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