Universal enveloping algebra of a Lie superalgebra

Definition

Let $\mathfrak g$ be a Lie superalgebra over a field of characteristic $0$ . Its universal enveloping algebra is the superalgebra

U(\mathfrak g) =T(\mathfrak g)\Big/ \left\langle x\otimes y-(-1)^{|x||y|}y\otimes x-[x,y] \right\rangle,

where $x,y$ range over homogeneous elements and $T(\mathfrak g)$ is the tensor algebra. The canonical map $\mathfrak g\to U(\mathfrak g)$ sends the Lie bracket to the supercommutator.

Universal property

If $A$ is an associative superalgebra and $\phi:\mathfrak g\to A$ is a Lie-superalgebra morphism into the supercommutator Lie superalgebra of $A$ , there is a unique superalgebra homomorphism

\widetilde\phi:U(\mathfrak g)\longrightarrow A

extending $\phi$ . Consequently, representations of $\mathfrak g$ are equivalently supermodules over $U(\mathfrak g)$ , with the same parity convention on morphisms.

Filtering $U(\mathfrak g)$ by tensor degree gives the super PBW theorem.

Clifford quotient

The Clifford algebra as a super-enveloping quotient makes precise how fixing a central even generator turns a quadratic odd bracket into a Clifford relation.

References

I. M. Musson, Lie Superalgebras and Enveloping Algebras, American Mathematical Society, 2012. DOI record. Relevant: Chapters 6–7.
M. Scheunert, The Theory of Lie Superalgebras, Lecture Notes in Mathematics 716, Springer, 1979. DOI record. Relevant: Chapter 2.