supercommutative algebra in nLab

Contents

Contents

Idea

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

a⋅b=(−1) deg(a)deg(b)b⋅a. 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

• supercommutative ring

• super-scheme

• commutative differential graded algebra