Superalgebra

From Exampleproblems

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.

Image:Associative.png

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.

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 $[\cdot,\cdot]$ instead. Category theoretically, $[\cdot,\cdot]\circ (id+\tau_{A,A})=0$ and $[\cdot,\cdot]\circ ([\cdot,\cdot]\otimes id)\circ(id+\sigma+\sigma^2)=0$ where σ is the cyclic permutation braiding $(id\otimes \tau_{A,A})\circ(\tau_{A,A}\otimes id)$ .