Alternating algebra explained

Alternating algebra should not be confused with Alternative algebra.

In mathematics, an alternating algebra is a -graded algebra for which for all nonzero homogeneous elements and (i.e. it is an anticommutative algebra) and has the further property that for every homogeneous element 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 is a subalgebra contained in the centre of, and is thus commutative.
• An anticommutative algebra over a (commutative) base ring in which 2 is not a zero divisor is alternating.^[1]

See also

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

Notes and References

1. Book: Nicolas Bourbaki. 1998. Algebra I. Springer Science+Business Media. 482.