# Questions tagged [ra.rings-and-algebras]

Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.

3,198
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### Orthogonality in Hilbert algebras and congruence

Consider a finite-dimensional Hilbert space $V$ (say, over $\mathbb{C}$) and a finite-dimensional Hilbert algebra $A$ (i.e., Hilbert space with a compatible associative unitary algebra structure). ...

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### Adjoining new factors for primes in UFDs

It is well-known that if we pass from a UFD to a new ring where we have factored one of the primes, it does not need to stay a UFD. The classic example is passing from $\mathbb{Z}$ to $\mathbb{Z}[\...

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### How to decompose a given polynomial by ideal generators

Given a finite set of polynomials $f_1, f_2,..., f_n$ of variables $x_1,...,x_m$, generating the ideal $I$, suppose that we have one more polynomial $g\in I$.
What is the algorythm for decomposing $g$ ...

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### What is known about finite dimensional modules over the nilCoxeter algebra?

Recall that the nilCoxeter algebra $\mathcal{N}_W$ for a Coxeter group $W$ is given by the $\mathbf{k}$-basis $x_w$ for each $w\in W$ and multiplication $x_ux_v=x_{uv}$ if $\ell(uv)=\ell(u)+\ell(v)$ ...

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### Equivalent definition of Spin group in terms of automorphisms

Let $\mathrm{Cl}(\mathbb{R}^n)$ denote the (real) Clifford algebra on $\mathbb{R}^n$ with respect to the Euclidean inner product. Let $\mathrm{Cl}^0({\mathbb{R}^n})$ denote the even part of $\mathrm{...

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### Tensor product of finite extensions of $\mathbb{Q}_p$

Consider the tensor product of finite extensions of a field $F$ of characteristic zero. (I am interested in the case $F=\mathbb{Q}_p$.)
$(1)$ If $M$ is a finite Galois extension of $F$ with Galois ...

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129
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### Vanishing (infinite) tensor products

Since the advent of free probabilities and QFT, infinite tensor products of $R$-associative algebras with units has become more familiar to the working mathematician.
Starting from the (permuting) ...

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### Decomposing an endomorphism as a tensor product

$\DeclareMathOperator\End{End}$Let $f$ be an endomorphism of the finite-dimensional vector space $V$, over the field $K$. The question of whether $f$ is decomposable, that is, whether $V$ can be ...

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### A standard name for the algebraic structure on a projective line?

Question: Is there any name for the natural algebraic structure of the projective line?
Algebraically, a projective line over a field is a set $L$ endowed with two binary operations $+$ and $\cdot$ ...

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### Automorphism groups for simple objects in abelian linear categories

Let $\mathcal{A}$ be an abelian category that is also $k$-linear, where $k$ is some algebraically closed field. Let $X$ be a simple object in $\mathcal{A}$. What can we say about $\mathrm{Aut}(X)$? I ...

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### Attempts to define a matrix exponential over (as much as possible) general fields

Given a $n \times n$ matrix $A$ over the complex numbers, the exponential of $A$ is defined as
$$\exp(A) := \sum_{k = 1}^\infty \frac1{k!} A^k , \qquad \tag{$\star$}\label{468645_star}$$
where ...

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### In a monoidal category with duals is the coevaluation map determined by the evaluation?

For a monoidal category $(\mathcal{M},\otimes,1)$ and an object $X$ with a left dual $X^*$. Let $(e,c)$ be a pair of (co)evaluation maps for the pair $(X,X^*)$. Is it possible to have another map $c': ...

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### Is it true that monomorphisms of local Artinian $\mathbb{R}$-algebras are regular?

A Weil algebra is a finite-dimensional real algebra, in which each element is the uniquely sum of a scalar and a nilpotent (so nilpotents constitute the only maximal ideal of codimension 1). In other ...

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### Does Manin's construction of non-commutative endomorphism algebra $\mathrm{End}(A)$ produce Koszul algebra, if $A$ is Koszul?

$\newcommand{\dual}{\mathrm{dual}}\DeclareMathOperator\End{End}\DeclareMathOperator\Fun{Fun}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\GL{GL}$Around 1986–7 Yu.Manin proposed natural and ...

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### The involutive structure on a division ring

This question is motivated by foundations of geometry, namely, by studying scalars in affine spaces.
Let $F$ be a field (or better a division ring). It has the operations of addition and ...

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### Hopf algebras actions

Can you write down a general type of Hopf algebra action? How do you justify the name "action", when it is already used for group actions?
There must be a common core, if the same term is ...

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### The evaluation and coevaluation maps for an object isomorphic to a dualisable object

Let $X$ be an object in a monoidal category with dual $X^*$ and evaluation and coevaluation maps $e$ and $c$. Now if we have an isomorphism $\sigma:X \to Y$, for some other object $Y$, then $Y$ must ...

4
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108
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### Second cohomology group of the contact Lie algebra $K_3$

Let $F$ be a field of characteristic zero and, for all $n>0$, consider the contact Lie algebra $K_{2n+1}$. It follows from Corollary 3 of the paper [V. Guillemin - S. Shnider: Some stable results ...

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### Is anything known about the center of the Fomin-Kirillov algebra?

Let $\mathcal{B}_{\mathbb{S}_m}$ be the quotient of the Fomin-Kirillov algebra so that its pairing becomes certainly nondegenerate. This algebra is conjecturally isomorphic to the Fomin-Kirillov ...

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### What is the best way to introduce tensor products to undergrads?

What is the best way to introduce vector space tensor products to undergrads? We have a number of options, given in no particular order.
The universal property proving uniqueness and not existence.
...

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### Is algebra: ac=ca, bd = db , ad - da = cb - bc ("Manin matrix algebra") - a Koszul algebra?

Question: Consider quadratic algebra with four generators $a,b,c,d,d$ and three relations $ac=ca,bd = db, ad-da = cb - bc$ . Is it a Koszul algebra ? (i.e. Koszul complex is resolution of ground field ...

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### Are the real numbers isomorphic to a nontrivial ultraproduct of fields?

Let $K_1, K_2, \dots$ be a countable sequence of fields, and let $\prod_{\mathcal F} K_i$ be the ultraproduct with respect to some nonprincipal ultrafilter $\mathcal F$.
Question: Can there be a field ...

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### Algebraic role of Dottie number

Consider an algebraic system, which we would call "anti-dual numbers", which is defined on a plane with a unit "circle" of the form of lemniscate (two adjanced circles) $r(\phi)=|\...

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### Reference for cocommutative coalgebras

I'm looking for references on cocommutative coalgebras where I can see them as kind of infinitesimal spaces. I'm trying to understand this post Why do Lie algebras pop up, from a categorical point of ...

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### Quasi-isomorphisms of P-algebras

In the paper "Homotopy algebras are homotopy algebras" from Markl a notion of strong homotopy morphism between strong homotopy P-algebras is defined. The author restricts to the case where $...

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### Real exponentiation in the quotients of rings of continuous functions by prime ideals

Consider the ring $C = C(X) = C(X; \mathbb{R})$ of continuous functions $f:X\to \mathbb{R}$ where $X$ is a Tychonoff space. This is naturally a lattice ordered ring by setting $f\geq 0$ iff $f(x)\geq ...

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### Examples of Noetherian integral group ring

I need to study the integral group ring of the fundamental group of a manifold. My knowledge of group and ring theory is very limited. I am looking for some examples of groups $G$ for which $\Bbb ZG$ ...

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### Particular example of a quadratic extension of a nonunital ring

I want to construct a concrete non-unital ring $R$ with the following properties:
$R$ is a noncommutative non-unital ring with a right unite $r$ i.e $t.r=t$ for any $t\in R$.
$S\subset R$ is a ...

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### Extending the proof of Maschke's Theorem from finite groups to algebras

In the theory of representations of a finite group there is Maschke's Theorem that any finite-dimensional representation of a finite group $G$ can be decomposed into a direct sum of irreducible ...

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### Concrete examples of derived categories

What examples of abelian categories $\mathcal{A}$ are there such that the derived category $\mathcal{D}(\mathcal{A})$ can be described concretely? For example, is there a concrete way of describing $\...

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### Representation of anti-commuting matrices in $\mathbb{C}^{2}$

This is a cross posting updated question from MSE. I have not got any answers there yet and I really want to understand this problem.
The basic question is the following. Let $V$ be a finite-...

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### When does the sum of squares/cubes of polynomials over finite field have less than maximum degree?

Given polynomials $p_1(x), p_2(x), \dots p_m(x) \in \mathbb{F}_p[x]/\langle x^p-x\rangle$ where $p$ is a prime, when does $\sum_{i=1}^m p^2_i(x)$ have degree $< p-1$? What about $\sum_{i=1}^m p^3_i(...

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### Isn't every algebraic operad equipped with a trivial weight?

In Loday–Vallette "Algebraic Operads" they state the following result (Theorem 6.6.2, Operadic twisting morphism fundamental theorem):
Let $P$ be a connected weight graded differential ...

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### An infinitely generated Lie algebra, its finitely generated envelope

If a Lie algebra $\mathfrak g$ is finitely generated, its enveloing algebra $U\mathfrak g$ is finitely generated as an associative algebra. In fact, taking the enveloping algebra of the surjection $\...

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### On a lemma of projective dimension

Let $R$ be a finite-dimensional algebra, and $A=R\oplus A_1\oplus A_2\oplus \dotsb$ be an $\mathbb{N}$-graded algebra which is locally finite (i.e. all $A_i$'s are of finite dimension). Let $\text{...

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### Property for bounding matrix exponential

Wikipedia states in the exponential map section about the exponential of a matrix that for any matrices $X$, $Y$ it holds that $\|e^{X+Y}-e^{X}\| \leq \|Y\|e^{\|X\|} e^{\|Y\|}$ where $\|\cdot\|$ ...

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### A question about the existence of rational functions

I am reading a paper Representations of shifted quantum affine algebras. I have a question about the existence of a rational function about the remark $4.4$
I'll briefly describe the problem.
We let $...

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### Let $a_1, \dots, a_n$ be a finite set of positive reals. Is there a $\mathbb Q$-basis of $\mathbb R$ where each $a_i$ has nonnegative coordinates?

Let $a_1, \dots, a_n$ be a finite set of positive reals. Is there a $\mathbb Q$-basis of $\mathbb R$ where each $a_i$ has nonnegative coordinates?
Playing around with the case $n = 2$, I’m pretty sure ...

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### Zero divisors in an algebra with two generators

Let $k$ be a field, and $R = k\langle x,y \mid x^2 = 0\rangle$. The generators $x$ and $y$ are not supposed to commute with each other. Is the only case where nonzero elements $a, b \in R$ satisfy $...

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### Finite étale cover of factorial ring

Let $A$ be a regular factorial ring.
Consider $B=A[X]/(P)$ such that $B$ is finite étale over $A$. When do we have that $B$ is also factorial?

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### On the irreducible submodules of adjoint representations $\text{ad}^{0}$

Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is ...

11
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### Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry

In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry.
For graphs this had been an open ...

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### Inner product of signatures of piecewise linear paths

It is a well-know observation that, given two points $x_1,x_2 \in \mathbb{R}^d$, the path signature associated to their linear interpolation is given by the tensor exponential. Precisely, if $\Delta x$...

6
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388
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### Tame-Wild dichotomy; why can't tame algebras be wild?

I would like to understand the Tame-Wild dichotomy, and in particular why an algebra cannot be tame and (semi-)wild at the same time. I've looked in the papers by Drozd and Crawley-Boevey [D80, CB88].
...

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### Reference request for a subfamily of regular graphs

[Repost of same question math stack exchange which got no answers]
I'm looking for literature on the following family of graphs:
Call a regular graph $G=(V,E)$ (of regularity degree $d$) nice if there ...

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### Names for product-like algebras involving a "duo of directed pseudoforests"

I am looking for the names (and/or for any information regarding) two algebras, one "free" and one "restricted" by an equivalence class.
In both cases, there is an (infix) binary ...

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### Resolutions of semi free (or almost free) commutative dg algebras with finitely generated cohomology

Let $A^{\bullet}:=\{ \cdots \rightarrow A^i \overset{d^i}{\rightarrow} A^{i+1} \rightarrow \cdots \rightarrow A^{-1} \rightarrow A^0 \rightarrow 0 \rightarrow \cdots \}$ be a non-positively graded ...

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### Problem 1.8 from Kirby's list

Context
I looked through a book called "Problems in Low-Dimensional Topology", where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...

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### Relations in a certain Lie algebra

Let ${\mathfrak g}$ be the (real) Lie algebra generated by infinitely many generators $D_i, E_i$ with $i=1,2,3,\dots$ subject to the following relations for any natural numbers $i,j$:
\begin{gather*}
[...

10
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1
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203
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### Matrix ring isomorphisms of different sizes

Do there exist (unital, associative, noncommutative) rings $R$ and $S$, where $\mathbb{M}_2(R)\cong \mathbb{M}_3(S)$, but these matrix rings are not isomorphic to $\mathbb{M}_6(T)$ for any ring $T$?