All Questions
393 questions with no upvoted or accepted answers
3
votes
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387
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Software for Combinatorial Algebra sought
I am looking for software which helps me do straightforward tasks in combinatorial algebra. Let me give an example of what I mean by a straightforward task:
I have two graded (generally ...
3
votes
0
answers
311
views
Tensor power- Notation question
Hi everyone
I have a notational question, which is written usually in papers, but I can not figure it out what could be. Let $M$ be an $A$-module. I have seen this notation
$$M^{\otimes -n}$$
I ...
- 2,508
3
votes
0
answers
163
views
Pulling out factors in a Noetherian Domain
Let $R$ be a Noetherian domain (not-necessarily commutative), and let $S$ be a Noetherian subring of $R$. An element $r\in R$ is left $S$-irreducible if, for any $s\in S$ and $r' \in R$ with $sr'=r$, ...
- 13k
2
votes
0
answers
92
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Geometric interpretation of flags and the role of the rook monoid and Kazhdan–Lusztig theory in $M_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, $B$ be its Borel subgroup, and $P$ a parabolic subgroup. The space $G/B$ corresponds to complete flags in $ \mathbb{C}^n$, and $G/P$ corresponds to partial flags. The ...
- 141
2
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0
answers
95
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Free, easy-to-use program for noncommutative algebra over finite fields
I am looking for a computer program that can handle computations in noncommutative algebra over a finite field of prime order $p$.
My requirements are:
The program should be free, as I do not have ...
- 1,013
2
votes
0
answers
66
views
Noncommutative transcendence degree of representation algebras
Let $G$ be a reductive group, for example $\text{GL}_n(\mathbb{C})$. Let $V$ denote its defining representation, and let $R$ denote the tensor algebra on the irreducible representations of $G$. It may ...
- 2,907
2
votes
0
answers
69
views
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
2
votes
0
answers
80
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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
2
votes
0
answers
91
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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
2
votes
0
answers
144
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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
2
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71
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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
2
votes
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answers
176
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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
2
votes
0
answers
111
views
Correct notion of "connected" for dga of bundle-valued forms
Consider a vector bundle $E$ over a manifold $M$ with flat connection, $\nabla$. From this data I can form the associative/unital differential graded algebra $\mathcal{A} = \left(\Omega^{\bullet}(M, ...
- 1,611
2
votes
0
answers
68
views
Semigroups related to iterated orthogonal complement
Let $R\subset V\times V$ be a relation on a set $V$. For a subset $S\subset V$, define its orthogonal complement with respect to $R$
as
$$S^l:=\{ x: \forall y\in S\ \ (x,y)\in R\},\ \ S^r:=\{y: \...
- 513
2
votes
0
answers
64
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A particular generalization of free partially commutative monoids
A trace monoid, or free partially commutative monoid, is one with the presentation $\langle \Sigma \mid a_1b_1 = b_1a_1, \dots, a_nb_n = b_na_n\rangle$. The theory of trace monoids has been well ...
- 21
2
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answers
73
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What should I call a log scheme with free reduced monoids?
This is a terminology question about a class of log varieties.
Given an fs (fine and saturated) log variety $(X, M)$ (for $M$ the defining sheaf of monoids), any geometric point $x\in X$ has a ...
- 7,962
2
votes
0
answers
161
views
Embedding a monoid into a group via its monoid ring
Suppose I have a monoid $(M,\, \cdot,\, e)$ equipped with a monoid homomorphism $\textrm{length} : M \rightarrow \mathbb{N}_+$ into the monoid of natural numbers under addition where $e$ is the only ...
user491484
2
votes
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answers
181
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So many types of subwords! How are they called?
Let $\mathscr F(X)$ be the free monoid on an alphabet $X$, the carrier set of $\mathscr F(X)$ being the union of $X^{\times k}$ (the Cartesian product of $k$ copies of $X$) as $k$ ranges over $\mathbb ...
- 10.5k
2
votes
0
answers
172
views
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^* ...
- 889
2
votes
0
answers
88
views
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, \...
- 889
2
votes
0
answers
101
views
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 \...
- 121
2
votes
0
answers
67
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Type of numerical semigroups is not bounded when embedding dimension is $\geq 4$
I am currently studying numerical semigroups. I know that there is no upper bound for the type of a numerical semigroup with embedding dimension greater or equal than $4$. There is a famous example by ...
- 121
2
votes
0
answers
174
views
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 ...
- 889
2
votes
0
answers
108
views
Left-elements of a numerical semigroup generated by two elements
A numerical semigroup $S$ is a semigroup in $\mathbb{N}$ such that $\mathbb{N}\backslash S$ is finite. It is known that there exists always a set $M$ such that an element in $S$ can be expressed as a ...
- 31
2
votes
0
answers
73
views
Nonzero idempotents in compact semitopological semigroups with zero
Let $S$ be a compact semitopological semigroup. Then, $S$ contains minimal idempotents by Ellis' theorem.
Ellis' Theorem: In a compact left-topological (resp. right-topological) semigroup, every ...
- 2,605
2
votes
0
answers
145
views
Semigroup ideals of a ring or an algebra
Let $R$ be a ring or an algebra. Suppose $B\subseteq R$ satisfies the property $BR \subseteq B$ and $RB\subseteq B$. Is there a general theory of subsets with this property in a ring (resp. algebra) ...
- 2,605
2
votes
0
answers
119
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The "matrix direct sum" monoid modulo unitary equivalence
Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the ...
- 4,943
2
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0
answers
203
views
Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?
It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G.
More generally here (MO275769) Qiaochu Yuan ...
- 24.9k
2
votes
0
answers
98
views
Do $r(a) \leq^\oplus R$ and $r(a) = r(a^2)$ imply $r(a) = eR$ and $aR \subseteq (1-e)R$ for some idempotent $e$?
Let $R$ be a (commutative or non-commutative, associative) ring with unity, and let $a$ be an element of $R$ such that $r(a) = r(a^2)$, where $r(\cdot)$ denotes a right annihilator. It follows that $r(...
- 10.5k
2
votes
0
answers
230
views
Reference request: the UEA of the LR-algebra of tangent vector fields on a smooth manifold coincides with the derivation ring and the ring of diff ops
Let $\mathcal{M}$ be a smooth real manifold and let $A:= \mathcal{C}\left(\mathcal{M}\right)$ be the real algebra of smooth functions on $\mathcal{M}$.
Recall from McConnell, Robson, Noncommutative ...
- 718
2
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0
answers
90
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,...
- 81
2
votes
0
answers
74
views
Terminology and notation for generated subgroups
I would like to think about formation of the smallest subgroup (or monoid, or whatever) $H$ of $G$ containing two given subgroups $A$ and $B$ as an operation on subgroups, and I wonder if there is a ...
- 12.5k
2
votes
0
answers
270
views
Road map: beyond Artin-Wedderburn theorem
For a noncommutative semisimple ring $R$, its structure and its category of representations can be largely understood using Artin-Wedderburn theorem. Such structure theory is useful, for example, in ...
- 5,230
2
votes
0
answers
89
views
Semigroups associated to binary necklaces and their semigroup algebra
I came across the following semi-group and the associated finite dimensional semi-group algebras over a field $K$ (which are Nakayama algebras) as they have very nice homological properties. My ...
- 26.5k
2
votes
0
answers
96
views
Non-commutative version of the order dimension of a poset
I view the order dimension of a poset $P$ as an inherently commutative notion. On the one hand, it can be defined via realizers, which I find fairly intuitive from an order-theoretic viewpoint. On the ...
- 51
2
votes
0
answers
50
views
Existence of nontrivial transfinite divisibility in $R$-modules
Let $R$ be a unital, possibly noncommutative ring and $s \in R$. For a right $R$-module $M$, define $Ms = \{ms \mid m \in M\}$; this is an additive subgroup of $M$, which is a module over the ...
- 64k
2
votes
0
answers
132
views
Rings whose finitely-generated modules are co-hopfian
Let $A$ be a unital, possibly noncommutative ring. Dischinger showed [1] that the following are equivalent:
For every $a \in A$, there exists $n \in \mathbb N$ such that $a^n A = a^{n+1} A$;
For ...
- 64k
2
votes
0
answers
98
views
Dimension of center of $k[G]/\mathrm{rad}k[G]$ when characteristic of $k$ divides the order of $G$
Let $G$ be a finite group and consider $k[G]$ where $k$ is a field. In the scenario where $\mathrm{char}(k)$ divides $|G|$, how can one show that the dimension of $Z(k[G]/\operatorname{rad}k[G])$ is ...
2
votes
0
answers
116
views
Bijection between monoid of nilpotent decreasing self-maps and local subsemigroup
Let $\mathcal{C}_X\cong\mathcal{C}_n$ be the monoid of self-maps $\alpha$ of $X=\{1\dots,n\}$ that are order-preserving ($\forall x,y$, $x\le y$ $\Rightarrow$ $\alpha(x)\le\alpha(y)$) and decreasing ($...
- 355
2
votes
0
answers
33
views
On the number of connected functional digraphs recoverable from the preimage set size structure
I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$.
For example,
$P_j=\left[f^{-j}(...
- 23
2
votes
0
answers
91
views
Is the natural action of the monoid of endomorphisms is a complete invariant for group?
Let $\alpha$ and $\beta$ be actions of semigroups $A$ and $B$ on sets $X$ and $Y$ respectively. Recall that $\alpha$ and $\beta$ are called isomorphic if there exists an isomorphism $\phi$ between ...
- 6,962
2
votes
0
answers
56
views
Non-singular rings which are Rickart
A ring $R$ is said to be a right Rickart ring if the right annihilator of any element in $R$ is of the form $eR$ for some idempotent $e \in R$.
It turns out that a ring $R$ is right Rickart iff every ...
- 1,065
2
votes
0
answers
48
views
Compute irreducibles of monoid
Given $n > 0$ and $w \in \mathbb{Z}^n$. Is there an efficient algorithm to compute the set of irreducible elements of the monoid $M_w = \{x \in \mathbb{N}^n \mid \langle x,w\rangle = 0 \}$?
Here, ...
2
votes
0
answers
70
views
Embedding problems on quantum groups？
We work over the field of complex numbers.
We have known that Lie algebra of type $A_2 $is a subalgebra of type $G_2$. However, when we consider their quantum groups, is this true i.e. does there ...
- 165
2
votes
0
answers
60
views
Integrals in noncommutative graded algebras which are not necessarily Hopf
Let $\mathbf{k}$ be a field. Let $A$ be a finite dimensional $\mathbb{Z}_{\geq 0}$-graded $\mathbf{k}$-algebra such that $A^0=\mathbf{k}1$. Let $m$ be the maximal non-negative integer such that $A^m\...
- 1,066
2
votes
0
answers
60
views
Are there finitely-presented astral monoids?
We say a semigroup $S$ is $k$-astral if there exists a finite set $F \subset S$ such that
whenever $s_1, s_2, ..., s_k \in S$ there exists $s \in S$ such that $\forall i: s_i \in sF$. Say $S$ is ...
- 6,652
2
votes
0
answers
103
views
Lattices with trivial coinvariants for finite groups
Let $G$ be a finite group. A $\mathbb{Z}G$-lattice is a $\mathbb{Z}G$-module that is (as abelian group) a free abelian group of finite rank.
Question: Is there a finite group $G$ and a $\mathbb{Z}...
- 2,160
2
votes
0
answers
42
views
Concerning $(x,y) \mapsto (x^{\frac{n}{r}+1}y + A,\mu x^{-\frac{n}{r}}+B)$
Let $r \in \mathbb{N}-\{0\}$.
Commutative case:
Let $f : (x,y) \mapsto (p,q)$ be a map from $\mathbb{C}[x,y]$ to $\mathbb{C}[x^{1/r},x^{-1/r},y]$ satisfying the following two conditions:
(i) $\...
- 2,837
2
votes
0
answers
44
views
Partially commutative elements in powers of augmentation ideal
Let $\vartheta$ a relation of parcial commutation over a set $X,$ and consider the respective free parcially commutative group $F(X, \vartheta).$ Let $K[F(X, \vartheta)]$ the parcially commutative ...
- 121
2
votes
0
answers
92
views
Monoidal structure on left dg-modules over a brace algebra
Relating to my other question: Modules over Hopf Algebras and $E_2$-algebras
Preliminary: Let $A$ be an associative dg-algebra that is also an algebra over the brace operad. Let $M$ and $N$ be left ...
- 527