Superalgebra - Knowlpedia

Definition

Let $k$ be a field of characteristic different from $2$ . A superalgebra over $k$ is an associative unital algebra object in $\mathbf{SuperVect}_k$ . Concretely, it is a super vector space

A=A_{\bar0}\oplus A_{\bar1}

with $1\in A_{\bar0}$ and $A_{\bar i}A_{\bar j}\subseteq A_{\bar i+\bar j}$ . Its multiplication and unit are even maps.

The supercommutator turns every associative superalgebra into a Lie superalgebra.

Examples and non-examples

An ordinary associative algebra placed entirely in even degree is a superalgebra. The Clifford algebra has its natural tensor-parity grading and is therefore a superalgebra.

A superalgebra need not be supercommutative. For instance, the Clifford relation usually gives a nonzero anticommutator of odd vectors.

Morphisms

A morphism of superalgebras is an even unital algebra homomorphism. Requiring evenness is part of the standard category: it preserves the stated grading, not merely the underlying ungraded multiplication.

References

V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, American Mathematical Society, 2004. DOI record. Relevant: Chapter 1.
P. Deligne and J. W. Morgan, “Notes on supersymmetry (following Joseph Bernstein),” in Quantum Fields and Strings: A Course for Mathematicians, Volume 1, American Mathematical Society, 1999. Relevant: Section 1.