Clasificación de álgebras de Lie reales de baja dimensión

Esta lista relacionada con las matemáticas proporciona la clasificación de Mubarakzyanov de álgebras de Lie reales de baja dimensión , publicada en ruso en 1963. ^[1] Complementa el artículo sobre álgebra de Lie en el área del álgebra abstracta .

^{Popovych et al. [2]} publicaron una versión en inglés y una revisión de esta clasificación en 2003.

Clasificación de Mubarakzianov

Sea un álgebra de Lie -dimensional sobre el cuerpo de números reales con generadores , . ^{[ aclaración necesaria ]} Para cada álgebra aducimos sólo conmutadores distintos de cero entre elementos base. ${\displaystyle {\mathfrak {g}}_{n}}$ ${\displaystyle n}$ ${\displaystyle e_{1},\dots ,e_{n}}$ ${\displaystyle n\leq 4}$ ${\displaystyle {\mathfrak {g}}}$

Unidimensional

• ${\displaystyle {\mathfrak {g}}_{1}}$, abeliano .

Bidimensional

• ${\displaystyle 2{\mathfrak {g}}_{1}}$, abeliano ; ${\displaystyle \mathbb {R} ^{2}}$
• ${\displaystyle {\mathfrak {g}}_{2.1}}$, solucionable , ${\displaystyle {\mathfrak {aff}}(1)=\left\{{\begin{pmatrix}a&b\\0&0\end{pmatrix}}\,:\,a,b\in \mathbb {R} \right\}}$

${\displaystyle [e_{1},e_{2}]=e_{1}.}$

Tridimensional

• ${\displaystyle 3{\mathfrak {g}}_{1}}$, abeliano, Bianchi I ;
• ${\displaystyle {\mathfrak {g}}_{2.1}\oplus {\mathfrak {g}}_{1}}$, descomponible soluble, Bianchi III;
• ${\displaystyle {\mathfrak {g}}_{3.1}}$, álgebra de Heisenberg-Weyl, nilpotente, Bianchi II,

${\displaystyle [e_{2},e_{3}]=e_{1};}$

• ${\displaystyle {\mathfrak {g}}_{3.2}}$, solucionable, Bianchi IV,

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{1}+e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{3.3}}$, solucionable, Bianchi V,

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{3.4}}$, resoluble, Bianchi VI, álgebra de Poincaré cuando , ${\displaystyle {\mathfrak {p}}(1,1)}$ ${\displaystyle \alpha =-1}$

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=\alpha e_{2},\quad -1\leq \alpha <1,\quad \alpha \neq 0;}$

• ${\displaystyle {\mathfrak {g}}_{3.5}}$, solucionable, Bianchi VII,

${\displaystyle [e_{1},e_{3}]=\beta e_{1}-e_{2},\quad [e_{2},e_{3}]=e_{1}+\beta e_{2},\quad \beta \geq 0;}$

• ${\displaystyle {\mathfrak {g}}_{3.6}}$, sencillo, Bianchi VIII, ${\displaystyle {\mathfrak {sl}}(2,\mathbb {R} ),}$

${\displaystyle [e_{1},e_{2}]=e_{1},\quad [e_{2},e_{3}]=e_{3},\quad [e_{1},e_{3}]=2e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{3.7}}$, sencillo, Bianchi IX, ${\displaystyle {\mathfrak {so}}(3),}$

${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2},\quad [e_{1},e_{2}]=e_{3}.}$

El álgebra puede considerarse como un caso extremo de , cuando , formando una contracción del álgebra de Lie. ${\displaystyle {\mathfrak {g}}_{3.3}}$ ${\displaystyle {\mathfrak {g}}_{3.5}}$ ${\displaystyle \beta \rightarrow \infty }$

Sobre las álgebras de campo , son isomorfas a y , respectivamente. ${\displaystyle {\mathbb {C} }}$ ${\displaystyle {\mathfrak {g}}_{3.5}}$ ${\displaystyle {\mathfrak {g}}_{3.7}}$ ${\displaystyle {\mathfrak {g}}_{3.4}}$ ${\displaystyle {\mathfrak {g}}_{3.6}}$

Cuatro dimensiones

• ${\displaystyle 4{\mathfrak {g}}_{1}}$, abeliano;
• ${\displaystyle {\mathfrak {g}}_{2.1}\oplus 2{\mathfrak {g}}_{1}}$, descomponible, soluble,

${\displaystyle [e_{1},e_{2}]=e_{1};}$

• ${\displaystyle 2{\mathfrak {g}}_{2.1}}$, descomponible, soluble,

${\displaystyle [e_{1},e_{2}]=e_{1}\quad [e_{3},e_{4}]=e_{3};}$

• ${\displaystyle {\mathfrak {g}}_{3.1}\oplus {\mathfrak {g}}_{1}}$, nilpotente descomponible,

${\displaystyle [e_{2},e_{3}]=e_{1};}$

• ${\displaystyle {\mathfrak {g}}_{3.2}\oplus {\mathfrak {g}}_{1}}$, descomponible, soluble,

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{1}+e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{3.3}\oplus {\mathfrak {g}}_{1}}$, descomponible, soluble,

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{3.4}\oplus {\mathfrak {g}}_{1}}$, descomponible, soluble,

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=\alpha e_{2},\quad -1\leq \alpha <1,\quad \alpha \neq 0;}$

• ${\displaystyle {\mathfrak {g}}_{3.5}\oplus {\mathfrak {g}}_{1}}$, descomponible, soluble,

${\displaystyle [e_{1},e_{3}]=\beta e_{1}-e_{2}\quad [e_{2},e_{3}]=e_{1}+\beta e_{2},\quad \beta \geq 0;}$

• ${\displaystyle {\mathfrak {g}}_{3.6}\oplus {\mathfrak {g}}_{1}}$, irresoluble,

${\displaystyle [e_{1},e_{2}]=e_{1},\quad [e_{2},e_{3}]=e_{3},\quad [e_{1},e_{3}]=2e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{3.7}\oplus {\mathfrak {g}}_{1}}$, irresoluble,

${\displaystyle [e_{1},e_{2}]=e_{3},\quad [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{4.1}}$, nilpotente indecomponible,

${\displaystyle [e_{2},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{4.2}}$, indecomponible, soluble,

${\displaystyle [e_{1},e_{4}]=\beta e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3},\quad \beta \neq 0;}$

• ${\displaystyle {\mathfrak {g}}_{4.3}}$, indecomponible, soluble,

${\displaystyle [e_{1},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{4.4}}$, indecomponible, soluble,

${\displaystyle [e_{1},e_{4}]=e_{1},\quad [e_{2},e_{4}]=e_{1}+e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3};}$

• ${\displaystyle {\mathfrak {g}}_{4.5}}$, indecomponible, soluble,

${\displaystyle [e_{1},e_{4}]=\alpha e_{1},\quad [e_{2},e_{4}]=\beta e_{2},\quad [e_{3},e_{4}]=\gamma e_{3},\quad \alpha \beta \gamma \neq 0;}$

• ${\displaystyle {\mathfrak {g}}_{4.6}}$, indecomponible, soluble,

${\displaystyle [e_{1},e_{4}]=\alpha e_{1},\quad [e_{2},e_{4}]=\beta e_{2}-e_{3},\quad [e_{3},e_{4}]=e_{2}+\beta e_{3},\quad \alpha >0;}$

• ${\displaystyle {\mathfrak {g}}_{4.7}}$, indecomponible, soluble,

${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{1},e_{4}]=2e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3};}$

• ${\displaystyle {\mathfrak {g}}_{4.8}}$, indecomponible, soluble,

${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{1},e_{4}]=(1+\beta )e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=\beta e_{3},\quad -1\leq \beta \leq 1;}$

• ${\displaystyle {\mathfrak {g}}_{4.9}}$, indecomponible, soluble,

${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{1},e_{4}]=2\alpha e_{1},\quad [e_{2},e_{4}]=\alpha e_{2}-e_{3},\quad [e_{3},e_{4}]=e_{2}+\alpha e_{3},\quad \alpha \geq 0;}$

• ${\displaystyle {\mathfrak {g}}_{4.10}}$, indecomponible, soluble,

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2},\quad [e_{1},e_{4}]=-e_{2},\quad [e_{2},e_{4}]=e_{1}.}$

El álgebra puede considerarse como un caso extremo de , cuando , formando una contracción del álgebra de Lie. ${\displaystyle {\mathfrak {g}}_{4.3}}$ ${\displaystyle {\mathfrak {g}}_{4.2}}$ ${\displaystyle \beta \rightarrow 0}$

Sobre el campo las álgebras , , , , son isomorfas a , , , , , respectivamente. ${\displaystyle {\mathbb {C} }}$ ${\displaystyle {\mathfrak {g}}_{3.5}\oplus {\mathfrak {g}}_{1}}$ ${\displaystyle {\mathfrak {g}}_{3.7}\oplus {\mathfrak {g}}_{1}}$ ${\displaystyle {\mathfrak {g}}_{4.6}}$ ${\displaystyle {\mathfrak {g}}_{4.9}}$ ${\displaystyle {\mathfrak {g}}_{4.10}}$ ${\displaystyle {\mathfrak {g}}_{3.4}\oplus {\mathfrak {g}}_{1}}$ ${\displaystyle {\mathfrak {g}}_{3.6}\oplus {\mathfrak {g}}_{1}}$ ${\displaystyle {\mathfrak {g}}_{4.5}}$ ${\displaystyle {\mathfrak {g}}_{4.8}}$ ${\displaystyle {2{\mathfrak {g}}}_{2.1}}$

Véase también

• Tabla de grupos de Lie
• Grupo de Lie simple#Clasificación completa

Notas

1. Mubarakzianov 1963
2. Popovich 2003

Referencias

• Mubarakzyanov, GM (1963). "Sobre álgebras de Lie solubles". Izv. Vys. Ucheb. Zaved. Matematika (en ruso). 1 (32): 114-123. Señor  0153714. Zbl  0166.04104.
• Popovych, RO; Boyko, VM; Nesterenko, MO; Lutfullin, MW; et al. (2003). "Realizaciones de álgebras de Lie reales de baja dimensión". J. Phys. A: Math. Gen . 36 (26): 7337–7360. arXiv : math-ph/0301029 . Código Bibliográfico :2003JPhA...36.7337P. doi :10.1088/0305-4470/36/26/309. S2CID  9800361.