nLab supercommutative algebra

Idea

A super-commutative algebra is a commutative algebra internal to the symmetric monoidal category of super vector spaces, hence a $\mathbb{Z}/2$-graded associative algebra such that for $a, b$ any two elements of homogeneous degree $deg(a), deg(b) \in \mathbb{Z}/2 = \{0,1\}$, we have

$$a \cdot b \;\;=\;\; (-1)^{deg(a) deg(b)} \; b \cdot a \,.$$

For more see at geometry of physics – superalgebra.

Examples

• The free supercommutative algebras are the Grassmann algebras.

• stable homotopy groups of homotopy commutative ring spectrum, see at Introduction to Stable homotopy theory the section 1-2 Homotopy commutative ring spectra

• Jordan superalgebra

Related concepts

• supercommutative ring

• super-scheme

• commutative differential graded algebra