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

\nabla:A\otimes A\rightarrow A

. 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. A supercommutative algebra is a superalgebra satisfying a graded version of commutivity. Category theoretically,

\nabla

and

\nabla\circ \tau_{A,A}

commute. The primary example being the exterior algebra on a vector space. A Lie superalgebra is nonassociative superalgebra which is the graded version of a ordinary Lie algebra. The product map is written as

\cdot,\cdot

instead. Category theoretically,

\circ (id+\tau_{A,A})=0

and

\otimes id)\circ(id+\sigma+\sigma^2)=0

where σ is the cyclic permutation braiding

(id\otimes \tau_{A,A})\circ(\tau_{A,A}\otimes id)

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