Questions tagged [associative-algebras]
For questions on algebras with an associative product.
88
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Stable module category of non-Frobenius algebras
It is often said that the stable module category $A-\underline{\operatorname{mod}}$ for an associative algebra $A$ is triangulated if $A$ is Frobenius (i.e. over $A$ we have projective = injective). ...
- 1,350
6
votes
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Explicit proof that $\mathbb{k}[x]/(x^n)$ is not derived discrete
In the question Explicit proof that algebra is derived wild it was asked whether there are examples of algebras $A$ where it is possible to show explicitly that $A$ is derived wild by finding an ...
- 1,350
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70
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Multiplicative bases, path algebras, and Ext algebras
I am interested in understanding when a multiplicative basis exists for finite dimensional algebras over an algebraically closed field, and, in particular, Ext-algebras that are finite dimensional.
It ...
- 71
0
votes
1
answer
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Orthogonality in Hilbert algebras and congruence
Consider a finite-dimensional Hilbert space $V$ (say, over $\mathbb{C}$) and a finite-dimensional Hilbert algebra $A$ (i.e., Hilbert space with a compatible associative unitary algebra structure). ...
- 327
5
votes
2
answers
208
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Enveloping algebra of affine Lie algebra is (not) noetherian
I work over an algebraically closed field of characteristic $0$. Let $\mathfrak{g}$ be a semisimple Lie algebra, $\hat{\mathfrak{g}}=\mathfrak{g}[t,t^{-1}]\oplus\mathbb{C}K\oplus\mathbb{C}D$ the ...
- 1,341
7
votes
2
answers
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Tensor product of irreducible representations of an algebra
Let $A$ be an associative algebra over $\mathbb{C}$ with irreducible finite-dimensional representations on $V$ and $W$. Then is the tensor product of representations on $V \otimes W$ semi-simple?
The ...
- 265
4
votes
1
answer
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Are polynomial algebras over fields (that are not algebraically closed) tame?
Let $A$ be an algebra over a field $K$. Loosely speaking, an algebra is said to be tame if for each $d \in \mathbb{Z}_{>0}$ all but finitely-many of the indecomposable $A$-modules of $K$-dimension $...
- 482
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List of automorphism groups of low-dimensional complex commutative algebras?
Let $\mathcal{A}$ be a finite-dimensional commutative associative unital $\mathbb{C}$-algebra. I am looking for a list (of further examples of) $\operatorname{Aut}_\mathbb{C}(\mathcal{A})$, the group ...
- 6,996
5
votes
1
answer
225
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Infinite-dimensional, non-unital Frobenius algebras
A Frobenius algebra is a tuple $(A,\mu,\delta,\eta,\varepsilon)$, where $A$ is a vector space over some field, $(A,\mu,\eta)$ a unital associative algebra, and $(A,\delta,\varepsilon)$ a counital ...
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4
votes
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Faithfully injective projective modules
An $R$-module I is called faithfully injective if it is injective and the functor $Hom_R(-, I)$ has the image of a complex being exact if and only if the original complex is exact.
I wonder if it is ...
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0
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Does the center of any finitely generated associative algebra over a field have finite type?
Consider the monoid algebra $R:=K\langle x_1,\dots,x_n\rangle$ generated by $n$ letters $x_1,\dots,x_n$ for $n>1$ over field $K$. Equivalently, $R$ is the tensor algebra $T(V)$ on the $n$-...
- 321
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Points and algebraic geometry on the quantum plane
The "quantum plane" is the "space" of the algebra $A=\Bbbk\langle X,Y\rangle/(YX-qXY)$, for a scalar $q$ (e.g. $\Bbbk=\mathbb C(q)$). I would like to know how much algebraic ...
- 2,489
1
vote
1
answer
114
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How to compute the associated reduced ring for this finitely generated algebra?
Let $k$ be a field, $m$ be a positive integer and $R$ be the subring $k[x,xy,xy^2,…,xy^m]$ of the polynomial ring $k[x,y]$. Let $B$ be the quotient ring $R/xR$. Then $B$ is the finitely generated $k$-...
- 569
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Structure and representation of a non-homogeneous quadratic algebra
Let $m$ be a positive integer and $s$ be a fixed $m\times m$ matrix. Let $U$ be the unital associative algebra (over $\mathbb{C}$) generated by $\{u^i_{j}\}_{1\leq i,j\leq m}$ quotient over the ...
- 575
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votes
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Bar constructions and pushouts
Suppose that $\mathsf S$ is a span of associative algebras (or, more generally, if you'd like, any type of object admitting a bar-cobar formalism) and let $A$ be its pushout.
Is there any hope of ...
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