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For questions on algebras with an associative product.

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143 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 ...
7
votes
3answers
526 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 ...
3
votes
1answer
243 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 ...
1
vote
0answers
15 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 ...
1
vote
0answers
54 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 $KG$, ...
2
votes
1answer
109 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 ...
1
vote
0answers
137 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 ...
1
vote
0answers
33 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 ...
3
votes
0answers
44 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 $kQ$. ...
3
votes
1answer
130 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....
1
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0answers
40 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?
4
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0answers
178 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 ...
4
votes
1answer
83 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 $A$-...
4
votes
1answer
261 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 ...
1
vote
1answer
74 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 ...
6
votes
0answers
72 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 ...
3
votes
1answer
141 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 ...
2
votes
1answer
128 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 ？
2
votes
0answers
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 ...
5
votes
0answers
184 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 ...
5
votes
0answers
225 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 ...
2
votes
0answers
30 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 $x\bullet y=y\bullet x$ ...
2
votes
0answers
51 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 $\prod$ ...
1
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0answers
66 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 ...
3
votes
2answers
237 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 ...
1
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0answers
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,...
6
votes
2answers
311 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 ...
0
votes
0answers
205 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 $n$ ...
1
vote
0answers
81 views

degree of associative algebra

Let $A$ be a finite dimensional associative algebra with unity over a field $F$. The degree of the algebra is the degree of its generic minimum polynomial (see Nathan Jacobson, Generic norm of an ...
5
votes
2answers
359 views

Conjugation in associative algebras over finite fields

Let $A$ be a finite dimensional associative algebra (with unity) over a finite field $F$. Let $L$ be a field extension of $F$. Suppose that after extending scalars to $L$, two elements $a,b$ of $A$ ...
6
votes
1answer
163 views

presentations of subalgebras

Assume that I have a finitely presented algebra $A$ over the complex numbers (by which I mean that $A$ is generated over $\mathbb{C}$ by finitely many elements $x_1,...,x_n$ subject to finitely many ...
4
votes
2answers
348 views

Relation between Associative algebra and group algebra

Let $A$ be an associative algebra over a filed $k$. Q) What are the condition we can impose on $A$ such that there exists a $G$ such that $A=k[G]$, the group algebra generated by $G$? I am ...
0
votes
0answers
259 views

Differences between primitive central idempotents and primitive orthogonal idempotents

If we have a complete set of primitive orthogonal idempotents of an algebra $A$, then we can obtain simple modules, indecomposable projective modules, indecomposable injective modules of $A$. If we ...
4
votes
1answer
243 views

Non-commutative normalization

Let $A$ be a (non-commutative) associative algebra with 1. Assume that $A$ contains a cental subalgebra $Z$ such that a) $Z$ is a noetherian domain b) $A$ is a finitely generated module over $Z$. ...
10
votes
1answer
189 views

Free subgroups in algebras of polynomial growth

What is known about free non-abelian subgroups in finitely generated associative algebras of polynomial growth (e.g., over finite fields, to avoid finite-dimensional free subgroups)? For example, are ...
14
votes
3answers
645 views

Is the Amitsur-Levitzki identity essentially unique?

Let us consider the matrix algebra. $Mat_n(\mathbb{C})$. The Amitsur-Levitzki identity states that for any matrices $X_1, X_2, ..., X_{2n} \in Mat_n(\mathbb{C})$ the sum $\Sigma_{\sigma \in S_{2n}} ...
3
votes
4answers
338 views

Nilradical of a Lie algebra associated to a associative algebra

Let $A$ be a finite dimensional (unital) $K$-Algebra. By $A^{\circ}$ we denote the associated $K$-Lie-algebra of $A$ with respect to the product $a\circ b:=ab-ba$. In addition, we denote by $rad(A^{\...
4
votes
1answer
384 views

Isomorphism of matrix ring over ore domain

Let $R_1,R_2$ be (left and right) ore domains. Does $ Mat_n(R_1)\cong Mat_m(R_2)$ implie m=n and $q.f.(R_1)\cong q.f.(R_2)$? An counter example, a proof or a reference is welcomed. Thanks
1
vote
0answers
61 views

extend derivations of ore domain to its quotient field

I wonder whether someone knows a good reference(textbook or paper) for the following result: Any derivation of ore domain may be extended unqiuely to a derivation of its quotient field. Thanks.
3
votes
1answer
131 views

Equivalence of star products on two differents Poisson algebras?

Let $A$, $B$ be two commutative and associative $\mathbb k$-algebras and let $A_\hbar:=A[[\hbar]]$, $B_\hbar:=B[[\hbar]]$ be the corresponding ring of formal series. Of sense [Deformation theory and ...
1
vote
2answers
513 views

Software for noncommutative Groebner bases over rational function fields

I am wondering whether there is any software package that can compute Groebner bases for noncommutative algebras defined over the field of rational functions $\mathbb{Q}(q)$. I have tried the GAP ...
3
votes
0answers
267 views

Hochschild homology of a tensor algebra modulo a two-sided ideal

Let $V$ is a module over a field $k$, and $A=T(V)$ the tensor algebra generated by $V$. The Hochschild homology $HH_*(A)$ has been determined by Loday and Quillen in their paper "Cyclic homology and ...
15
votes
4answers
967 views

If tensor product of representations is a representation, must we have a bialgebra?

Hopf algebras and bialgebras are sometimes introduced by saying that you've got an associative algebra $A$ and want to introduce the structure of an $A$-module on $V \otimes W$ where $V,W$ are $A$-...
3
votes
0answers
245 views

simple tensor product of modules over algebras

Let $M$, $N$ be simple modules over associative algebras $A$ and $B$ (over $\mathbb{C}$), respectively. When is $M\otimes N$ simple as a $A\otimes B$-module? It is right if $A$ or $B$ has a ...
-4
votes
2answers
303 views

Representing quaternions as matrices [closed]

Assume F is a field of characteristic different than 2. Let a, b be invertible elements in F, and let A(a,b) be the generalised quaternions. Using the Artin–Wedderburn theorem, there is a ...
0
votes
0answers
268 views

Isomorphic maximal commutative semi-simple sub algebras of complex matrices

When giving $A_1,A_2$ two isomorphic maximal commutative semi-simple sub algebras of $M_n(\mathbb{C})$, are these algebras conjugate in $M_n(\mathbb{C})$? Namely, does there exists a regular matrix $P$...
-2
votes
2answers
442 views

Anick resolution [closed]

I would like to know some applications of Anick's resolution in non-commutative algebras.
3
votes
1answer
722 views

Reference for Clifford theory of algebras

Clifford theory relates the representation theory of a group to that of a normal subgroup. A good reference for this is Curtis and Reiner's "Methods in Representation theory II", Theorem 11.1. ...
7
votes
1answer
227 views

Description of modules over self-injective algebras of finite representation type

Is there any description of indecomposable modules and irreducible morphisms over self-injective algebras of finite representation type? I am interested mainly in such a description for nonstandard ...
1
vote
0answers
205 views

Algebra out of a set of modules of a Lie algebra? Fusion

The problem I faced is how to organize a set of finite-dimensional irreducible representations $U_\alpha$ of some simple Lie algebra $g$ into an Lie algebra $A$ that contains $g$ as a Lie subalgebra ...

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