Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups.

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8
votes
1answer
163 views

If a faithfully flat extension of dg/A_$\infty$-algebra is formal, is the original algebra formal (over positive characteristic)?

Proposition 6.2 of Formality of DG algebras (after Kaledin) by Lunts reads (with a few additions to clarify notation): Let $k$ be a field of characteristic 0. Let $A$ be an $A_\infty$ algebra ...
0
votes
1answer
96 views

Example of noncommutative central reduced rings which is not reduced

A ring $R$ is called central reduced if every nilpotent element is central. Ungor et al. math.RA 14 Dec 2013 has given an example of a commutative ring which is central reduced but not reduced. Can we ...
0
votes
1answer
94 views

Definition of a cosemisimple Hopf algebra

A cosemisimple Hopf algebra is usually defined to a Hopf algebra which is the sum of its cosimple subcoalgebras. Does this definition assume that each cosimple subcoalgebra appears only once in the ...
1
vote
1answer
82 views

Jordan algebra of $3 \times 3$ quaternionic hermitian matrices

Let $\mathbb H = \mathbf H \otimes_{\mathbf R} \mathbf C$ be the tensor product of the quaternions with $\mathbf C$, and let $\mathcal J_3(\mathbb H)$ denote the set of $\mathbb H$-hermitian $3 \times ...
5
votes
0answers
164 views

A question on symmetric functions

Let $0 \leq m \leq n$ be integers. The group $S_n$ of permutations acts on the ring $\mathbb{Z}[X_1,\dots,X_n]$ by permuting the coordinates, with fixed subring $\mathbb{Z}[\sigma_1,\dots,\sigma_n]$, ...
-4
votes
1answer
90 views

Free algebra and ring of quotients [closed]

I am reading of example T.Y.Lam 'A first Course in Noncommutative Rings': Let $R=\mathbb{Z}\langle x,y\rangle/(y^2,yx)$. To work with $R$, we shall confuse $x,y$ with their images in $R$. Thus,we ...
1
vote
0answers
63 views

Does this system of equations have a closed form solution? [closed]

I am faced with the following system of equations and I'm looking for tools that allow me to characterize its solutions. The unknowns are $\{x_i\}$, $\{y_i\}$, and $\lambda$, for $i=1,...,N$. So there ...
7
votes
0answers
240 views

How bad can $SK_1$ of a commutative ring be?

For a commutative ring $R$ define $\mathrm{SK}_1(n, R)=\mathrm{SL}(n, R)/\mathrm{E}(n, R)$, the quotient of the special linear group by its subgroup generated by the elementary matrices. When $n\...
1
vote
0answers
58 views

Completion of an algebra

Based on arXiv:math/9802041v1, there is a definition for $NC$-filtration and $NC$-completion of an associated algebra over the complex numbers: Let $R$ be an associative algebra and $R^{\rm Lie} = (R,...
3
votes
0answers
68 views

False optima for control on Lie groups?

Consider the equation $\frac{d Y_t}{dt} = (A + w(t)B) Y_t$ evolving on a compact semi-simple Lie group $G$ where $\frak{g}$ is the Lie algebra of $G$ and $A,B \in \frak{g}$ and: $J:G \rightarrow [0,...
1
vote
0answers
88 views

Transitivity for algebraic extensions of integral domains?

I'm trying to prove the following result. Let $R_1\subseteq R_2\subseteq R_3$ be integral domains such that $R_1\subseteq R_2$ and $R_2\subseteq R_3$ are algebraic extensions (note: I do not want to ...
8
votes
0answers
230 views

Conjecture on matrix with reciprocal principal minors

Some notation: $A(\alpha|\beta)$ is the submatrix of $A \in \mathbb{R}^{n \times n}$ with with rows $\alpha$ and columns $\beta$. $\textrm{det } A(\alpha|\alpha) =: \textrm{det } A(\alpha)$ are the ...
9
votes
3answers
500 views

is every element in a C* algebra a product of normal elements?

I have the following question and since I am not an expert on C*-algebras, I thought I ask it here: I know that in general the sum and product of normal elements need not be normal. It is even true ...
2
votes
0answers
65 views

Is the positive part of an algebraic bilateral p-adic convergent power series algebraic?

Let $\mathbb{Z}_p \{ X\}$ and $\mathbb{Z}_p \{ X , X^{-1}\}$ be the henselizations of $\mathbb{Z}_p [X]$ and $\mathbb{Z}_p [ X , X^{-1}]$ with respect to the ideals $p\mathbb{Z}_p [X]$ and $p\mathbb{Z}...
0
votes
1answer
93 views

Finitely generated projective modules over matrix rings [closed]

Is every (left) finitely generated projective modules over the matrix ring $M_n(\mathbb{C})$ isomorphic to a trivial module? Is there a good reference to look at this problem? Apologies for asking ...
2
votes
0answers
166 views

Unique product groups (and semigroups)

A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a ...
7
votes
1answer
187 views

Hopfian modules

My question is slightly motivated by basic results in linear algebra, for example, that if $F$ is a field then a surjective linear map $F^n \rightarrow F^n$ is injective. More generally, any ...
2
votes
1answer
73 views

Dimension of preprojective algebra of Dynkin type

Fix a field $\Bbbk$. Let $Q$ be a Dynkin quiver and let $\Pi(Q)$ be its preprojective algebra. It is well-known that in this case $\Pi(Q)$ is finite-dimensional, but I've been unable to find a ...
1
vote
0answers
60 views

The projective resolution of a direct summand

For an $R$-module $M$ fix a projective resolution $P^\bullet\to M$. If $N$ is a direct summand of $M$, that is there is $L$ such that $M=N\oplus L$, then is there a projective resolution of $N$ which ...
9
votes
0answers
159 views

Detecting a module for the free group algebra on a finite quotient

Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module $M = R / (ax + by + c) R$. I am ...
4
votes
0answers
105 views

Does a given bound quiver algebra admit an algebra analogous to the preprojective algebra of a path algebra?

Let $Q=(Q_0,Q_1,s,e)$ be a finite quiver. In the formal construction of the preprojective algebra associated to $Q$, we set $\ Q^*_1=\{\alpha^*:y\rightarrow x| \alpha\in Q_1, \alpha:x\rightarrow y \}...
3
votes
0answers
76 views

Poincaré-Birkhoff-Witt theorem for Leibniz algebras

Leibniz algebras can be seen as a non-skew-symmetric generalization of Lie algebras. I have already taken a look at some papers related to Leibniz algebras and extending main results of Lie algebras ...
6
votes
1answer
138 views

Trivial algebras given by generators and relations

Let $V$ be a finite dimensional vector space over a field $K$ of characteristic zero. Assume that we are given a set of (not necessarily homogeneous) elements $f_1,\ldots f_n$ in the tensor algebra $T(...
1
vote
2answers
88 views

A Lie-like product in rings with involution

Suppose $R$ is a ring with involution $*$ and $x,y\in R$. Does the quantity $xy-y^{*}x^{*}$ have a standard name? Has this product undergone systematic study in the ring-theory literature, and if so, ...
0
votes
1answer
175 views

About the Wedderburn-Malcev Theorem

The Wedderburn-Malcev theorem states that for every (associative unital) finite-dimensional algebra $A$ over field $F$, there exist a semisimple subalgebra $S$, such that $A=\mathrm{Rad}(A)\oplus S$ ...
2
votes
2answers
439 views

For a ring R, does $GL_n(R)$ embed into $GL_m(F)$ for some field F?

Suppose $R$ is a noncommutative ring. What is the sufficient and necessary condition for $GL_n(R)$ embedding into $GL_m(F)$ for some field $F$? In particular, what if $R$ is the group ring $\mathbb{Z}...
3
votes
0answers
89 views

Is there a name for a noncommutative generalization of Poisson algebra?

Is there a name for an associative algebra which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e., $$[a, b \circ c]=[a,b]\circ c+b\circ [a,c], \hspace{15mm}(*)$$ ...
0
votes
0answers
150 views

When Noetherian ring is Artinian

Let R be a ring with1. It is known that if R is left Noetherian and every finitely generated left R-module embeddable in a free left R-module, then R is Artinian. It is known also that if every ...
2
votes
0answers
89 views

Irreducible representations in BGG category $\mathcal{O}$ over (finitely) direct sum of general linear Lie superalgebra

Let $\mathfrak{g} = \oplus_i^k\mathfrak{gl}(m_i|n_i)$ be a direct sum of general linear Lie superalgebras $\mathfrak{gl}(m_i|n_i)$'s with the Cartan subalgebra $\mathfrak{h} = \oplus_i^k \mathfrak{h}...
7
votes
1answer
141 views

Can every trace preserving isomorphism of unital self-adjoint matrix algebras be realized as conjugation by a unitary?

In this paper, Friedland shows (in Lemma 3.4) that if $\phi$ is an isomorphism of coherent algebras, then there exists a unitary $U$ such that $$ \phi(M) = UMU^\dagger$$ for all $M$. I am wondering if ...
1
vote
1answer
52 views

Under what assumptions can endomorphisms of $M/IM$ be realized as a subquotient of endomorphisms of $M$?

Suppose we have an algebra $A$ (unital, associative), with an ideal $I \leq A$ and a finitely generated module $M$ over $A$. It is possible to obtain both $\mathrm{End}_A(M)$ and $\mathrm{End}_A(M/IM)...
2
votes
1answer
97 views

An isomorphic invariant in ring theory

Let $R$ be a unital ring. We define the Murray Von Neumann relation $M$ on $R$ as follows: We say $a M b$ iff $a=xy,\;b=yx$ for some $x,y\in R$. (This is inspired by the usual Murray Von Neumann ...
1
vote
0answers
190 views

Semidirect product of semidirect products

For algebraic objects, say for groups $ N,K$ and algebra $A$, if we have the semidirect product of a semidirect product, $(A \rtimes_{\gamma} N ) \rtimes_{\theta} K$, are there conditions that would ...
3
votes
1answer
127 views

How to obtain an quantization of the algebra of functions of a given space?

My question is inspired by quantum spheres and quantum projective spaces (studied by Dijkhuizen & Noumi, Vaksman & Soibelman, etc., i.e. "A FAMILY OF QUANTUM PROJECTIVE SPACES AND RELATED $q$-...
5
votes
1answer
322 views

A Hom-Tensor identity - $\text{Hom}_{R}(P,B)\otimes _SC \cong \text{Hom}_{R}(P,B \otimes_S C) $

let $R,S$ be associative algebras over $\mathbb{C}$. Let $\mathcal{C} \subseteq$ $R$-Mod be a full abelain subcategory of $R$-Mod which is the category of $R$-modules. Let $B$ and $C$ be, a $(R,S)$-...
3
votes
2answers
203 views

On rings $R$ such that $xR\cap yR$ is non zero whenever $x$ and $y$ are non zero

I've isolated a property of rings (integral domains, associative, unitary, non necessarily commutative) that is useful to me : $$xR\cap yR\neq\{0\}\quad\text{ whenever $x$ and $y$ are non zero.}$$ ...
1
vote
0answers
167 views

A functor on the category of rings, algebras or compact Hausdorff topological space

Assume that $R$ is a unital ring or a complex or real (Banach or $C^{*}$) algebra. We define a relation $M$ on $R$ as follows: $$a\;M b \;\;\; \text{iff}\;\; a=xy,\;b=yx \;\; \text{for ...
7
votes
0answers
119 views

Associated graded of double Koszul dual

Let $k$ be a field, and let $A$ be a graded, connected, augmented, locally finite $k$-algebra. If $\Omega^* A$ denotes the cobar complex of $A$ (i.e., the dual $Hom_k(B_*(A), k)$ of the bar complex ...
3
votes
1answer
91 views

ideal generated from a truncated “real radical-like” set are still real radical?

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1...x_n$. $\mathbb{R}[X]_t$ is the truncated ring such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], deg(f) \leq ...
7
votes
1answer
755 views

The saturation of Murray von Neumann relation

Edit: According to comment of Pace Nielsen, I remove question 2 of the previous version: Let $R$ be a unital ring. We define Murray Von Neumann relation $M$ on $R$ as follows: We say $a M b$ iff $...
15
votes
1answer
804 views

Has anything ever been done with the set $\{1,2,3,4,\ldots\}$ equipped with the operation $a \oplus b = a+b-1$ and the usual notion of multiplication?

Definition. $$\mathbb{J} = \{1,2,3,\ldots\}.$$ We can refer to the elements of $\mathbb{J}$ as "joiners." The product of joiners is inherited from $\mathbb{Z}$. The sum of joiners will ...
7
votes
1answer
144 views

Why do we want $p$-permutation modules in splendid equivalences?

First Rickard (in Splendid Equivalences: Derived Categories and Permutation Modules ) and then Rouquier (Block theory via stable and Rickard equivalences, Appendix A.1) define splendid equivalences ...
8
votes
2answers
247 views

Sets of matrices which are irreducible but not strongly irreducible

A set of $d \times d$ real or complex matrices is commonly called irreducible if those matrices do not jointly preserve a linear subspace with dimension strictly between zero and $d$. A stronger ...
3
votes
2answers
503 views

Primary structures in $\mathbb Q$

I'll formulate a topic restricted here to the positive rational numbers $\ \mathbb Q_{_{>0}},\ $, then will pose a question (Q2) plus some related, to which I don't know the answers nor reference. ...
1
vote
1answer
34 views

On finite Uniform (Goldie) dimensions

1) Is there any characterization of $\Bbb{Z}$-modules with finite uniform dimensions? 2) Find two $\Bbb{Z}$-modules $N, M$ such that $N\leq M$ and $M\hookrightarrow N$ but $N$ is not isomorphic to $M$...
3
votes
0answers
98 views

Systematic treatment of folding and valued graphs

I'm going to say beforehand that this question has something of a "am I missing something?" flavor. I'm in that odd position mathematicians often find themselves, where a topic has been addressed ...
0
votes
0answers
65 views

Structure theorem for non connected graded Hopf algebras

Let $V^{\bullet}$ be a graded vector space over a field of char $0$. Assume that $V^{i}=0$ for $i<1$. Then the tensor algebra has the structure of a connected cocommutative Hopf algebra and in ...
12
votes
4answers
698 views

Applications of Jordan algebras

Jordan algebras are non-associative algebras satisfying a somewhat strange (to me) list of axioms, see wikipedia. Basic examples are real symmetric and complex hermitian matrices with the product $A\...
1
vote
0answers
65 views

A property of minimal prime ideals in rings with finite chromatic number

Let $R$ be a commutative ring with identity. There are so many ways to associate a graph to $R$. Consider this: take the elements of $R$ (All elements including zero) as vertices an two distinct ...
10
votes
1answer
181 views

How “nondegenerate” are amalgamated free products of C*-algebras?

In the following, I assume all algebras are unital. Let $A$ and $B$ be C*-algebras that each contain (isomorphic copies of) a common C*-subalgebra $C$. Let $A *_C B$ denote the amalgamated free ...