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Results tagged with ra.rings-and-algebras
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user 160378
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
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An open problem about simple Noetherian rings
The following is a well-known open problem in ring theory (see, for instance, Goodearl, Warfield, 'An introduction to noncommutative Noetherian rings, Appendix, Problem 19)
Question: Let $R$ be a left …
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A naive looking question about Gelfand-Kirillov dimension
Let $A$ and $B$ two affine algebras, $A$ a subalgebra of $B$. If we have a left $A$-module $M$ we can extend the scalars: $B \otimes_A M$. I will denote the resulting $B$-module by $N$
How are $\opera …
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Gelfand-Kirillov dimension and tensor products
$\DeclareMathOperator\GK{GK}$Let $k$ be the base field.
The Gelfand-Kirillov dimension was introduced by Gelfand and Kirillov in their seminal paper on the Gelfand-Kirillov conjecture.
A very famous p …
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Most general filtered algebras with Hilbert polynomials and multiplicities
Let $k$ be any base field and $A$ an affine infinite dimensional $k$-algebra.
Let $\mathcal{F}= \{ A_i \}_{i \geq 0}$ be a finite dimensional filtration for $A$: that is, $k \subset A_0$ and each $A_i …
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Gelfand-Kirillov dimension for non-associative algebras
Let $A$ be any finitely generated algebra - non necessarely unital neither associative - over a base field $k$. Let us denote the product $*$. Suppose $A$ is finitely generated by $S$, and introduce $ …
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non-associative but commutative algebra
The class of Jordan algebras are the most important class of algebras in this direction.
They are defined by the two identities,
(commutativity): $xy=yx$,
(Jordan identity): $(xy)(xx)=x(y(xx))$.
They …
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3
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answer
201
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Relation between enveloping algebras and algebras of differential operators
I asked this question on math stack exchange about 3 years ago, but received no answer.
Our base field $\mathsf{k}$ will be algebraically closed of zero characteristic. Let $X$ be an smooth affine var …
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1
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answer
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Polynomial identities satisfied by the Weyl algebra in prime characteristic
The rank $n$ Weyl $A_n(\mathsf{k})$ algebra over a field $\mathsf{k}$ of zero characteristic does not satisfies any polinomial identity. If it were a PI-algebra, Kaplansky theorem would apply (since t …
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Can the Weyl algebra be free over its invariant subalgebra?
Let $k$ be an algebraically closed field of zero characteristic, let $P_n$ denote the polynomial algebra in $n$ indeterminates, and let $G$ be a finite group of linear automorphisms. Then, by Chevalle …
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Accepted
Wedderburn–Artin like theorem for infinite dimensional Lie algebras?
I had the opportunity to chat with an expert in nonassociative algebras, and I report here what he told me.
The short answer is: as yet, there is no Wedderburn–Artin theory for Lie algebras. The detai …
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64
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Proof of a folkloric result about PI-algebras [duplicate]
I am not not an specialist in PI-algebras, but I can say I have a rather good understanding on the subject.
It is, of course, interesting to discover if an algebra $A$ is a PI-algebra. But it is also …
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3
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1
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Wedderburn–Artin like theorem for infinite dimensional Lie algebras?
The Wedderburn–Artin Theorem is one of the cornerstones of the structure theory of (associative) rings.
Wedderburn–Artin Theorem : Let $R$ be a left Artinian ring with zero Jacobson radical. Then $R$ …
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Non-holonomic modules for $\mathcal{D}(X)$, where $X$ is an affine open subspace of the affi...
Let $k$ be any algebraically closed field of zero characteristic.
Let $A_n$ be the n rank Weyl algebra, and $M$ a finitely generated module. We have that $GK(M)$ is always a positive integer and
(Bers …
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My category is rigid: what this implies for representation theory?
I am studying a subcategory $\mathcal{C}$ of modules for an associative noncommutative algebra $A$ (which is in fact also a Hopf algebra).
It is clear from our definition of $\mathcal{C}$ that it is a …
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3
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217
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What is a Gelfand-Tsetlin subalgebra?
For context on general Gelfand-Tsetlin theory, see for instance this MO post.
Let's work over $\mathbb{C}$. Fix $n>0$. There is a natural chain of embeddings of the general linear Lie algebras $\mathf …
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