Superalgebra

From Academic Kids

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 <math>\nabla:A\otimes A\rightarrow A<math>.

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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Associative.png
Image:Associative.png

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

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Commutative.png
Image:Commutative.png

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

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Liealgebra.png
Image:Liealgebra.png

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