Alternating_algebra Knowpia

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 $x 2 = 0$ for every homogeneous element $x$ of odd degree.^[1]

Examples edit

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 edit

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]

References edit

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

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

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

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