Definition
Formally, a Lie superalgebra is a (nonassociative) Z2-graded algebra, or superalgebra, over a commutative ring (typically R or C) whose product, called the Lie superbracket or supercommutator, satisfies the two conditions (analogs of the usual Lie algebra axioms, with grading):
Super skew-symmetry:
The super Jacobi identity:
where x, y, and z are pure in the Z2-grading. Here, |x| denotes the degree of x (either 0 or 1). The degree of is the sum of degree of x and y modulo 2.
One also sometimes adds the axioms for |x|=0 (if 2 is invertible this follows automatically) and for |x|=1 (if 3 is invertible this follows automatically). When the ground ring is the integers or the Lie superalgebra is a free module, these conditions are equivalent to the condition that the Poincaré-Birkhoff-Witt theorem holds (and, in general, they are necessary conditions for the theorem to hold).
Just as for Lie algebras, the universal enveloping algebra of the Lie superalgebra can be given a Hopf algebra structure.
Read more about this topic: Lie Superalgebra
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