alternating algebra
{{Short description|Algebra with a graded anticommutativity property on multiplication}}
{{distinguish|Alternative algebra}}
In mathematics, an alternating algebra is a {{math|Z}}-graded algebra for which {{math|1=xy = (−1){{sup|deg(x)deg(y)}}yx}} for all nonzero homogeneous elements {{math|x}} and {{math|y}} (i.e. it is an anticommutative algebra) and has the further property that {{math|1=x2 = 0}} (nilpotence) for every homogeneous element {{math|x}} of odd degree.{{cite book|author=Nicolas Bourbaki|year=1998|title=Algebra I|publisher=Springer Science+Business Media|page=482}}
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 {{math|A}} is a subalgebra contained in the centre of {{math|A}}, and is thus commutative.
- An anticommutative algebra {{math|A}} over a (commutative) base ring {{math|R}} in which 2 is not a zero divisor is alternating.