Alternating algebra
In mathematics, an alternating algebra is a -graded algebra for which for all nonzero homogeneous elements and and has the further property that for every homogeneous element 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 is a subalgebra contained in the centre of, and is thus commutative.
- An anticommutative algebra over a base ring in which 2 is not a zero divisor is alternating.