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 (nilpotence) for every homogeneous element x of odd degree.
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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. /wiki/Springer_Science%2BBusiness_Media ↩
Nicolas Bourbaki (1998). Algebra I. Springer Science+Business Media. p. 482. /wiki/Springer_Science%2BBusiness_Media ↩