# Alternating algebra

From HandWiki

__: Algebra with a graded anticommutativity property on multiplication__

**Short description**In mathematics, an **alternating algebra** is a **Z**-graded algebra for which *xy* = (−1)^{deg(x)deg(y)}*yx* for all nonzero homogeneous elements *x* and *y* (i.e. it is an anticommutative algebra) and has the further property that *x*^{2} = 0 for every homogeneous element *x* of odd degree.^{[1]}

## Examples

- The differential forms on a differentiable manifold form an alternating algebra.
- The exterior algebra is an alternating algebra.
- The cohomology ring of a topological space is an alternating algebra.

## Properties

- The algebra formed as the direct sum of the homogeneous subspaces of even degree of an anticommutative algebra
*A*is a subalgebra contained in the centre of*A*, and is thus commutative. - An anticommutative algebra
*A*over a (commutative) base ring*R*in which 2 is not a zero divisor is alternating.^{[2]}

## See also

## References

- ↑ Nicolas Bourbaki (1998).
*Algebra I*. Springer Science+Business Media. p. 482. - ↑ Nicolas Bourbaki (1998).
*Algebra I*. Springer Science+Business Media. p. 482.

Original source: https://en.wikipedia.org/wiki/Alternating algebra.
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