldots ,a_{n}}$⁠, that is ⁠ ${\displaystyle \textstyle \bigwedge _{i=1}^{n}a_{i}}$⁠, has a magnitude equal to the volume of the ⁠ ${\displaystyle n}$⁠-parallelotope. An ⁠ ${\displaystyle n}$⁠-vector does not necessarily have a shape of a parallelotope – this is a convenient visualization. It could be any shape, although the volume equals that of the parallelotope.

Intersection of a line and a plane

A line L defined by points T and P (which we seek) and a plane defined by a bivector B containing points P and Q.

We may define the line parametrically by ⁠ ${\displaystyle p=t+\alpha \ v}$⁠, where ⁠ ${\displaystyle p}$⁠ and ⁠ ${\displaystyle t}$⁠ are position vectors for points P and T and ⁠ ${\displaystyle v}$⁠ is the direction vector for the line.

Then

${\displaystyle B\wedge (p-q)=0}$ and ${\displaystyle B\wedge (t+\alpha v-q)=0}$

so

${\displaystyle \alpha ={\frac {B\wedge (q-t)}{B\wedge v}}}$

and

${\displaystyle p=t+\left({\frac {B\wedge (q-t)}{B\wedge v}}\right)v.}$

Rotating systems

A rotational quantity such as torque or angular momentum is described in geometric algebra as a bivector. Suppose a circular path in an arbitrary plane containing orthonormal vectors ⁠ ${\displaystyle {\widehat {u}}}$⁠ and ⁠ ${\displaystyle {\widehat {\ \!v}}}$⁠ is parameterized by angle.

${\displaystyle \mathbf {r} =r({\widehat {u}}\cos \theta +{\widehat {\ \!v}}\sin \theta )=r{\widehat {u}}(\cos \theta +{\widehat {u}}{\widehat {\ \!v}}\sin \theta )}$

By designating the unit bivector of this plane as the imaginary number

${\displaystyle {i}={\widehat {u}}{\widehat {\ \!v}}={\widehat {u}}\wedge {\widehat {\ \!v}}}$

${\displaystyle i^{2}=-1}$

this path vector can be conveniently written in complex exponential form

${\displaystyle \mathbf {r} =r{\widehat {u}}e^{i\theta }}$

and the derivative with respect to angle is

${\displaystyle {\frac {d\mathbf {r} }{d\theta }}=r{\widehat {u}}ie^{i\theta }=\mathbf {r} i.}$

The cross product in relation to the exterior product. In red are the unit normal vector, and the "parallel" unit bivector.

For example, torque is generally defined as the magnitude of the perpendicular force component times distance, or work per unit angle. Thus the torque, the rate of change of work ⁠ ${\displaystyle W}$⁠ with respect to angle, due to a force ⁠ ${\displaystyle F}$⁠, is

${\displaystyle \tau ={\frac {dW}{d\theta }}=F\cdot {\frac {dr}{d\theta }}=F\cdot (\mathbf {r} i).}$

Rotational quantities are represented in vector calculus in three dimensions using the cross product. Together with a choice of an oriented volume form ⁠ ${\displaystyle I}$⁠, these can be related to the exterior product with its more natural geometric interpretation of such quantities as a bivectors by using the dual relationship

${\displaystyle a\times b=-I(a\wedge b).}$

Unlike the cross product description of torque, ⁠ ${\displaystyle \tau =\mathbf {r} \times F}$⁠, the geometric algebra description does not introduce a vector in the normal direction; a vector that does not exist in two and that is not unique in greater than three dimensions. The unit bivector describes the plane and the orientation of the rotation, and the sense of the rotation is relative to the angle between the vectors ⁠ ${\displaystyle {\widehat {u}}}$⁠ and ⁠ ${\displaystyle {\widehat {\ \!v}}}$⁠.

Geometric calculus

Geometric calculus extends the formalism to include differentiation and integration including differential geometry and differential forms.^[47]

Essentially, the vector derivative is defined so that the GA version of Green's theorem is true,

${\displaystyle \int _{A}dA\,\nabla f=\oint _{\partial A}dx\,f}$

and then one can write

${\displaystyle \nabla f=\nabla \cdot f+\nabla \wedge f}$

as a geometric product, effectively generalizing Stokes' theorem (including the differential form version of it).

In 1D when ⁠ ${\displaystyle A}$⁠ is a curve with endpoints ⁠ ${\displaystyle a}$⁠ and ⁠ ${\displaystyle b}$⁠, then

${\displaystyle \int _{A}dA\,\nabla f=\oint _{\partial A}dx\,f}$

reduces to

${\displaystyle \int _{a}^{b}dx\,\nabla f=\int _{a}^{b}dx\cdot \nabla f=\int _{a}^{b}df=f(b)-f(a)}$

or the fundamental theorem of integral calculus.

Also developed are the concept of vector manifold and geometric integration theory (which generalizes differential forms).

History

Before the 20th century

Although the connection of geometry with algebra dates as far back at least to Euclid's Elements in the third century B.C. (see Greek geometric algebra), GA in the sense used in this article was not developed until 1844, when it was used in a systematic way to describe the geometrical properties and transformations of a space. In that year, Hermann Grassmann introduced the idea of a geometrical algebra in full generality as a certain calculus (analogous to the propositional calculus) that encoded all of the geometrical information of a space.^[48] Grassmann's algebraic system could be applied to a number of different kinds of spaces, the chief among them being Euclidean space, affine space, and projective space. Following Grassmann, in 1878 William Kingdon Clifford examined Grassmann's algebraic system alongside the quaternions of William Rowan Hamilton in (Clifford 1878). From his point of view, the quaternions described certain transformations (which he called rotors), whereas Grassmann's algebra described certain properties (or Strecken such as length, area, and volume). His contribution was to define a new product – the geometric product – on an existing Grassmann algebra, which realized the quaternions as living within that algebra. Subsequently, Rudolf Lipschitz in 1886 generalized Clifford's interpretation of the quaternions and applied them to the geometry of rotations in ⁠ ${\displaystyle n}$⁠ dimensions. Later these developments would lead other 20th-century mathematicians to formalize and explore the properties of the Clifford algebra.

Nevertheless, another revolutionary development of the 19th-century would completely overshadow the geometric algebras: that of vector analysis, developed independently by Josiah Willard Gibbs and Oliver Heaviside. Vector analysis was motivated by James Clerk Maxwell's studies of electromagnetism, and specifically the need to express and manipulate conveniently certain differential equations. Vector analysis had a certain intuitive appeal compared to the rigors of the new algebras. Physicists and mathematicians alike readily adopted it as their geometrical toolkit of choice, particularly following the influential 1901 textbook Vector Analysis by Edwin Bidwell Wilson, following lectures of Gibbs.

In more detail, there have been three approaches to geometric algebra: quaternionic analysis, initiated by Hamilton in 1843 and geometrized as rotors by Clifford in 1878; geometric algebra, initiated by Grassmann in 1844; and vector analysis, developed out of quaternionic analysis in the late 19th century by Gibbs and Heaviside. The legacy of quaternionic analysis in vector analysis can be seen in the use of ⁠ ${\displaystyle i}$⁠, ⁠ ${\displaystyle j}$⁠, ⁠ ${\displaystyle k}$⁠ to indicate the basis vectors of ⁠ ${\displaystyle \mathbf {R} ^{3}}$⁠: it is being thought of as the purely imaginary quaternions. From the perspective of geometric algebra, the even subalgebra of the Space Time Algebra is isomorphic to the GA of 3D Euclidean space and quaternions are isomorphic to the even subalgebra of the GA of 3D Euclidean space, which unifies the three approaches.

20th century and present

Progress on the study of Clifford algebras quietly advanced through the twentieth century, although largely due to the work of abstract algebraists such as Élie Cartan, Hermann Weyl and Claude Chevalley. The geometrical approach to geometric algebras has seen a number of 20th-century revivals. In mathematics, Emil Artin's Geometric Algebra^[49] discusses the algebra associated with each of a number of geometries, including affine geometry, projective geometry, symplectic geometry, and orthogonal geometry. In physics, geometric algebras have been revived as a "new" way to do classical mechanics and electromagnetism, together with more advanced topics such as quantum mechanics and gauge theory.^[5] David Hestenes reinterpreted the Pauli and Dirac matrices as vectors in ordinary space and spacetime, respectively, and has been a primary contemporary advocate for the use of geometric algebra.

In computer graphics and robotics, geometric algebras have been revived in order to efficiently represent rotations and other transformations. For applications of GA in robotics (screw theory, kinematics and dynamics using versors), computer vision, control and neural computing (geometric learning) see Bayro (2010).

See also

• Comparison of vector algebra and geometric algebra
• Clifford algebra
• Grassmann–Cayley algebra
• Spacetime algebra
• Spinor
• Quaternion
• Algebra of physical space
• Universal geometric algebra

Notes

1. A 'universal' algebra is the most "complete" or least degenerate algebra that satisfies all the defining equations. In this article, by 'Clifford algebra' we mean the universal Clifford algebra.
2. The term inner product as used in geometric algebra refers to the symmetric bilinear form on the ⁠ ${\displaystyle 1}$⁠-vector subspace, and is a synonym for the scalar product of a pseudo-Euclidean vector space, not the inner product on a normed vector space. Some authors may extend the meaning of inner product to the entire algebra, but there is little consensus on this. Even in texts on geometric algebras, the term is not universally used.
3. It may be replaced by the condition that^[11] the product of any set of linearly independent vectors in ⁠ ${\displaystyle V}$⁠ must not be in ⁠ ${\displaystyle F}$⁠ or that^[12] the dimension of the algebra must be ⁠ ${\displaystyle 2^{\dim V}}$⁠.
4. The term outer product used in geometric algebra conflicts with the meaning of outer product elsewhere in mathematics
5. Given ⁠ ${\displaystyle u^{2}=1}$⁠, we have that ${\textstyle ({\tfrac {1}{2}}(1+u))^{2}}$ ${\displaystyle ={\tfrac {1}{4}}(1+2u+uu)}$ ${\displaystyle ={\tfrac {1}{4}}(1+2u+1)}$ ⁠ ${\displaystyle ={\tfrac {1}{2}}(1+u)}$⁠, showing that ${\textstyle {\tfrac {1}{2}}(1+u)}$ is idempotent, and that ${\displaystyle {\tfrac {1}{2}}(1+u)(1-u)}$ ${\displaystyle ={\tfrac {1}{2}}(1-uu)}$ ⁠ ${\displaystyle ={\tfrac {1}{2}}(1-1)=0}$⁠, showing that it is a nonzero zero divisor.
6. Grade is a synonym for degree of a homogeneous element under the grading as an algebra with the exterior product (a ⁠ ${\displaystyle \mathrm {Z} }$⁠-grading), and not under the geometric product.
7. "reviving and generalizing somewhat a term from hamilton's quaternion calculus which has fallen into disuse" Hestenes defined a ⁠ ${\displaystyle k}$⁠-versor as a multivector which can be factored into a product of ${\displaystyle k}$ vectors.^[20]
8. Only the outermorphisms of linear transformations that respect the bilinear form fit this description; outermorphisms are not in general expressible in terms of the algebraic operations.
9. [...] the exterior product operation and the join relation have essentially the same meaning. The Grassmann–Cayley algebra regards the meet relation as its counterpart and gives a unifying framework in which these two operations have equal footing [...] Grassmann himself defined the meet operation as the dual of the exterior product operation, but later mathematicians defined the meet operator independently of the exterior product through a process called shuffle, and the meet operation is termed the shuffle product. It is shown that this is an antisymmetric operation that satisfies associativity, defining an algebra in its own right. Thus, the Grassmann–Cayley algebra has two algebraic structures simultaneously: one based on the exterior product (or join), the other based on the shuffle product (or meet). Hence, the name "double algebra", and the two are shown to be dual to each other.^[26]
10. This should not be confused with Hestenes's irregular generalization ⁠ ${\displaystyle \textstyle C\bullet _{\text{H}}D:=\sum _{r\neq 0,s\neq 0}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{\vert s-r\vert }}$⁠, where the distinguishing notation is from Dorst, Fontijne & Mann (2007), p. 590, §B.1, which makes the point that scalar components must be handled separately with this product.
11. The condition that ${\displaystyle {\underline {\mathsf {f}}}(1)=1}$ is usually added to ensure that the zero map is unique.
12. This definition follows Dorst, Fontijne & Mann (2007) and Perwass (2009) – the left contraction used by Dorst replaces the ("fat dot") inner product that Perwass uses, consistent with Perwass's constraint that grade of ⁠ ${\displaystyle A}$⁠ may not exceed that of ⁠ ${\displaystyle B}$⁠.
13. Dorst appears to merely assume ${\displaystyle B^{+}}$ such that ⁠ ${\displaystyle B\;\rfloor \;B^{+}=1}$⁠, whereas Perwass (2009) defines ⁠ ${\displaystyle B^{+}=B^{\dagger }/(B\;\rfloor \;B^{\dagger })}$⁠, where ${\displaystyle B^{\dagger }}$ is the conjugate of ⁠ ${\displaystyle B}$⁠, equivalent to the reverse of ${\displaystyle B}$ up to a sign.
14. That is to say, the projection must be defined as ⁠ ${\displaystyle {\mathcal {P}}_{B}(A)=(A\;\rfloor \;B^{+})\;\rfloor \;B}$⁠ and not as ⁠ ${\displaystyle (A\;\rfloor \;B)\;\rfloor \;B^{+}}$⁠, though the two are equivalent for non-null blades ⁠ ${\displaystyle B}$⁠.
15. This generalization to all ⁠ ${\displaystyle A}$⁠ is apparently not considered by Perwass or Dorst.

Citations

1. Hestenes 1986, p. 6.
2. Li 2008, p. 411.
3. Hestenes 1966.
4. Hestenes 2003.
5. Doran 1994.
6. Lasenby, Lasenby & Doran 2000.
7. Hildenbrand et al. 2004.
8. Hestenes & Sobczyk 1984, p. 3–5.
9. Aragón, Aragón & Rodríguez 1997, p. 101.
10. Lounesto 2001, p. 190.
11. Lounesto 2001, p. 191.
12. Vaz & da Rocha 2016, p. 58, Theorem 3.1.
13. Hestenes 2005.
14. Penrose 2007.
15. Wheeler, Misner & Thorne 1973, p. 83.
16. Wilmot 1988a, p. 2338.
17. Wilmot 1988b, p. 2346.
18. Chevalley 1991.
19. Wilmot 2023.
20. Hestenes & Sobczyk 1984, p. 103.
21. Dorst, Fontijne & Mann 2007, p. 204.
22. Dorst, Fontijne & Mann 2007, pp. 177–182.
23. Lundholm & Svensson 2009, pp. 58 et seq.
24. Lundholm & Svensson 2009, p. 58.
25. Francis & Kosowsky 2008.
26. Kanatani 2015, pp. 112–113.
27. Dorst & Lasenby 2011, p. 443.
28. Vaz & da Rocha 2016, §2.8.
29. Hestenes & Sobczyk 1984, p. 31.
30. Doran & Lasenby 2003, p. 102.
31. Dorst & Lasenby 2011, p. vi.
32. "Electromagnetism using Geometric Algebra versus Components". Retrieved 2013-03-19.
33. Selig 2005.
34. Hadfield & Lasenby 2020.
35. "Projective Geometric Algebra". projectivegeometricalgebra.org. Retrieved 2023-10-03.
36. Lengyel 2024.
37. Hrdina, Návrat & Vašík 2018.
38. Wu 2022.
39. Sokolov 2013.
40. Lasenby 2004.
41. Dorst 2016.
42. Perwass 2009.
43. Breuils et al. 2019.
44. Easter & Hitzer 2017.
45. Dorst, Fontijne & Mann 2007, §3.6 p. 85.
46. Perwass 2009, §3.2.10.2 p. 83.
47. Hestenes & Sobczyk 1984.
48. Grassmann 1844.
49. Artin 1988.

References and further reading

Arranged chronologically

• Grassmann, Hermann (1844), Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert, Leipzig: O. Wigand, OCLC 20521674
• Clifford, Professor (1878), "Applications of Grassmann's Extensive Algebra", American Journal of Mathematics, 1 (4): 350–358, doi:10.2307/2369379, JSTOR 2369379
• Artin, Emil (1988) [1957], Geometric algebra, Wiley Classics Library, Wiley, doi:10.1002/9781118164518, ISBN 978-0-471-60839-4, MR 1009557
• Hestenes, David (1966), Space–time Algebra, Gordon and Breach, ISBN 978-0-677-01390-9, OCLC 996371
• Wheeler, J. A.; Misner, C.; Thorne, K. S. (1973), Gravitation, W.H. Freeman, ISBN 978-0-7167-0344-0
• Bourbaki, Nicolas (1980), "Ch. 9 "Algèbres de Clifford"", Eléments de Mathématique. Algèbre, Hermann, ISBN 9782225655166
• Hestenes, David; Sobczyk, Garret (1984), Clifford Algebra to Geometric Calculus, a Unified Language for Mathematics and Physics, Springer Netherlands, ISBN 978-90-277-1673-6
• Hestenes, David (1986), "A Unified Language for Mathematics and Physics", in J.S.R. Chisholm; A.K. Commons (eds.), Clifford Algebras and Their Applications in Mathematical Physics, NATO ASI Series (Series C), vol. 183, Springer, pp. 1–23, doi:10.1007/978-94-009-4728-3_1, ISBN 978-94-009-4728-3
• Wilmot, G.P. (1988a), The Structure of Clifford algebra. Journal of Mathematical Physics, vol. 29, pp. 2338–2345
• Wilmot, G.P. (1988b), "Clifford algebra and the Pfaffian expansion", Journal of Mathematical Physics, 29: 2346–2350, doi:10.1063/1.528118
• Chevalley, Claude (1991), The Algebraic Theory of Spinors and Clifford Algebras, Collected Works, vol. 2, Springer, ISBN 3-540-57063-2
• Doran, Chris J. L. (1994), Geometric Algebra and its Application to Mathematical Physics (PhD thesis), University of Cambridge, doi:10.17863/CAM.16148, hdl:1810/251691, OCLC 53604228
• Baylis, W. E., ed. (2011) [1996], Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering, Birkhäuser, ISBN 9781461241058
• Aragón, G.; Aragón, J.L.; Rodríguez, M.A. (1997), "Clifford Algebras and Geometric Algebra", Advances in Applied Clifford Algebras, 7 (2): 91–102, doi:10.1007/BF03041220, S2CID 120860757
• Hestenes, David (1999), New Foundations for Classical Mechanics (2nd ed.), Springer Verlag, ISBN 978-0-7923-5302-7
• Lasenby, Joan; Lasenby, Anthony N.; Doran, Chris J. L. (2000), "A Unified Mathematical Language for Physics and Engineering in the 21st Century" (PDF), Philosophical Transactions of the Royal Society A, 358 (1765): 21–39, Bibcode:2000RSPTA.358...21L, doi:10.1098/rsta.2000.0517, S2CID 91884543, archived (PDF) from the original on 2015-03-19
• Lounesto, Pertti (2001), Clifford Algebras and Spinors (2nd ed.), Cambridge University Press, ISBN 978-0-521-00551-7
• Baylis, W. E. (2002), Electrodynamics: A Modern Geometric Approach (2nd ed.), Birkhäuser, ISBN 978-0-8176-4025-5
• Dorst, Leo (2002), "The Inner Products of Geometric Algebra", in Dorst, L.; Doran, C.; Lasenby, J. (eds.), Applications of Geometric Algebra in Computer Science and Engineering, Birkhäuser, pp. 35–46, doi:10.1007/978-1-4612-0089-5_2, ISBN 978-1-4612-0089-5
• Doran, Chris J. L.; Lasenby, Anthony N. (2003), Geometric Algebra for Physicists (PDF), Cambridge University Press, ISBN 978-0-521-71595-9, archived (PDF) from the original on 2009-01-06
• Hestenes, David (2003), "Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics" (PDF), Am. J. Phys., 71 (2): 104–121, Bibcode:2003AmJPh..71..104H, CiteSeerX 10.1.1.649.7506, doi:10.1119/1.1522700{{citation}}: CS1 maint: url-status (link)
• Hildenbrand, Dietmar; Fontijne, Daniel; Perwass, Christian; Dorst, Leo (2004), "Geometric Algebra and its Application to Computer Graphics" (PDF), Proceedings of Eurographics 2004, doi:10.2312/egt.20041032, archived (PDF) from the original on 2015-09-06
• Lasenby, Anthony (2004), "Conformal Models of de Sitter Space, Initial Conditions for Inflation and the CMB", AIP Conference Proceedings, vol. 736, pp. 53–70, arXiv:astro-ph/0411579, doi:10.1063/1.1835174, S2CID 18034896
• Hestenes, David (2005), Introduction to Primer for Geometric Algebra
• Selig, J.M. (2005). Geometric Fundamentals of Robotics. Monographs in Computer Science. New York, NY: Springer New York. doi:10.1007/b138859. ISBN 978-0-387-20874-9.
• Bain, J. (2006), "Spacetime structuralism: §5 Manifolds vs. geometric algebra", in Dennis Dieks (ed.), The ontology of spacetime, Elsevier, p. 54 ff, ISBN 978-0-444-52768-4
• Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007), Geometric algebra for computer science: an object-oriented approach to geometry, Elsevier, ISBN 978-0-12-369465-2, OCLC 132691969
• Penrose, Roger (2007), The Road to Reality, Vintage books, ISBN 978-0-679-77631-4
• Francis, Matthew R.; Kosowsky, Arthur (2008), "The Construction of Spinors in Geometric Algebra", Annals of Physics, 317 (2): 383–409, arXiv:math-ph/0403040v2, Bibcode:2005AnPhy.317..383F, doi:10.1016/j.aop.2004.11.008, S2CID 119632876
• Li, Hongbo (2008), Invariant Algebras and Geometric Reasoning, World Scientific, ISBN 9789812770110. Chapter 1 as PDF
• Vince, John A. (2008), Geometric Algebra for Computer Graphics, Springer, ISBN 978-1-84628-996-5
• Lundholm, Douglas; Svensson, Lars (2009), "Clifford Algebra, Geometric Algebra and Applications", arXiv:0907.5356v1 [math-ph]
• Perwass, Christian (2009), Geometric Algebra with Applications in Engineering, Geometry and Computing, vol. 4, Bibcode:2009gaae.book.....P, doi:10.1007/978-3-540-89068-3, ISBN 978-3-540-89067-6
• Selig, J. M. (2000), "Clifford algebra of points, lines and planes" (PDF), Robotica, 18 (5): 545–556, doi:10.1017/S0263574799002568, S2CID 28929170
• Bayro-Corrochano, Eduardo (2010), Geometric Computing for Wavelet Transforms, Robot Vision, Learning, Control and Action, Springer Verlag, ISBN 9781848829299
• Bayro-Corrochano, E.; Scheuermann, Gerik, eds. (2010), Geometric Algebra Computing in Engineering and Computer Science, Springer, ISBN 9781849961080 Extract online at https://davidhestenes.net/geocalc/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
• Goldman, Ron (2010), Rethinking Quaternions: Theory and Computation, Morgan & Claypool, Part III. Rethinking Quaternions and Clifford Algebras, ISBN 978-1-60845-420-4
• Dorst, Leo.; Lasenby, Joan (2011), Guide to Geometric Algebra in Practice, Springer, ISBN 9780857298119
• Macdonald, Alan (2011), Linear and Geometric Algebra, CreateSpace, ISBN 9781453854938, OCLC 704377582
• Snygg, John (2011), A New Approach to Differential Geometry using Clifford's Geometric Algebra, Springer, ISBN 978-0-8176-8282-8
• Hildenbrand, Dietmar (2012), "Foundations of Geometric Algebra computing", Numerical Analysis and Applied Mathematics Icnaam 2012: International Conference of Numerical Analysis and Applied Mathematics, AIP Conference Proceedings, 1479 (1): 27–30, Bibcode:2012AIPC.1479...27H, doi:10.1063/1.4756054
• Sokolov, Andrey (2013), Clifford algebra and the projective model of Minkowski (Pseudo-Euclidean) spaces, arXiv:1307.4179
• Bromborsky, Alan (2014), An introduction to Geometric Algebra and Calculus (PDF), archived (PDF) from the original on 2019-10-15
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• Wilmot, G.P. (2023). "The Algebra Of Geometry". GitHub.
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External links

• A Survey of Geometric Algebra and Geometric Calculus Alan Macdonald, Luther College, Iowa.
• Imaginary Numbers are not Real – the Geometric Algebra of Spacetime. Introduction (Cambridge GA group).
• Geometric Algebra 2015, Masters Course in Scientific Computing, from Dr. Chris Doran (Cambridge).
• Maths for (Games) Programmers: 5 – Multivector methods. Comprehensive introduction and reference for programmers, from Ian Bell.
• IMPA Summer School 2010 Fernandes Oliveira Intro and Slides.
• University of Fukui E.S.M. Hitzer and Japan GA publications.
• Google Group for GA
• Geometric Algebra Primer Introduction to GA, Jaap Suter.
• Geometric Algebra Resources curated wiki, Pablo Bleyer.
• Applied Geometric Algebras in Computer Science and Engineering 2018 Early Proceedings
• bivector.net Geometric Algebra for CGI, Vision and Engineering community website
• AGACSE 2021 Videos

English translations of early books and papers

• G. Combebiac, "calculus of tri-quaternions" (Doctoral dissertation)
• M. Markic, "Transformants: A new mathematical vehicle. A synthesis of Combebiac's tri-quaternions and Grassmann's geometric system. The calculus of quadri-quaternions"
• C. Burali-Forti, "The Grassmann method in projective geometry" A compilation of three notes on the application of exterior algebra to projective geometry
• C. Burali-Forti, "Introduction to Differential Geometry, following the method of H. Grassmann" Early book on the application of Grassmann algebra
• H. Grassmann, "Mechanics, according to the principles of the theory of extension" One of his papers on the applications of exterior algebra.

Research groups

• Geometric Calculus International. Links to Research groups, Software, and Conferences, worldwide.
• Cambridge Geometric Algebra group. Full-text online publications, and other material.
• University of Amsterdam group
• Geometric Calculus research & development (archive of Hestenes's website at Arizona State University).
• GA-Net blog and newsletter archive. Geometric Algebra/Clifford Algebra development news.
• Geometric Algebra for Perception Action Systems. Geometric Cybernetics Group (CINVESTAV, Campus Guadalajara, Mexico).