All Questions
1,123 questions
2
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69
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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 ...
- 1,066
1
vote
1
answer
110
views
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 ...
- 223
2
votes
0
answers
80
views
An alternative definition for finitely generated (and principal) ideals in a semigroup
Let $S$ be a semigroup. An ideal (of $S$) is a subset $I$ of $S$ such that $SI$ and $IS$ are both contained in $I$. The non-empty ideals constitute a subsemigroup, $\mathfrak I(S)$, of the power ...
- 10.5k
1
vote
0
answers
69
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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{...
- 157
6
votes
0
answers
632
views
Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
10
votes
1
answer
221
views
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 $T$?
- 18.7k
3
votes
0
answers
137
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Composition of Frobenius $n$-homomorphisms, characteristic-free?
This question is, as so often, a crossbreed of curiosity and laziness. The
former has led me to an interesting, but somewhat unsatisfactory paper by
Khudaverdian and Voronov
(arXiv:2002.02395v2) and, ...
- 33.8k
2
votes
0
answers
91
views
A recursive description of the smallest divisor-closed subsemigroup containing a set
Let $S$ be a semigroup and $\widehat{S}$ be its unitization, i.e., the monoid obtained from $S$ by adjoining an identity element if necessary (so that $\widehat{S} = S$ when $S$ is already a monoid).
...
- 10.5k
3
votes
1
answer
171
views
Every homomorphism between (rational) Puiseux monoids is multiplication by a non-negative rational
Let a (rational) Puiseux monoid be a non-trivial submonoid of the non-negative rational numbers under (the usual operation of) addition. It is not difficult to show that, if $f \colon H \to K$ is a (...
- 10.5k
1
vote
1
answer
368
views
Wedderburn theorem for finite-dimensional algebras over the complex numbers
I'm trying to understand how to apply the Wedderburn theorem in the context of unitary algebras over $\mathbb{C}$ that are finite-dimensional and semisimple. Let $\mathcal{A}$ be a $\mathbb{C}$-...
7
votes
1
answer
129
views
Semi-simple algebras over operads
I believe people thought about this questions, however I couldn't find any reference. I appreciate if someone could direct me to some detailed discussions about it.
The categories of associative ...
- 449
9
votes
1
answer
236
views
Formal smoothness of path algebras and connections
Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if
$$
\Omega^1_kA = \operatorname{Ker}(\...
- 985
2
votes
2
answers
226
views
Minimal ideals and subalgebras of semisimple algebras
I'm considering an algebra to be a ring which is also a vector space over some field $F$, and the algebra $A$ is said to be semisimple if it is semisimple as a ring, i.e., $A$ can be written as a ...
6
votes
1
answer
173
views
References on semigroup actions
I posted this question on Math Stack Exchange about 10 days ago, but received no answer (https://math.stackexchange.com/q/4843881/1223994).
I would like to ask for references on semigroup actions on ...
- 339
7
votes
2
answers
1k
views
Is there a notion of point in noncommutative geometry?
It is not clear to me whether there is a general notion of point in NCG. I have heard (more through physics) that the notion of a point becomes meaningless or ill-defined in noncommutative spaces, but ...
- 136
4
votes
0
answers
143
views
On the conditions for Artin-Schelter Gorenstein algebras
Let $ k $ be a field and $ A $ a connected graded $ k $-algebra ($ A $ is associative, but not assumed to be commutative).
The algebra $ A $ is called Artin-Schelter Gorenstein* of dimension $ d $ if ...
- 936
3
votes
0
answers
75
views
Are the automorphisms of the power semigroup of a cancellative semigroup cardinality-preserving?
Let $S$ be a semigroup (written multiplicatively) and $f$ be an automorphism of the power semigroup $\mathcal P(S)$ of $S$, that is, a bijective function on the family of all non-empty subsets of $S$ ...
- 10.5k
1
vote
1
answer
84
views
Simple-direct-injective modules
A right $R$-module $M$ is called a simple-direct-injective module if it satisfies any of the following equivalent conditions:
For any simple submodules $A,B$ of $M$ with $A \cong B \subseteq^{\oplus} ...
- 324
8
votes
2
answers
577
views
Faithful flatness and non-commutative algebras
$\DeclareMathOperator\Spec{Spec}$When dealing with commutative algebras, a usefull criterion for faithful flatness is the following:
Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then $...
- 541
1
vote
1
answer
142
views
Congruences that aren't "finite from above," take 2: semigroups
This is a hopefully less trivial version of this question. Briefly, say that a congruence is parafinite if it is the largest congruence contained in some equivalence relation with finitely many ...
- 20.5k
3
votes
0
answers
103
views
An isomorphism problem for semigroups of ideals
An ideal of a semigroup $S$ (written multiplicatively) is a set $I \subseteq S$ such that $IS$ and $SI$ are both contained in $I$ (here, $XY$ means, for all $X, Y \subseteq S$, the setwise product of $...
- 10.5k
14
votes
4
answers
742
views
Prove or disprove: $R^{n+1} \supseteq R \cap R^2 \cap \cdots \cap R^n$ for every binary relation $R$ on a set of size $n$
Prove or disprove: $R^{n+1} \supseteq R \cap R^2 \cap \cdots \cap R^n$ for every binary relation $R$ on a set of size $n$.
I have verified the statement for $n \leq 4$ with a Mathematica code.
I have ...
- 143
12
votes
1
answer
1k
views
A problem in commutative algebra whose solution requires algebraic geometry (resp., noncommutative algebra)?
One can argue that commutative algebra is affine algebraic geometry. However, a great deal of commutative algebra generalizes to non-commutative algebra, and in that setting there is little geometry, ...
- 4,985
0
votes
0
answers
92
views
What is the highest $n\in\Bbb N$ for which a complete classification of inverse semigroups of order up to $n$ is known?
What is the highest $n\in\Bbb N$ for which a complete classification of inverse semigroups of order up to $n$ is known?
Given that there are $3{,}684{,}030{,}417$ semigroups of order $8$, I guess $n\...
- 379
2
votes
1
answer
194
views
Continuity of Moore-Penrose generalized inversion
Any matrix $A\in\mathbb{C}^{m\times n}$ has a unique generalized inverse $A^{\dagger}\in\mathbb{C}^{n\times m}$ with the properties $$AA^{\dagger}A=A,\qquad A^{\dagger}AA^{\dagger}=A^{\dagger},\qquad (...
- 1,093
5
votes
0
answers
305
views
Arithmetic derivatives and non-commutative generalizations
In the theory of arithmetic derivatives, in the simplest case an arithmetic derivative on $\mathbb{N}$ is defined via the rule $(a \times b)'= a \times b' + a' \times b$, mirroring the product rule ...
- 1,347
5
votes
0
answers
135
views
Confusion around a (necklace) cobracket in Ginzburg's article Calabi-Yau Algebras
Something has been puzzling me for quite a while in Ginzburg's article Calabi-Yau Algebras.
At some point he considers the free graded algebra $\mathbb{C}\langle x_1, \dots, x_n, \theta_1, \dots \...
- 437
5
votes
0
answers
160
views
$S$ and $T$ globally isomorphic semigroups, with $S$ (commutative and) cancellative, iff $S$ is isomorphic to $T$?
Denote by $\mathcal P(S)$ the semigroup obtained by equipping the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication ...
- 10.5k
2
votes
1
answer
271
views
Apropos of two groups being globally isomorphic iff they are isomorphic
Denote by $\mathcal P(S)$ the semigroup obtained by endowing the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication induced ...
- 10.5k
0
votes
1
answer
152
views
Name for a monoid on the basis of a vector space?
Is there a name for the structure of a vector space with a monoid defined on its basis?
Given a vector space V over a field F, we can choose a basis and define a monoid on it. Now we can use each ...
- 481
0
votes
1
answer
327
views
Can we generalise groupoids to monoid-oids? [closed]
Groups correspond to one object categories where every morphism is an isomorphism. Monoids correspond to one object categories.
Groupoids correspond to small categories where every morphism is an ...
7
votes
0
answers
227
views
On the structure of an algebra as a bimodule
$\DeclareMathOperator\End{End}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ker{Ker}\newcommand{\bi}{\mathrm{bi}}\newcommand{\op}{\mathrm{op}}$Let $K$ be a field (say of characteristic zero), and $...
- 541
2
votes
1
answer
265
views
Gluing data for modules over a ring with idempotents
Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $A\text{-}\mathrm{Mod}$ and $eAe\text{-}\mathrm{Mod}$. This is Example 2.7 in Homological ...
- 1,071
5
votes
2
answers
396
views
Algebra with three anti-commutator relations
Let $u,v,w \in \mathbb{F}_p^{\times}$. Consider the $\mathbb{F}_p$-algebra $V$ generated by $ a,b,c$ and the relations
$$u a^2 = bc + cb$$
$$v b^2 = ac + ca$$
$$w c^2 = ab + ba$$
Is $V$ generated by ...
- 63.1k
5
votes
0
answers
293
views
On the deformation theory of associative algebras
Let us start by recalling the notion of a formal deformation:
Let $K$ be a field of characteristic zero and $A$ be an associative $K$-algebra. Consider a commutative augmented $K$-algebra $R$, with ...
- 541
12
votes
0
answers
542
views
Does Wedderburn's Little Theorem hold constructively?
Wedderburn's Little Theorem states that every finite division ring is commutative. Perhaps even more surprising, this implies that every finite reduced ring is commutative.
The proofs that I am aware ...
- 63.1k
5
votes
2
answers
478
views
Generalization of the concept of a measure
Consider the following generalization of the concept of a measure:
Let $L = (X, \lor, \land, \bot)$ be a semi-bounded lattice.
Let $M = (Y, \bullet, e)$ be a commutative monoid.
An $(L, M)$-measure is ...
- 2,203
4
votes
1
answer
270
views
Kaplansky inverse element theorem on group C-star algebra
In a class talking about $C^*$ algebra and (higher) index theory, I heard a theorem
(related to Kaplansky, proved?), that is
Suppose $\Gamma$ is a group (admitting Haar measure if necessary) while $\...
- 193
2
votes
0
answers
144
views
Zero divisors in the extra-special group algebra $\mathbb{R}[2^{1+6}_+]$
Can you characterize the unit-group of the real group-algebra of the extraspecial plus-type 2-group of order 128? (That is $\mathbb{R}[2_+^{1+6}]$ using Conway's notation.)
(Please choose any irrep ...
- 21
3
votes
0
answers
176
views
The monoid of stably-free modules over integral group rings
Fix a torsion-free group G, let $M_G$ be the monoid of stably-free $\mathbb{Z}G$-modules under operation $\oplus$, the direct sum of modules.
In studying objects related to Wall’s D2 problem on CW-...
- 187
2
votes
0
answers
71
views
Classification of polynomials leading to finite dimensional admissible algebras
Let $K \langle x , y \rangle $ ($K$ a field, we can assume it has only two elements if it helps) be the non-commutative polynomial ring in 2 variables.
Question 1: For which non-commutative ...
- 26.5k
7
votes
1
answer
183
views
Is lambda calculus polymorphism a type of generalized monad?
Let $\mathbf{C}$ be a Cartesian closed category. Then simply typed lambda calculus in $\mathbf{C}$ in one type variable can be interpreted as a category $\mathbf{STLC}_{\mathbf C}$ where the objects ...
1
vote
0
answers
86
views
Inner product on Standard form of von Neumann algebra
Let $(M, H, H_+,J)$ be a standard form of a von Neumann algebra $M$ acting on a complex Hilbert space $H$ endowed with a self-dual cone $H_+$. Is it true that
$$\langle x,yz\rangle=\langle zx,y\rangle$...
- 131
2
votes
0
answers
176
views
On the origin of power semigroups
Let $S$ be a (multiplicatively written) semigroup. Equipped with the (binary) operation of setwise multiplication $(X, Y) \mapsto \{xy \colon x \in X, \, y \in Y\}$, the family of all non-empty ...
- 10.5k
8
votes
3
answers
1k
views
Are all free monoids residually finite?
I cannot manage to prove that a free monoid with operation concatenation, and with at least two generators is residually finite. If there is just one generator, the free monoid $\{a\}^*$ is isomorphic ...
- 81
1
vote
1
answer
87
views
Semigroup algebras with one dimensional center
Let $S$ be a finite semigroup and $K$ a field of characteristic 0 (we can assume the complex numbers for simplicity).
Question: Is there a characterization when the center of the semigroup algebra $...
- 26.5k
1
vote
1
answer
129
views
Cycles in almost breakable semigroups
Last October, I learned from Benjamin Steinberg's answer to another question of mine that a semigroup $S$ is called breakable if $xy \in \{x, y\}$ for all $x, y \in S$. Let's now say that $S$ is an ...
- 10.5k
4
votes
0
answers
80
views
Indecomposable injectives over Weyl algebras
Let $A=A_n(\mathbb{C})$ be the $n$-th Weyl algebra over the complex field. Then $A$ is a left Noetherian noncommutative ring. Is there a complete classification of indecomposable injective $A$-modules?...
- 615
10
votes
0
answers
248
views
What is the tiling semigroup for an einstein "hat" tiling?
My undergraduate dissertation was on inverse semigroups and the key text I used for it was Lawson's, "Inverse Semigroups: The Theory of Partial Symmetries". In said book, Lawson describes ...
- 379
0
votes
0
answers
63
views
A construction that sort of merges two semigroups to build a new one
Suppose $H$ and $K$ are semigroups and assume without loss of generality that (the underlying sets of) $H$ and $K$ are disjoint. We can then extend the operations of both $H$ and $K$ to a binary ...
- 10.5k