Graded Lie Superalgebras
A graded Lie superalgebra over a field k (not of characteristic 2) consists of a graded vector space E over k, along with a bilinear bracket operation
such that the following axioms are satisfied.
-
- respects the gradation of E:
- .
- (Symmetry.) If x ε Ei and y ε Ej, then
- (Jacobi identity.) If x ε Ei, y ε Ej, and z ε Ek, then
- .
- (If k has characteristic 3, then the Jacobi identity must be supplemented with the condition for all x in Eodd.)
Note, for instance, that when E carries the trivial gradation, a graded Lie superalgebra over k is just an ordinary Lie algebra. When the gradation of E is concentrated in even degrees, one recovers the definition of a (Z-) graded Lie algebra.
Read more about this topic: Graded Lie Algebra
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