In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } ‑grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. The notion of Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } grading used here is distinct from a second Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } grading having cohomological origins. A graded Lie algebra (say, graded by Z {\displaystyle \mathbb {Z} } or N {\displaystyle \mathbb {N} } ) that is anticommutative and has a graded Jacobi identity also has a Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } grading; this is the "rolling up" of the algebra into odd and even parts. This rolling-up is not normally referred to as "super". Thus, supergraded Lie superalgebras carry a pair of Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } ‑gradations: one of which is supersymmetric, and the other is classical. Pierre Deligne calls the supersymmetric one the super gradation, and the classical one the cohomological gradation. These two gradations must be compatible, and there is often disagreement as to how they should be regarded.
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:
[ x , y ] = − ( − 1 ) | x | | y | [ y , x ] . {\displaystyle [x,y]=-(-1)^{|x||y|}[y,x].\ }
The super Jacobi identity:
( − 1 ) | x | | z | [ x , [ y , z ] ] + ( − 1 ) | y | | x | [ y , [ z , x ] ] + ( − 1 ) | z | | y | [ z , [ x , y ] ] = 0 , {\displaystyle (-1)^{|x||z|}[x,[y,z]]+(-1)^{|y||x|}[y,[z,x]]+(-1)^{|z||y|}[z,[x,y]]=0,}
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 [x,y] is the sum of degree of x and y modulo 2. One also sometimes adds the axioms [ x , x ] = 0 {\displaystyle [x,x]=0} for |x| = 0 (if 2 is invertible this follows automatically) and [ [ x , x ] , x ] = 0 {\displaystyle [[x,x],x]=0} 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.
Comments Lie superalgebras show up in physics in several different ways. In conventional supersymmetry, the even elements of the superalgebra correspond to bosons and odd elements to fermions. This corresponds to a bracket that has a grading of zero:
| [ a , b ] | = | a | + | b | {\displaystyle |[a,b]|=|a|+|b|}
This is not always the case; for example, in BRST supersymmetry and in the Batalin–Vilkovisky formalism, it is the other way around, which corresponds to the bracket of having a grading of -1:
| [ a , b ] | = | a | + | b | − 1 {\displaystyle |[a,b]|=|a|+|b|-1}
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