Alternating algebra

Not to be confused with Alternative 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 x² = 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.

See also

• Alternating multilinear map
• Exterior algebra
• Graded-symmetric algebra