Associative superalgebra over the complex numbers

A unital associative superalgebra over $\mathbb{C}$ is an associative unital complex algebra $A$ together with a direct-sum grading $A=A_{\bar{0}}\oplus A_{\bar{1}}$ for which $A_{\bar{i}}{A}_{\bar{j}}\subseteq A_{\overline{i+j}}$. Its unit belongs to $A_{\bar{0}}$. Elements of one graded summand are homogeneous, with parity $\mathrm{\mid }a\mathrm{\mid }\in \{0,1\}$. Multiplication is complex-bilinear and associative. A homomorphism of associative superalgebras is a unital algebra homomorphism preserving the grading.

The supercommutator of homogeneous $a,b\in A$ is $[a,b{]}_s:=ab-(-1{)}^{\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }}ba$. This notation is used only for the commutator induced by an associative superalgebra; a general Lie-superalgebra bracket is denoted $[a,b]$.