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Questions tagged [associative-algebras]

For questions on algebras with an associative product.

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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 ...
Projective injective's user avatar
1 vote
0 answers
80 views

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 $n$-...
GiS's user avatar
  • 321
1 vote
0 answers
101 views

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 ...
grok's user avatar
  • 2,339
1 vote
1 answer
79 views

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 $k$-...
Boris's user avatar
  • 331
1 vote
0 answers
65 views

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 ...
Lagrenge's user avatar
  • 271
5 votes
0 answers
145 views

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 ...
Pedro's user avatar
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2 votes
0 answers
84 views

Is there a standard reference for: taking projective covers of simple modules commutes with finite Galois field extensions?

Let $\Lambda$ be an Artin algebra over a finite field $k$. Is there a standard reference for the fact that taking projective covers of simple $\Lambda$-modules commutes with finite Galois field ...
Stein Chen's user avatar
1 vote
1 answer
104 views

Non-rigid modules and Auslander-Reiten quiver

I have a question about proving that a module is non-rigid using Auslander-Reiten quiver. Suppose that we have some algebra $A$ and the components of its Auslander-Reiten quivers are tubes like the ...
Jianrong Li's user avatar
  • 5,961
1 vote
0 answers
66 views

A sufficient condition for automorphism of an exact sequence

I asked A sufficient condition for Automorphism of an exact sequence earlier on Math.StackExchange but did not get any response so am posting it here. I am given the following commutative diagram with ...
Subham Jaiswal's user avatar
2 votes
1 answer
113 views

A problem about extensions of division rings

For a division ring $D$ with center field $F:=Z(D)$ such that $\dim_F D = n^2$, there is a classical result saying that $D\otimes_{F}\bar{F}\cong M_n(\bar{F})$ as $\bar{F}$-algebras, where $\bar{F}$ ...
GiS's user avatar
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8 votes
1 answer
396 views

Trying to understand "a refinement of the Peter–Weyl theorem" by Lusztig

"A refinement of the Peter–Weyl theorem" is the title of Chapter 29 in Lusztig's "Introduction to quantum groups" (Birkhäuser 2010, reprint of the 1994 edition). This chapter is ...
მამუკა ჯიბლაძე's user avatar
4 votes
1 answer
388 views

Zhu's algebra for the Virasoro VOA

I am trying to understand the proof in the appendix of the following paper: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.110.8757&rep=rep1&type=pdf The paper discusses Zhu's ...
clvolkov's user avatar
  • 193
3 votes
1 answer
234 views

Number of rings with additive group $(\mathbb{Z}_{16})^2$. A341547(16) in OEIS

I would like to know if somewhere the number of non-isomorphic rings with additive group $(\mathbb{Z}_{16})^2$ is mentioned. If not, is someone able to calculate it? And (easier) the commutative case? ...
José María Grau Ribas's user avatar
1 vote
0 answers
83 views

Are there non-semisimple complex "non-unital special Frobenius algebras"?

I'm interested in "non-unital special Frobenius algebras", consisting of two linear maps (morphisms in the symmetric monoidal category of finite-dimensional complex vector spaces) $$\mu: V\...
Andi Bauer's user avatar
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1 vote
0 answers
59 views

A list of low-dimensional associative $\mathbb{C}$-algebras with non-trivial centers

I am looking for a list of (examples of) low(est)-dimensional non-commutative associative unital $\mathbb{C}$-algebras $A$ with non-trivial centers $Z(A)$ (hence non-semisimple). For our purposes $Z(A)...
M.G.'s user avatar
  • 6,469
4 votes
0 answers
179 views

divided powers of a deformation class

Let $A$ be a (unital, associative) $k$-algebra where $k$ is a field. Given a flat deformation of $A$ one gets the deformation class $h$ in the second Hochschild cohomology $HH^2(A)$. Suppose $k$ has ...
Roman's user avatar
  • 1,331
3 votes
0 answers
50 views

Continuous differentiations of functional algebras

Let $A$ be some algebra (infinite-dimensional) of analytic functions on $\mathbb{C}^n$, and $D$ be some derivation of $A$, i.e. $D(fg)=Df \cdot g + f \cdot Dg)$ (so A may be considered as a ...
Vladimir47 's user avatar
5 votes
1 answer
257 views

Is there a short proof for the permutation invariance of this combinatorial map?

Consider a positive integer $n$ and integers $(c_i)_{1\le i \le 4}$, with $1 \le c_i \le n$. Conside the map: $$f_n: (c_1,c_2,c_3,c_4) \mapsto \delta_{c_1,c_2}\delta_{c_3,c_4} - \# \{ |2n+1-2|x||, \ x ...
Sebastien Palcoux's user avatar
2 votes
0 answers
48 views

Free resolutions of universal enveloping algebras for simple, finite dimensional Lie algebras

I'm currently studying Anick's resolution on the context of universal enveloping algebras for certain Lie algebras, namely some of the smallest cases: $A_1,A_2,A_3,B_2,G_2$, and so on. What are some ...
Hilario Fernandes's user avatar
4 votes
1 answer
91 views

Literature on the polynomials and equations, in structures with zero-divisors

I need literature about zeroes of polynomials and equation resolution in associative algebraic structures with zero-divisors, but I am having difficulties to find it. For example, there is literature ...
Dragon lala lalo's user avatar
3 votes
1 answer
578 views

The inner product of a Clifford Algebra

Any Clifford algebra $\operatorname{Cl}(k, p)$ carries an induced inner product, which is the "trace" on its 0-blade: $\langle AB\rangle_0$ for given elements $A, B$ of the algebra. This inner ...
Mirco A. Mannucci's user avatar
1 vote
1 answer
113 views

References about transfinite socle series

I would like to know if there is any literature that discusses "transfinite socle series" of a ring module. Below is my attempt at defining the series. Let $R$ be an associative unital ring and $...
Batominovski's user avatar
1 vote
2 answers
330 views

Classification of finite-dimensional (nilpotent) associative algebras

What is known about the classification of finite-dimensional (nilpotent) associative algebras? I am assuming that algebras are over a field of characteristic zero. If it is simple, then it has to be ...
Eugene Starling's user avatar
7 votes
3 answers
708 views

What's an illustrative example of a tame algebra?

A finite-dimensional associative $\mathbf{k}$-algebra $\mathbf{k}Q/I$ is of tame representation type if for each dimension vector $d\geq 0$, with the exception of maybe finitely many dimension vectors ...
Mike Pierce's user avatar
  • 1,129
5 votes
1 answer
316 views

"Non-associative" standard polynomials

I saw somewhere (I appreciate if anyone has any references to proof of this fact) that if $A$ is a finite dimensional associative algebra such that $\textrm{dim}(A)<n$, then $A$ satisfies the ...
cl4y70n____'s user avatar
1 vote
0 answers
18 views

Exponents in unit groups of modular group algebras

Let $p$ be a prime number, $G=:G_1$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_2:=1+\operatorname{rad}(KG)$ is a p-group ...
Sven Wirsing's user avatar
1 vote
0 answers
63 views

Number of conjugacy classes of unit groups of modular group algebras

Let $n$ be a natural number, $p$ a prime number, $G$ a finite $p$-group and $K$ a finite field with $p^n$ elements. We focus on the group $1+J(KG)$, where $J(KG)$ is the Jacobson radical of $KG$, ...
Sven Wirsing's user avatar
2 votes
1 answer
135 views

A weak Schur's lemma for non-semisimple finite dimensional algebras

Let $B \subseteq C$ be an inclusion of finite dimensional (associative) algebras over a field $k$. Assume that $C$ is a free $B$-module. Let $\bigoplus_i U_i$ be a decomposition of $B$ into ...
Sebastien Palcoux's user avatar
1 vote
0 answers
276 views

Infinite-dimensional representation theory of $K[x]$

Let $K$ be an algebraically closed field. The finite-dimensional representation theory of the polynomial algebra $K[x]$ is tame and completely understood, which I shall first summarise. It's ...
Iteraf's user avatar
  • 430
1 vote
0 answers
40 views

Rigid $Hom$-orthogonal modules in wild hereditary algebras

Let $Q$ be a simply-laced wild quiver with at least one multiple edge, $k$ be an algebraically closed field, $1$ be the source of and $2$ be the sink of one set of such edges. Can we find rigid ...
Ying Zhou's user avatar
  • 417
3 votes
0 answers
48 views

$c$-matrix reduction in hereditary algebras

Let $k$ be an algebraically closed field, $Q$ be a finite connected quiver and $Q'$ a subquiver of $Q$. Let $C$ be the $c$-matrix of a chamber of the scattering diagram/semi-invariant picture of $kQ$. ...
Ying Zhou's user avatar
  • 417
3 votes
1 answer
181 views

Operation of a p'-group on a set of p-power order and fix points

The question is related to Taft's Theorem about G-invariant radical complements. Let $A$ be an associative unitary finite-dimensional $K$-Algebra posessing a separable factor Algebra by ist nilradical....
Sven Wirsing's user avatar
2 votes
0 answers
58 views

Maximal separable subalgebras of semisimple algebras

Is anything known about maximal separable subalgebras of semisimple algebras in finite-dimension? Are those subalgebras of unique dimension or isomorphic or conjugate?
Sven Wirsing's user avatar
5 votes
0 answers
247 views

Higher Braces algebra and operads

1) In [HIGHER OPERATIONS ON HOCHSCHILD COMPLEX], Gerstenhaber and Voronov showed that the Hochschild complex $C_1(\mathcal A)$ of any associative algebra (or e_1 algebra) $\mathcal A$ is naturally ...
thingsthatmighthavebeen's user avatar
5 votes
1 answer
148 views

Given a representation-infinite algebra, when is every AR component infinite?

Let $A$ be a finite dimensional algebra over an algebraically closed field $K$. The Auslander-Reiten quiver $\Gamma_A$ of $A$ is a means of presenting the category of finitely generated right $A$-...
Iteraf's user avatar
  • 430
4 votes
1 answer
418 views

Elementary proof that a central simple algebra over a field having a maximal subfield is a cyclic algebra

I'm currently reading the book "Central Simple Algebras and Galois Cohomology" written by Philippe Gille and Tamas Szamuely. In the book, I don't understand a computational proof of the theorem that ...
gualterio's user avatar
  • 1,033
1 vote
1 answer
77 views

Jordan-Hölder series of $k$-subalgebras?

I am reading Shurygin's survey "Smooth Manifolds over Local Algebras and Weil Bundles" (Journal of Math. Sciences, Vol. 108, No. 2, 2002) and it mentions the following basic fact which I don't quite ...
M.G.'s user avatar
  • 6,469
6 votes
0 answers
78 views

Non-rigid indecomposable summands of simple-minded collections in bounded derived category of hereditary algebras

Let $\Lambda$ be a hereditary algebra over an algebraically closed field $k$. Let $S$ be one of the indecomposable summands of one simple-minded collection in $D^b(\Lambda)$. Is it true that $S$ is ...
Ying Zhou's user avatar
  • 417
3 votes
1 answer
230 views

A differential graded Lie algebra with the Hochschild differential

Let $(V,\cdot)$ be an associative algebra and $W$ be a vector space endowed with a bimodule structure $\triangleright:V\otimes W\to W$ and $\triangleleft:W\otimes V\to W$ such that the following ...
thingsthatmighthavebeen's user avatar
2 votes
1 answer
148 views

Injection of the Universal enveloping algebra

Let L1 and L2 be two Lie algebras.If U(L1)is isomorphic to U(L2)as associative algebra,then L1 is isomorphic to L2 ?
double-function's user avatar
2 votes
0 answers
36 views

Morphisms from quasi-simple regular rigid $\Lambda$-module $M$ to $\tau^m M$ when $\Lambda$ is wild hereditary

Let $\Lambda$ be a finite dimensional basic wild hereditary algebra and $M$ be an indecomposable regular quasi-simple right-$\Lambda$ that is rigid in a regular component $\mathcal{R_i}$ of the ...
Ying Zhou's user avatar
  • 417
5 votes
0 answers
206 views

Which groups can occur as the group of units of finite-dimensional noncommutative algebras?

This is a continuation of a previous question: Connectedness of groups of units in finite-dimensional commutative algebras. Let $k$ be an algebraically closed field of characteristic $0$. Which ...
M.G.'s user avatar
  • 6,469
6 votes
0 answers
281 views

Representation-finiteness vs. $\tau$-tilting-finiteness

Setting: Throughout, $\Lambda$ is a finite dimensional associative algebra, $\operatorname{mod} \Lambda$ is the category of all finitely generated left $\Lambda$-modules, and all subcategories are ...
Kaveh's user avatar
  • 473
2 votes
0 answers
33 views

Is -1 a sum of Hermitian squares in associative *-envelopes of formally real Jordan algebras?

Let $J$ be an unital Jordan algebra (over $\mathbb{R}$) - recall that this means that $J$ is an unital $\mathbb{R}$-algebra (whose product we denote by $\bullet$) satisfying $x\bullet y=y\bullet x$ ...
Pedro Lauridsen Ribeiro's user avatar
2 votes
0 answers
62 views

Question about the mutation of a cluster seed associated to any word of the braid semigroup

Let $G$ be a semisimple Lie group with the set of positive simple roots $\prod$. Let $\prod^{-}$ be the set of negative simple roots and let $\mathfrak{M}$ be the semigroup freely generated by $\prod$ ...
Daisy's user avatar
  • 338
1 vote
0 answers
72 views

Projective modules over a Hochschild extension of algebras

Let $k$ be a commutative ring and let $R$ be an associative (not necessarily commutative) $k$-algebra with unit. Also, let $M$ be an $R$-bimodule. It is well known that on $S:=R+M$ we can define an ...
Adrian's user avatar
  • 11
3 votes
2 answers
333 views

Unique dimension of Cartan subalgebras in modular Lie algebras

If $L$ is a Lie algebra over an algebraic closed field $K$ of characteristic zero, then all Cartan subalgebras are conjugated. Hence, they have all the same dimension. If $K$ is not algebraic closed ...
Sven Wirsing's user avatar
1 vote
0 answers
81 views

If $A$ is an integer ring such that each $P \in A_L[X]$ has a finite number of zeros in $A$, is $A$ commutative?

Let $A$ be a ring in which the product of any two nonzero elements is nonzero (we shall say that $A$ is an integral domain, even if $A$ is non commutative). It is well-known that if $A$ is commutative,...
Stabilo's user avatar
  • 1,345
6 votes
2 answers
355 views

Connectedness of units in finite-dimensional commutative complex algebras

In the following, an algebra will always mean a finite-dimensional associative commutative unital algebra (over some field $k$). Let $A$ be a $\mathbb{C}$-algebra. I am trying to understand how its ...
M.G.'s user avatar
  • 6,469
0 votes
0 answers
210 views

Can we drop commutativity assumption?

Let $A$ be an associative algebra with a unit over a field $k$. fix $n > 1$. Define a $k$-algebra structure on the vector space $A^{\otimes n} = A \otimes_k \dots \otimes_k A$ (where there are $n$ ...
Ed Fischer's user avatar