Talk:Superalgebra

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[edit] Removed diagrams

I've removed the following material from the article as I'm not sure what to call these diagrams. I've heard them called tangle diagrams but they don't seem to have any relation to the notion discussed there. Also, I personally don't find these diagrams very illuminating.

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 an 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).

Image:Liealgebra.png

-- Fropuff 08:13, 7 February 2006 (UTC)

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