For questions on algebras with an associative product.

**10**

votes

**1**answer

158 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 ...

**11**

votes

**2**answers

299 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

**4**answers

201 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

**1**answer

159 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

**0**answers

37 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

**1**answer

87 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 ...

**0**

votes

**1**answer

79 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

**0**answers

158 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 ...

**8**

votes

**4**answers

383 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

**0**answers

140 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 ...

**0**

votes

**0**answers

213 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**

vote

**0**answers

198 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

**2**answers

373 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

**0**answers

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

**1**answer

294 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 ...

**12**

votes

**1**answer

493 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

**1**answer

607 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

**1**answer

453 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

**2**answers

864 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 ...