Algebra over a field

:This article is about a particular kind of vector space. For other uses of the term "algebra" see algebra (disambiguation).

Properties

For algebras over a field, the bilinear multiplication from A × A to A is completely determined by the multiplication of basis elements of A.

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Conversely, once a basis for A has been chosen, the products of basis elements can be set arbitrarily, and then extended in a unique way to a bilinear operator on A, i.e. so that the resulting multiplication will satisfy the algebra laws.

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Thus, given the field K, any algebra can be specified up to isomorphism by giving its dimension (say n), and specifying n3 structure coefficients ci,j,k, which are scalars.

Related Topics:
Up to - Isomorphism - Dimension - Scalar

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These structure coefficients determine the multiplication in A via the following rule:

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: mathbf{e}_{i} mathbf{e}_{j} = sum_{k=1}^n c_{i,j,k} mathbf{e}_{k}

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where e1,...,en form a basis of A.

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The only requirement on the structure coefficients is that, if the dimension n is an infinite number, then this sum must always converge (in whatever sense is appropriate for the situation).

Related Topics:
Infinite number - Converge

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Note however that several different sets of structure coefficients can give rise to isomorphic algebras.

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When the algebra can be endowed with a metric, then the structure coefficients are written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations. Specifically, lower indices are covariant indices, and transform via pullbacks, while upper indices are contravariant, transforming under pushforwards. Thus, in mathematical physics, the structure coefficients are often written ci,jk, and their defining rule is written using the Einstein notation as

Related Topics:
Metric - Covariant - Pullback - Contravariant - Pushforward - Mathematical physics - Einstein notation

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: eiej = ci,jkek.

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If you apply this to vectors written in index notation, then this becomes

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: (xy)k = ci,jkxiyj.

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If K is only a commutative ring and not a field, then the same process works if A is a free module over K. If it isn't, then the multiplication is still completely determined by its action on a generating set of A; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism.

Related Topics:
Free module - Generating set

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