# The Unapologetic Mathematician

## Algebras

May 8, 2007 - Posted by | Ring theory

1. […] one that we haven’t considered directly: let be a commutative ring with unit and let be an algebra over with unit. Then we have a homomorphism of rings sending to — the action of on the […]

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2. […] mention the Bracket polynomial and the Jones polynomial. Jones was studying a certain kind of algebra when he realized that the defining relations for these algebras were very much like those of the […]

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3. […] go any further into linear algebra. Specifically, we’ll need to know a few things about the algebra of polynomials. Specifically (and diverging from the polynomials discussed earlier) we’re […]

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4. […] And, of course, once we’ve got monoids and -linearity floating around, we’re inexorably drawn — Serge would way we have an irresistable compulsion — to consider monoid objects in the category of -modules. That is: -algebras. […]

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5. […] or not. We know that the linear maps from a vector space (of finite dimension ) to itself form an algebra over . We can pick a basis and associate a matrix to each of these linear transformations. It turns […]

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6. […] We’re about to talk about certain kinds of algebras that have the added structure of a “grading”. It’s not horribly important at the […]

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7. […] and Symmetric Algebras There are a few graded algebras we can construct with our symmetric and antisymmetric tensors, and at least one of them will be […]

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8. […] we proceed with the differential geometry: Lie algebras. These are like “regular” associative algebras in that we take a module (often a vector space) and define a bilinear operation on it. This much is […]

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9. […] Algebras from Associative Algebras There is a great source for generating many Lie algebras: associative algebras. Specifically, if we have an associative algebra we can build a lie algebra on the same […]

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10. […] called and give it a bilinear operation which we write as . We often require such operations to be associative, but this time we impose the following two […]

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11. […] off, we need an algebra over a field . This doesn’t have to be associative, as our algebras commonly are; all we […]

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