# Questions tagged [noncommutative-algebra]

Non-commutative rings and algebras, non-associative algebras. Can be used in combination with ra.rings-and-algebras

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### Number of right divisors of a central skew polynomial

Let $\mathbb{F}$ be a finite field of $p$ elements, $\sigma \in \operatorname{Aut}(F)$ of order $m$, $\mathbb{F}^\sigma$ be the fixed field of $\sigma$, and $\mathbb{F}[x,\sigma]$ be a skew polynomial ...

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### A question about surjective maps between quadratic algebras

Let $V$ be a finite-dimensional vector space and
$$
U \subseteq W \subseteq V \otimes V
$$
be a proper inclusion of vector subspaces. Then take the tensor algebra
$$
T(V) = \bigoplus_{i=1}^{\infty} V^{...

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### A note on orders in quaternion algebras

Definition. An algebra $B$ over a field $F$ is a quaternion algebra if there exists $i,j\in B$ such that $1,i,j,ij$ is a basis for $B$ as a vector space over $F$, where $i^2=a,j^2=b;a,b\in F^\times$.
...

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### What is the order of these binary trees?

In the book "Free Lie algebras" by the author Christophe Reutenauer, Example 4.2 (in subsection 4.1) gives the trees of degree $\le 5$ of a Hall set in magma $M(A)$, where $A=\{a,b\}$ as the ...

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### Polynomial identities satisfied by the Weyl algebra in prime characteristic

The rank $n$ Weyl $A_n(\mathsf{k})$ algebra over a field $\mathsf{k}$ of zero characteristic does not satisfies any polinomial identity. If it were a PI-algebra, Kaplansky theorem would apply (since ...

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### Convolution algebra of a simplicial set

Consider a simplicial set $X^\bullet$ with face maps $d_i$ (assume the set is finite in each degree so there are no measure issues). Then given two functions $f,g:X^1\to \mathbb{C}$ one can form their ...

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### Structure of tame concealed algebra of Euclidean type

I wanted to know some references where people have studied the representation theory of tame concealed algebra of Euclidean type. What do we know about the structure of their module category? What ...

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### 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 $n$-...

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### Extension of work by Novelli and Thibon on noncommutative symmetric functions and Lagrange inversion

(Edit May 12, 2023: I just put up a brief summary of some of my notes on the partition polynomials described below in my WordPress mini-arXiv at "As Above, So Below". It contains multinomial ...

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### Smallest faithful representation of an upper-triangular matrix quotient

This is a curiosity question that came out of teaching abstract algebra.
Let $F$ be a field, and $n>1$ an integer.
Let $F^{n \leq n}$ be the $F$-algebra of all upper-triangular $n\times n$-matrices ...

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### Finitistic dimension conjecture — why artin algebras?

As I understand it, the finitistic dimension conjecture says that if a ring $A$ is finitely generated over a commutative artinian ring $K$, then the finitistic dimension of $A$ is finite.
My question ...

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### Condition for equality of modules generated by columns of matrices

Let $R$ be a commutative ring with unit. Let $M_A$ denote the submodule of $R^m$ generated by columns of a matrix $A$ with entries in $R$. Suppose we are given two matrices $A,B \in R^{m \times k}$. I ...

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### Projective dimension of group ring

Assume that $G$ is a group and $R$ is a p.i.d. What can we say about the projective dimension of $R[G]$? For example can we say that this dimension is at most $1$ for reductive groups? (I think if $...

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### Relation(s) between units and nilpotent elements in graded noncommutative rings

In Commutative Algebra we have the following standard facts which I am going to state in a slightly different form than usually found in textbooks. Namely, let $A$ be a commutative unital ring of ...

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### Cubical vs. simplicial Hochschild cohomology

Simplicial Hochschild cohomology.
$\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\B}{\mathrm{B}}\newcommand{\Obj}{\mathrm{Obj}}\newcommand{\HH}{\mathrm{HH}}\newcommand{\Mod}{\mathsf{Mod}}$One way to ...

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### Are perfect complexes the same as compact objects in D(R) for noncommutative rings?

The Stacks Project proves Thomason's insight that
compact objects of the derived category $\simeq$ bounded complexes of finitely generated projective modules
in Section 15.78, but the running ...

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### Existence of finite dimensional representation of an algebra

Let $m>1$ be an integer and let $A$ be the algebra generated by the elements
$\{u^i_j,v^i_j,\bar{u}^i_j, \bar{v}^i_j| 1\leq i,j\leq m\}$ quotient over the relations
\begin{eqnarray}
u^i_j v^k_l&...

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### Regular coherence of tensor algebras

Under what conditions on a bimodule $M$ over a noetherian commutative ring $R$ is the tensor algebra $T=T_R(M)$ regular coherent? A theorem of Gersten says this is true for a free bimodule $M$. If $M$ ...

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### Is there a $3$-commutative algebra?

Let $k$ be a field of characteristic $0$. If $m\ge2$, I denote $P_m$ the standard polynomial in $m$ non-commutating indeterminates:
$$P_m(X_1,\dotsc,X_m)=\sum_{\sigma\in\mathfrak S_m}\epsilon(\sigma)...

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### Abelianization of the group of invertible elements in a finite local ring

Let $R$ be a finite local $\mathbb{F}_q$-algebra. Assume that $R\cong R^*$ as left $R$-modules. Are there any known results about the abelianization $(R^{\times})_{\mathrm{ab}}$?
(We can factor $R$ be ...

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### Injective modules

Let $A$ be a finite dimensional $k$-algebra and let $I$ be an injective module。My question is whether $I$ is a direct sum of finite-dimensional injective modules。

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### Graded global dimension of a graded algebra

Let $k$ be an algebraically closed field of characteristic $0$.
Let $A := k \langle x,x^{-1},y \rangle /(xy-qyx, x^{d_1}-ay^{d_2})$, where deg$(x)>0$, deg$(y)>0$, $q,a \in k^*$ and $d_1\text{deg}...

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### Symmetry of unique generator property

In this article:
Canfell, M. J. "Completion of Diagrams by Automorphisms and Bass′ First Stable Range Condition." Journal of algebra 176.2 (1995): 480-503.
the author defines a ring $R$ to ...

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### An action on multiplicatively antisymmetric matrix

A matrix $ Q=(q_{ij})$ is called multiplicatively antisymmetric over a field $ F $ if $ q_{ii}=1 $ and $ q_{ij}={q_{ji}}^{-1} $.Let $ \mathcal{Q} $ be the set of all $ n \times n $ multiplicatively ...

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### Simple modules of quantum planes

Let $k$ be an algebraically closed field.
Let $R := k\langle x,y \rangle/(yx-qxy) (q \in k^*)$.
We often call $R$ a quantum plane.
If $q$ is a primitive $n$-th root, then for any $(\zeta, \xi) \in k^* ...

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### The generators of twisted homogeneous coordinate rings

Let $X$ be a projective scheme over an algebraically closed field $k$ of characteristic $0$.
Let $\sigma$ is an automorphism of $X$ and $\mathcal{L}$ be an invertible sheaf on $X$.
Let $B := B(X, \...

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### Does the choice of the algebraically closed field of characteristic $p$ have influence on the module category?

Let $G$ be a finite group and $p$ be a prime number dividing $|G|$.
Let $k$ be the algebraic closure of $\mathbb{F}_p$.
Let $K$ be another algebraically closed field of characteristic $p$ which is not ...

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### Noncommutative group schemes corresponding to quantum groups

I'm not an expert on quantum groups by any stretch, so forgive me if this question seems overly naive. That said, I was wondering if there is a way (or if there has been any attempt in the literature) ...

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### About a recent paper of Rickard on finitistic dimension

Apologizes if this is a basic question, but I am new to the area of finite dimensional algebras. I am reading the paper "Unbounded derived categories and the finitistic dimension conjecture" ...

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### What is the relationship between ramification in central simple algebras and in fields?

Suppose $K$ is the field of fractions of a Dedekind domain $R$, and let $L$ be a finite extension of $K$. There is a notion of ramification of primes of $K$ in $L$, which describes how $\mathfrak p \...

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### On continuous seminorms on Fréchet-Stein algebras

Let $K$ be a discretely valued complete non-archimedean field and $U$ be a left Fréchet-Stein algebra as defined in Algebras of p-adic distributions and admissible representations, with a Fréchet-...

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### Reference request: Left $R/k$-modules [closed]

In the paper titled:
On the module of differentials of a noncommutative algebra and symmetric biderivations of a semiprime algebra
I found the following definition:
Let $k$ be a commutative ring with ...

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### On the definition of the Cherednik algebra of a variety with a finite group action

Let $X$ be a connected complex smooth affine variety, acted on by a finite group $G$. We define a reflection hypersurface $(Y,g)$ as a smooth codimension one subvariety $Y\subset X$ which is fixed by $...

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### Quasi-coherent cohomology in non-commutative algebraic geometry

In non-commutative algebraic geometry, the motto so to speak is to replace the study of a scheme $X$ with the study of the category $D_{qcoh}(X)$ of quasi-coherent sheaves and study the properties ...

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### Proving the spectrum of the Young-Jucys-Murphy elements by formal computation in the degenerate affine Hecke algebra

This is really a followup to Why are Jucys-Murphy elements' eigenvalues whole numbers? , specifically to Igor Makhlin's beautiful answer. I'm trying to make it even more beautiful by getting rid ...

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### When are simple holonomic D-modules of the form $\mathcal{D}/\mathcal{D}L$?

Let $\mathcal{D}=\mathcal{D}_X$ be the sheaf of rings of differential operators on a smooth algebraic curve $X$.
Since $\dim X=1$, the D-modules of the form $\mathcal{D}/\mathcal{D}L$ are necessarily ...

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### The combinatorics of $(f \partial)^n$ in the noncommutative setting?

This is a noncommutative version of these three previous questions:
differential operator power coefficients
Сlosed formula for $(g\partial)^n$
A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
...

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### How to go from primitive idempotents in $\text{End}_A(M)$ to primitive idempotents in $A$?

Let $K$ be a large enough finite field, let $A$ be a finite-dimensional $K$-algebra.
Moreover, let $M$ be a finitely generated $A$-module and let $M = M_1\oplus ... \oplus M_n$ be a decomposition of $...

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### Nullstellensatz for maximal left ideals of quantum plane

Let $R=\mathbb{C}\langle x,y\rangle/\langle xy=qyx\rangle$ be the quantum plane algebra. Does some sort of Nullstellensatz holds for the maximal left ideals of $R$? By this we mean all maximal left ...

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### Moduli spaces of stable sheaves on noncommutative projective schemes

In noncommutative algebraic geometry in the sense of Artin and Zhang, can we construct moduli spaces of stable sheaves on noncommutative projective schemes as (commutative)schemes ?
I would appreciate ...

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### Finding non-inner derivations of simple $\mathbb Q$-algebras

What's a good example of a simple algebra over a field of characteristic $0$ which has a non-inner derivation but also has the invariant basis number property (IBN)?
I'm under the impression that when ...

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### Pseudo-coherent complexes over sheaves of non-commutative rings

I am posing a question on derived categories to which I was not able to find an answer anywhere in the literature. I would appreciate any answer, hint or suggestion.
Assume that $\mathcal{R}_X$ is a ...

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### Explicit separability idempotent for the center of a separable algebra

Let $A$ be a $k$-algebra for some commutative ring $k$. Recall that $A$ is said to be separable over $k$ if the multiplication map $A\otimes_k A^{\operatorname{op}}\to A$ has a section as a map of $A\...

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### Existence of reduced norms for CSAs using fpqc descent

Let $k$ be a field and $A$ be a central simple algebra over $k$. It's known that $A$ has a splitting field (i.e. a field $K/k$ such that $A_K\cong M_n(K)$ for some $n$) which is finite and Galois.
...

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### A non-commutative, left duo ring whose only unit is the identity

Let $R$ be a ring (here, rings are always associative, unital, and non-zero). We say that $R$ is a left duo ring if $aR \subseteq Ra$ for every $a \in R$.
Question. Is there a non-commutative, left ...

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### Are there any central simple algebras admitting a standard basis?

Are there any central simple algebras admitting a standard basis?
By a standard basis I mean a normal basis that has a cyclic property generalizing that of the familiar basis $1, i, j, k$ for ...

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### Rings s.t. each element has a power lying in the center (and their completely prime ideals)

Let $R$ be a ring (throughout, all rings are associative and unital). We say $R$ satisfies condition (C) if, for every $a \in R$, there exists an integer $n \ge 1$ (depending on $a$) such that $a^n$ ...

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### Non-commutative rings where every non-unit is contained in a completely prime ideal

Below, all rings are associative and unital; and the word "ideal" always refers to a two-sided ideal.
Let's stipulate that a ring $R$ has property (P) if every non-unit of $R$ is contained ...

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### 'Lie correspondence' for formal power series in non-commuting indeterminates

This is related to an earlier question of mine. I would like an argument or a reference (or a missing hypothesis if needed) for the following.
Let $\mathbb{F}\langle\langle \alpha\rangle\rangle$ and ...

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### Question concerning the coefficients of block idempotents

Let $G$ be a finite group. Let $p$ be a prime number such that $p \mid |G|$.
Let Irr$(G)$ denote the set of ordinary irreducible characters of $G$.
For $\chi\in$ Irr$(G)$ define $e_{\chi} := \frac{\...