Lie superalgebra - Knowlpedia

Definition

Let $k$ be a field of characteristic $0$ . A Lie superalgebra over $k$ is a super vector space $\mathfrak g=\mathfrak g_{\bar0}\oplus\mathfrak g_{\bar1}$ with an even bilinear bracket such that homogeneous $x,y,z$ satisfy

[x,y]=-(-1)^{|x||y|}[y,x]

and

(-1)^{|x||z|}[x,[y,z]] +(-1)^{|y||x|}[y,[z,x]] +(-1)^{|z||y|}[z,[x,y]]=0.

The first formula is graded skew-symmetry and the second is the super Jacobi identity.

Even and odd parts

The even part $\mathfrak g_{\bar0}$ is an ordinary Lie algebra. The odd part $\mathfrak g_{\bar1}$ is a representation of $\mathfrak g_{\bar0}$ , and the bracket on two odd elements is a symmetric equivariant map

\mathfrak g_{\bar1}\otimes\mathfrak g_{\bar1} \longrightarrow\mathfrak g_{\bar0}.

It is symmetric because graded skew-symmetry gives $[x,y]=[y,x]$ when $x$ and $y$ are odd.

Morphisms

A morphism of Lie superalgebras is an even linear map that preserves brackets. A Lie superalgebra can act on a super vector space through a representation of a Lie superalgebra.

Associative source of examples

Every associative superalgebra $A$ becomes a Lie superalgebra under its supercommutator. In particular, endomorphisms of a super vector space form $\mathfrak{gl}(V)$ .

Small-characteristic warning

The displayed sign definition is cleanest in characteristic $0$ . In characteristic $2$ , graded skew-symmetry loses its usual meaning; in characteristic $3$ , one must take care with the additional behavior of $[x,[x,x]]$ for odd $x$ . Definitions over those fields therefore include extra axioms or divided-power data and are not being silently identified with the characteristic-zero theory here.

References

M. Scheunert, The Theory of Lie Superalgebras, Lecture Notes in Mathematics 716, Springer, 1979. DOI record. Relevant: Chapter 1.
I. M. Musson, Lie Superalgebras and Enveloping Algebras, American Mathematical Society, 2012. DOI record. Relevant: Chapters 1–2.