Alternating algebra

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.

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.