Super PBW theorem - Knowlpedia

Statement

Let $\mathfrak g$ be a Lie superalgebra over a field of characteristic $0$ . Filter its universal enveloping algebra $U(\mathfrak g)$ by tensor degree. The super Poincaré–Birkhoff–Witt theorem gives a canonical graded-superalgebra isomorphism

\operatorname{gr}U(\mathfrak g) \cong \operatorname{Sym}_{\mathrm s}(\mathfrak g),

where the right side is the symmetric superalgebra.

After choosing ordered homogeneous bases, a PBW basis consists of ordered monomials in which even basis elements have arbitrary nonnegative exponents and odd basis elements have exponent $0$ or $1$ . Consequently there is a noncanonical vector-space isomorphism

U(\mathfrak g) \cong U(\mathfrak g_{\bar0})\otimes\Lambda(\mathfrak g_{\bar1}).

This last isomorphism is generally not an algebra isomorphism.

References

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