Alternating algebra

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Short description: Algebra with a graded anticommutativity property on multiplication

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 for every homogeneous element x of odd degree.[1]

Examples

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

  1. Nicolas Bourbaki (1998). Algebra I. Springer Science+Business Media. p. 482. 
  2. Nicolas Bourbaki (1998). Algebra I. Springer Science+Business Media. p. 482.