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superalgebra (Definition)

A graded algebra $ A$ is said to be a superalgebra if it has a $ \mathbb{Z}/2\mathbb{Z}$ grading. As a vector space, a superalgebra has a decomposition into two homogeneous subspaces, $ A = A_0\oplus A_1$. The homogeneous subspace $ A_0$ is known as the space of even elements of $ A$, and $ A_1$ is known as the space of odd elements. Let $ \vert a\vert$ denote the degree of a homogeneous element. That is, $ \vert a\vert = 0$ if $ a \in A_0$ and $ \vert a\vert = 1$ if $ a \in A_1$. The degree satisfies $ \vert ab\vert = \vert a\vert + \vert b\vert$.



"superalgebra" is owned by mhale. [ full author list (2) | owner history (1) ]
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See Also: supernumber, supercommutative, Lie super algebra, Lie superalgebra, graded algebra

Other names:  super algebra
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Cross-references: homogeneous element, degree, odd, even, subspaces, homogeneous, decomposition, vector space, grading, graded algebra
There are 14 references to this entry.

This is version 4 of superalgebra, born on 2002-06-09, modified 2007-12-17.
Object id is 3082, canonical name is SuperAlgebra.
Accessed 4700 times total.

Classification:
AMS MSC: 16W55 (Associative rings and algebras :: Rings and algebras with additional structure :: ``Super'' structure)
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