# Questions tagged [associative-algebras]

For questions on algebras with an associative product.

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### 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 ...

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### 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 ...

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### 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 ...

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### $c$-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 $kQ$. ...

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### 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....

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### 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?

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### 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 ...

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### Elementary proof about a criterion of a central simple algebra to be a cyclic algebra on the book “Central Simple Algebras and Galois Cohomology”

I've been reading the book "Central Simple Algebras and Galois Cohomology" by Gille and Szamuely.
And I'm stuck with the proof of the Lemma 2.5.4.
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enter image ...

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### 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 $A$-...

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### 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 ...

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72 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 ...

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### 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 ...

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### 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 ...

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### 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 ？

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $x\bullet y=y\bullet x$ ...

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### 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 $\prod$ ...

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### 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 ...

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### 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 ...

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### 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,...

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### 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 ...

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### 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 ...

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### 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 $A$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $Z$.
...

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### 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 ...

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### 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}} ...

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### 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^{\...

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### 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

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### 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.

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $A$-...

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### 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 ...

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### 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 ...

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### 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 $P$...

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### Anick resolution [closed]

I would like to know some applications of Anick's resolution in non-commutative algebras.

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### 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.
...

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### 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 ...

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### 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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### surjectivity of irreducible representation

I don't know how to show the following:
Let $A$ be an associative algebra (not necessary finite-dimensional) and $p\colon A\to End(V)$ be it irreducible finite-dimensional representation. Then $p$ in ...

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### Preservation of direct sums and finite generation

I asked this question on Mathematics - Stack Exchange (MSE). Having figured out out how to handle the problem in an extremely particular case, I also posted it as an answer (in the technical sense of ...

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### Algebras with a degenerate trace form

Let the bilinear trace form of a finite-dimensional associative algebra be defined as:
$(u,v) \mapsto Tr(L_u L_v)$
For $L_u$ the linear map given by multiplication on the left by $u$. In the ...

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### Does $\mathbb{K}[G]\simeq\mathbb{K}[H]$ for some field $\mathbb{K}$ of characteristic $p$, imply $\mathbb{F}_p[G]\simeq\mathbb{F}_p[H]$?

Due to the first (and very helpful) answer I received, I've reformulated the question a little: $G$ and $H$ are now assumed to be $p$-groups.
Let $p$ be a prime, and let $\mathbb{F}_p$ be the field ...

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### $A_{\infty}$ structure of (co)homology of a space

Let $X$ be a topological space, and $Homeo(X)$ the group of self-homeomorphisms of $X$.
(1) What is the exact meaning of: $H^*(X)$ is a an $A_\infty$-module over $Homeo(X)$?
(2) Does $H_*(X)$ also ...