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.

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4
votes
1answer
63 views

Tensor product of positive linear maps is positive

Let $\pi_1: A_1 \to B_1$ and $\pi_2: A_2 \to B_2$ be positive linear maps between complex $*$-algebras. Is the mapping $$\pi_1 \otimes \pi_2: A_1 \otimes A_2 \to B_1 \otimes B_2$$ again positive? I.e.,...
9
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2answers
169 views

Global homological dimension of group rings

In all that follows, let $k$ be a field and $G$ be a finite group. It is well-known that the order of $G$ is invertible in $k$ iff the group ring $k[G]$ is semisimple, which is equivalent, inter alia, ...
3
votes
0answers
76 views

Adic completion of quotient

Let $R$ be a ring and $I=(a,b)$ be the ideal inside $R$ generated by $a,b\in R$. We denote $\hat{R}$ to be the $I$-adic completion of $R$. Do we have $\hat{R}/a$ is just the $\overline{b}$-adic ...
1
vote
1answer
61 views

If, in a unit-regular ring, the right annihilator of $a$ equals the right annihilator of $b$, then $aR = bR$?

Recall that a (unital) ring $R$ is von Neumann regular (VNR) if, for each $x \in R$, there exists $y \in R$ such that $x = xyx$; and unit-regular if such an element $y$ can be taken to be a unit. ...
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0answers
37 views

quantum affine $gl_2$

There are many sources of the relations and Hopf algebra structure of quantum affine $sl_2$ as a deformed enveloping algebra. However, for an application to integrable systems I need to look at ...
3
votes
2answers
192 views

Low dimensional noncommutative non-cocommutative Hopf algebras

Sweedler's Hopf algebra (see here) is the lowest dimesnional ($4$-dimensional) Hopf algebra that is noncommutative and non-cocommutative. What are the next examples? Are there noncommutative, ...
0
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0answers
91 views

Is there a roadmap to learning representation theory of finite group over finite field?

I've been wanted to learn some basic theories of the (non-semisimple) representation of the finite group over a finite field. I have been guessing that the materials might be contained in the books on ...
31
votes
8answers
3k views

Uncountable counterexamples in algebra

In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
13
votes
1answer
279 views

Existence of a translation-invariant basis of $\ell^2$

This question is heavily inspired by this other one, but is meant to be a hopefully more accessible variant of it (and I think slightly more natural). I give four equivalent formulations of the same ...
4
votes
0answers
117 views

Rings such that torsion-free/flat/projective modules are flat/projective/free

While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, ...
7
votes
0answers
74 views

Is this “semi-tensor product” something recently invented? Are there other usages of it?

The context: I was reading a paper in which they used the following definition called "Semi-Tensor Product" (STP) or "Cheng" product (In honor to its "inventor" D. Cheng):...
17
votes
3answers
661 views

Existence of translation-invariant basis on $C_c(\mathbb R)$

Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
2
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0answers
31 views

Lie Groups and Lie algebras related to Jordan algebras

Let $J$ be a Jordan algebra. I knew three relative Lie groups/Lie algebras to $J$. In the paper "The Capelli Identity, Tube Domains, and the Generalized Laplace Transform" Jacobson [J] has ...
2
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0answers
75 views

On the use of the fundamental exact sequence of K\"ahler differentials in a paper of Lyubeznik

Let $k$ be a field, $R := k[x_1, \cdots , x_n]$ the polynomial ring in $n$ indeterminates over $k$ and $f$ a nonzero element of $R$. The following paper of Lyubeznik which I have been recently reading,...
9
votes
1answer
98 views

Special nilpotent elements

Let $R$ be a (noncommutative, associative) ring. Set $N_2:=\{x\in R : x^2=0\}$, the set of nilpotent elements of degree $2$ (also called the square-zero elements). If $x,y\in R$ satisfy $xy=0$, then $...
0
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0answers
53 views

Levi decomposition for associative algebras

Does there exist an analogue of Lie algebra Levi decomposition for some/any class of associative algebras?
2
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0answers
101 views

Base change to algebraic closure commutes with quotient of polynomial ring by maximal ideal

Let $k$ be a field, $R:=k[x_1, \cdots , x_n]$ and $\mathfrak m$ be a maximal ideal such that $R/\mathfrak m$ is a finite separable field extension of $k$. Consider the algebraic closure $\overline k$ ...
2
votes
0answers
66 views

Socle of a quotient of the ring of differential operators of a polynomial ring

I have been reading the following paper: https://www.sciencedirect.com/science/article/pii/S002240491000263X Proposition 2.4(ii) shows that if $\mathfrak D$ is a ring of $k$-linear differential ...
10
votes
2answers
428 views

Are the trace relations among matrices generated by cyclic permutations?

Let $X_1,\dots,X_n$ be non commutative variables such that $\operatorname{tr} f(X_1,\dots,X_n) = 0$ whenever the $X_i$ are specialized to square matrices in $M_r(k)$ for any $r \geq 1$. Does this ...
4
votes
1answer
123 views

Can all finite-dimensional noncommutative algebras with trace be trace-preserving embedded into matrix rings?

Suppose I have a finite (non-)commutative ring $R/k$ (over a field $k$ of char $0$) with a linear "trace" function $t: R \to k$. Can I always find an embedding $f: R \to M_r(k)$ compatible ...
-3
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0answers
64 views

Did anyone ever propose a hypercomplex numbers system with more than one anisotropic axis? [closed]

The real number axis is asymmetric against zero: for instance, multiplication of two negative or two positive numbers will produce a positive number, a square root of a negative number is not real, ...
0
votes
0answers
118 views

Can the notion of algebraic closedness be generalized to the rings with zero divisors?

Is there a notion of rings that are algebraically closed except for the roots of polynomials with coefficients that are divisors of zero? For instance, it seems that any polynomial of non-zero-divisor-...
0
votes
0answers
89 views

How to classify rings by combinatorial structures?

There are many ways to encode information about algebraic structures such as groups, rings, etc... in combinatorial form. For example the Cayley graph of a group with a subset of generators, or the ...
9
votes
1answer
323 views

Mistake and correction-Drozd, Y.A., Kirichenko, V.V.: Finite Dimensional Algebras

In the book: Drozd, Y.A., Kirichenko, V.V.: Finite Dimensional Algebras. Springer, Berlin (1994), there is an exrcise which suggest a positive answer to the next question, and Ryszard R. ...
1
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0answers
50 views

Faithfull flatness of a module containing the ring as a direct summand

Let $R$ be a not necessarily commutative ring, and let $M$ be a projective left $R$-module. Question. If $R$ is a direct summand of $M$ as a left $R$-module, then is it true that $M$ is faithfully ...
1
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0answers
90 views

Free monoids on posets

I've suddenly found myself working with some free monoids $F(S)$ in which the set $S$ is a poset, and the order extends to an order $F(S)$, satisfying if (but not only if) $s_1, s_2, \ldots, s_r, t_1, ...
1
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0answers
39 views

Construction, similar to Chow's EL-numbers? Is it valid? What are the properties?

The idea of EL-numbers, proposed by Chow, impressed me very much, so I decided to build something similar and look what this will turn out. Instead of $\exp(x)$ and $\ln(x)$ functions as the building ...
2
votes
1answer
151 views

English translation of Emmy Noether's Hyperkomplexe Grössen und Darstellungstheorie

I'm wondering if anybody knows where one can find an English translation of Emmy Noether's classical paper E. NOETHER, Hyperkomplexe Grössen und Darstellungstheorie, Math. Zeit. 30(1929), 641–692 ?...
7
votes
0answers
164 views

Is there a reasonable definition of an octonionic manifold?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\End{End}$ Todorov and Dubois-Violette have recently shown how to understand the structural gauge group of the standard model via octonions. Q. Is there ...
18
votes
2answers
979 views

Why does Drinfeld Unitarization work?

In Drinfeld's paper "Quasi-Hopf Algebras" he illuminates a process by which you can replace the $R \in A \otimes A$ associated to a quasi-Hopf QUE-algebra $(A, \Delta, \varepsilon, \Phi)$ ...
3
votes
1answer
311 views

Is a comultiplication structure unique?

Let $A$ be an $R$-algebra. Suppose $A$ has a $R$-coalgebra structure compatible with the algebra structure. (I.e. there is a comultiplication map $\Delta$ and counit map $\epsilon$ compatible with the ...
5
votes
3answers
278 views

Unital *-homomorphisms between matrices

It is mentioned on Wikipedia that every unital *-homomorphism $\Phi:M_i\to M_j$ is necessarily of the form $\Phi(a)=U^*(a\otimes I_r)U$ for some unitary $U$ and some $r$. (Here $M_i$ are the $i\times ...
3
votes
2answers
242 views

Is there any simple formula for the character of $S_{n}$ represented by the set of $k$-tuples of $\{1,2,…,n\}$?

I'm interested in the representation theory of symmetric groups. I'm now trying to search for the formula for the characters of $\Omega^{k}$, the set of $k$-tuple of elements of $\Omega$ a set of $n$ ...
5
votes
1answer
282 views

Explicit construction of division algebras of degree 3 over $\mathbb{Q}$

In his book Introduction to arithmetic groups, Dave Witte Morris implicitly gives a construction of central division algebras of degree 3 over $\mathbb{Q}$ in Proposition 6.7.4. More precisely, let $L/...
3
votes
0answers
33 views

Bound quiver algebras with relations of the form $x_ix_j=$sum of paths of length $\geqslant 3$

While working with homotopes and isotopes of finite dimensional algebras I often encounter algebras isomorphic to a path algebra of a bound quiver, i.e. $k[\Gamma]/I$, where the relations $I$ have the ...
1
vote
0answers
25 views

Morphism of R-algebras between R-adic algebras

Let $R$ be a $f$-adic ring. Let $A$ and $B$ be $R$-adic algebras. I would like to show that any morphism of $R$-algebras between $A$ and $B$ is actually adic.
2
votes
1answer
63 views

About extensions between morphisms on the multiplier algebra

Let $A$ be a non-degenerate algebra and let $\Delta: A \to M(A \otimes A)$ be a non-degenerate morphism. We can extend the algebra morphism $$\iota \otimes \Delta: M(A \otimes A) \to M(A \otimes A \...
1
vote
1answer
66 views

Non-degeneracy of comultiplication (multiplier Hopf algebras)

Consider the following fragment from the paper "Multiplier Hopf-algebras" by Van Daele. Can someone explain how the coassociativity in definition 2.2 (ii) and the requirement $(\Delta \...
5
votes
1answer
160 views

Invariant ideal generated by invariant elements

Let $G$ be a complex reductive group acting linearly on $\mathbb{C}^n$ and let $X$ be a $G$-invariant closed subvariety of $\mathbb{C}^n$. Is $X$ the zero-set of finitely many $G$-invariant functions? ...
1
vote
1answer
68 views

Antipode on a multiplier Hopf-algebra

Probably an easy question, but here goes: I'm reading the paper Multiplier Hopf algebras by Van Daele. Let $(A, \Delta)$ be a multiplier Hopf algebra. Let $L(A), R(A), M(A)$ be the left, right and ...
2
votes
0answers
58 views

Examples of multiplier Hopf algebras

A multiplier Hopf-algebra (introduced by Van Daele) is a pair $(A, \Delta)$ where $A$ is a non-degenerate algebra $A$ together with a non-degenerate algebra morphism $\Delta: A \to M(A \otimes A)$ ...
5
votes
0answers
131 views

Representation theory terminology question

For a paper I'm writing, I need a term for a representation-theoretic concept that I'm sure someone has thought of before, so I thought I'd ask here rather than just make something up. Let $G$ be a ...
4
votes
2answers
151 views

Embedding of a division algebra into a matrix algebra over its centre

Let $K$ be a number field and let $D$ be a central division algebra over $K$. Let $d$ be the index so that $[D:K]=d^2$. What is the minimal $n$ such that there exists an embedding of $D$ into $\mathrm{...
2
votes
0answers
78 views

Is there a characterisation of Cayley–Dickson Algebras?

The Cayley–Dickson construction takes an algebra with involution and produces another algebra with involution of twice the dimension. Starting from the reals (with trivial involution), we ...
1
vote
0answers
87 views

How can we prove a specific isomorphism between graded Lie algebra and graded universal enveloping algebra?

$\DeclareMathOperator\gr{gr}$Let $L$ be a Lie algebra over field of characteristic different from $2$, and let $L_n$ be its descending central series. Therefore, we consider the associated graded Lie ...
1
vote
1answer
109 views

A comodule algebra map from a Hopf algebra to itself

Let $H$ be a cosemisimple Hopf algebra (or just a bialgebra). Considering $H$ as a left $H$-comodule (i.e. take $\Delta_H$ to be the coaction), can there exist a (non-identity) algebra map $\sigma:H \...
3
votes
1answer
212 views

Number of rings with additive group $(\mathbb{Z}_{16})^2$. A341547(16) in OEIS

I would like to know if somewhere the number of non-isomorphic rings with additive group $(\mathbb{Z}_{16})^2$ is mentioned. If not, is someone able to calculate it? And (easier) the commutative case? ...
2
votes
0answers
62 views

Explicit example of prime ideal not an intersection of maximal ideals, in universal enveloping algebra

Let $A$ be a $\mathbb k$-algebra. If $A$ is affine commutative, by Nullstellensatz, then every prime ideal of $A$ is an intersection of maximal ideals. To justify the notion of being primitive in ...
3
votes
1answer
94 views

What is the $p$-regular partition corresponding to the sign representation of $S_{n}$ over a field of characteristic $p$?

I'm now interested in the modular representation of symmetric groups. It is well-known that for a fixed prime $p$, there is a bijection between the irreducible representations of $S_{n}$ over a field ...
3
votes
1answer
85 views

Subalgebras of the Temperley-Lieb algebra

I've recently met with the Temperley-Lieb algebra in my work. I'm in no way a specialist, and it's seems like a pretty simple question, but nevertheless. I'm interested in the subalgebra generated by ...

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