Lie superalgebra
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In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around).
Formally, a Lie superalgebra is a (nonassociative) Z2-graded algebra, or superalgebra, over a field of characteristic 0 (typically R or C) whose product [·, ·], called the Lie superbracket or supercommutator, satisfies
- <math>[x,y]=-(-1)^{|x| |y|}[y,x]</math>
and
- <math>(-1)^{|z| |x|}[x,[y,z]]+(-1)^{|x| |y|}[y,[z,x]]+(-1)^{|y| |z|}[z,[x,y]]=0</math>
where x, y, and z are pure in the Z2-grading. Here, |x| denotes the degree of x (either 0 or 1).
Lie superalgebras are a natural generalization of normal Lie algebras to include a Z2-grading. Indeed, the above conditions on the superbracket are exactly those on the normal Lie bracket with modifications made for the grading. The last condition is sometimes called the super Jacobi identity.
Note that the even subalgebra of a Lie superalgebra forms a (normal) Lie algebra as all the funny signs disappear, and the superbracket becomes a normal Lie bracket.
One way of thinking about a Lie superalgebra is to consider its even and odd parts, L0 and L1 separately. Then, L0 is a Lie algebra, L1 is a linear rep of L0, and there exists a symmetric L0-intertwiner <math>\{.,.\}:L_1\otimes L_1\rightarrow L_0</math> such that for all x,y and z in L1,
- <math>\left\{x, y\right\}[z]+\left\{y, z\right\}[x]+\left\{z, x\right\}[y]=0</math>
A * Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map from itself to itself which respects the Z2 grading and satisfies [x,y]*=[y*,x*] for all x and y in the Lie superalgebra. Its universal enveloping algebra would be an ordinary *-algebra.
The universal enveloping algebra of the Lie superalgebra can be given a Hopf algebra structure.
Examples
Given any associative superalgebra A one can define the supercommutator on homogeneous elements by
- <math>[x,y] = xy - (-1)^{|x||y|}yx</math>
and then extending by linearity to all elements. The algebra A together with the supercommutator then becomes a Lie superalgebra.
Category-theoretic definition
In category theory, a Lie superalgebra can be defined as a nonassociative superalgebra whose product satisfies
- <math>[\cdot,\cdot]\circ (id+\tau_{A,A})=0</math>
- <math>[\cdot,\cdot]\circ ([\cdot,\cdot]\otimes id)\circ(id+\sigma+\sigma^2)=0</math>
where σ is the cyclic permutation braiding <math>(id\otimes\tau_{A,A})\circ(\tau_{A,A}\otimes id)</math>. In diagrammatic form:
