Premium
Classical Lie Superalgebras Over Simple Associative Algebras
Author(s) -
Benkart Georgia,
Xu Xiaoping,
Zhao Kaiming
Publication year - 2006
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1017/s0024611505015583
Subject(s) - mathematics , lie conformal algebra , non associative algebra , associative property , simple (philosophy) , pure mathematics , universal enveloping algebra , algebra over a field , associative algebra , algebra representation , lie algebra , cellular algebra , philosophy , epistemology
Over arbitrary fields of characteristic not equal to 2, we construct three families of simple Lie algebras and six families of simple Lie superalgebras of matrices with entries chosen from different one‐sided ideals of a simple associative algebra. These families correspond to the classical Lie algebras and superalgebras. Our constructions intermix the structure of the associative algebra and the structure of the matrix algebra in an essential, compatible way. Many examples of simple associative algebras without an identity element arise as a by‐product. The study of conformal algebras and superalgebras often involves matrix algebras over associative algebras such as Weyl algebras, and for that reason, we illustrate our constructions by taking various one‐sided ideals from a Weyl algebra or a quantum torus. 2000 Mathematics Subject Classification 17B20, 17B65, 17B67, 17B68.