Lie superalgebra

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).

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::$[x,y]\; =-(-1)^\; [y,\; [z,x]\; +(-1)^\; [z,\; [x,y]\; =0$

where "x", "y", and "z" are pure in the Z₂-grading. Here, |"x"| denotes the degree of "x" (either 0 or 1).

One also sometimes adds the axioms $[x,x]\; =0$ for |"x"|=0 (if 2 is invertible this follows automatically) and $[x,x]\; ,x]\; =0$ for |"x"|=1 (if 3 is invertible this follows automatically).

Just as for Lie algebras, the universal enveloping algebra of the Lie superalgebra can be given a Hopf algebra structure.

Distinction from graded Lie algebra

A graded Lie algebra (say, graded by Z or N) that is commutative and Jacobi in the graded sense also has a $Z\_2$ grading (which is called "rolling up" the algebra into odd and even parts), but is not referred to as "super". See note at graded Lie algebra for discussion.

Even and odd parts

Note that the even subalgebra of a Lie superalgebra forms a (normal) Lie algebra as all the 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 representation of L₀, and there exists a symmetric L₀-equivariant linear map $\{cdot,cdot\}:L\_1otimes\; L\_1\; ightarrow\; L\_0$ such that for all x,y and z in L₁,

:$left\{x,\; y\; ight\}\; [z]\; +left\{y,\; z\; ight\}\; [x]\; +left\{z,\; x\; ight\}\; [y]\; =0.$

Involution

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. (Some authors prefer the convention [x,y] ^*=(−1)^|x||y| [y^*,x^*] ; changing * to −* switches between the two conventions.) Its universal enveloping algebra would be an ordinary ^*-algebra.

Examples

Given any associative superalgebra "A" one can define the supercommutator on homogeneous elements by:$[x,y]\; =\; xy\; -\; (-1)^yx$and then extending by linearity to all elements. The algebra "A" together with the supercommutator then becomes a Lie superalgebra.

The Whitehead product on homotopy groups gives many examples of Lie superalgebras over the integers.

Classification

The simple complex finite dimensional Lie superalgebras were classified by Victor Kac.

Category-theoretic definition

In category theory, a Lie superalgebra can be defined as a nonassociative superalgebra whose product satisfies

*$[cdot,cdot]\; circ\; (id+\; au\_\{A,A\})=0$
*$[cdot,cdot]\; circ\; (\; [cdot,cdot]\; otimes\; id)circ(id+sigma+sigma^2)=0$where σ is the cyclic permutation braiding $(idotimes\; au\_\{A,A\})circ(\; au\_\{A,A\}otimes\; id)$. In diagrammatic form:

:

See also

* Anyonic Lie algebra
* Grassmann algebra
* Representation of a Lie superalgebra
* Superspace
* Supergroup

References

*Kac, V. G. "Lie superalgebras." Advances in Math. 26 (1977), no. 1, 8--96.
* Manin, Yuri I. "Gauge field theory and complex geometry." Grundlehren der Mathematischen Wissenschaften, 289. Springer-Verlag, Berlin, 1997. ISBN 3-540-61378-1