For questions on algebras with an associative product.

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4
votes
2answers
283 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
72 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 ...
3
votes
1answer
230 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
174 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 ...
12
votes
3answers
404 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
214 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 ...
4
votes
1answer
171 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
40 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.
2
votes
1answer
91 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
135 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
177 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 ...
9
votes
4answers
403 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 ...
1
vote
0answers
149 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
243 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
217 views

Isomorphic maximal commutative semi-simple sub algebras of M_n(C).

When giving A_1,A_2 two Isomorphic maximal commutative semi-simple sub algebras of M_n(C). Are these algebras conjugate in M_n(C). Namely, is there exists a regular matrix P such that P^{-1}A_1P=A_2. ...
-1
votes
2answers
270 views

Anick resolution [closed]

I would like to know some applications of Anick's resolution in non-commutative algebras.
3
votes
1answer
334 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. ...
3
votes
0answers
99 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
200 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 ...
4
votes
2answers
405 views

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 ...
1
vote
0answers
114 views

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 ...
0
votes
1answer
313 views

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 ...
13
votes
1answer
500 views

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 ...
3
votes
1answer
625 views

$A_{\infty}$ 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 ...
1
vote
1answer
455 views

Is the multiplication beetween even numbers an associative algebra?

We were discussing about the possibility of having an algebra over a field which is associative but has not the unity. Does it exist? It has been proposed as a counterexample the set of even numbers. ...
7
votes
2answers
878 views

What's the sense in which A_\infty algebras are “deformable”?

I realise this is a very vague question! I've heard people say that A∞ algebras are the right homotopy-theoretic generalization of usual associative algebras, because you can deform them. What ...