Alternating algebra
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In mathematics, an alternating algebra is a ZScript error: No such module "Check for unknown parameters".-graded algebra for which xy = (−1)deg(x)deg(y)yxScript error: No such module "Check for unknown parameters". for all nonzero homogeneous elements xScript error: No such module "Check for unknown parameters". and yScript error: No such module "Check for unknown parameters". (i.e. it is an anticommutative algebra) and has the further property that x2 = 0Script error: No such module "Check for unknown parameters". (nilpotence) for every homogeneous element xScript error: No such module "Check for unknown parameters". 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 AScript error: No such module "Check for unknown parameters". is a subalgebra contained in the centre of AScript error: No such module "Check for unknown parameters"., and is thus commutative.
- An anticommutative algebra AScript error: No such module "Check for unknown parameters". over a (commutative) base ring RScript error: No such module "Check for unknown parameters". in which 2 is not a zero divisor is alternating.[1]
See also
References
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