# Superalgebra

In mathematics and theoretical physics, a superalgebra over a field K generally refers to a Z2-graded algebra over K (here Z2 is the cyclic group of order 2).

Category theoretically, a superalgebra is an object A of the category of Z2-graded vector spaces together with an even morphism [itex]\nabla:A\otimes A\rightarrow A[itex].

An associative superalgebra (or Z2-graded associative algebra) is one whose product is associative. Category theoretically, this means the commutative diagram expressing associativity commutes. Principal examples are Clifford algebras.

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A supercommutative algebra is a superalgebra satisfying a graded version of commutivity. Category theoretically, [itex]\nabla[itex] and [itex]\nabla\circ \tau_{A,A}[itex] commute. The primary example being the exterior algebra on a vector space.

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A Lie superalgebra is nonassociative superalgebra which is the graded version of a ordinary Lie algebra. The product map is written as [itex][\cdot,\cdot][itex] instead. Category theoretically, [itex][\cdot,\cdot]\circ (id+\tau_{A,A})=0[itex] and [itex][\cdot,\cdot]\circ ([\cdot,\cdot]\otimes id)\circ(id+\sigma+\sigma^2)=0[itex] where σ is the cyclic permutation braiding [itex](id\otimes \tau_{A,A})\circ(\tau_{A,A}\otimes id)[itex].

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