All Questions
Tagged with modules rt.representation-theory
121 questions
4
votes
0
answers
212
views
When does the canonical $t$-structure restrict to perfect complexes?
I am interested in non-Noetherian(!) rings such that the canonical $t$-structure on $D(R)$ (the derived category of left $R$-modules) restricts to perfect complexes i.e. to the subcategory of ...
- 16.9k
2
votes
1
answer
640
views
Simple modules for direct sum of simple Lie algebras
I think that the following statement is true, but I do not know how to prove it.
Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be two real simple Lie algebras. If $M$ is a (infinite dimensional) complex ...
- 951
2
votes
1
answer
143
views
A weak Schur's lemma for non-semisimple finite dimensional algebras
Let $B \subseteq C$ be an inclusion of finite dimensional (associative) algebras over a field $k$. Assume that $C$ is a free $B$-module. Let $\bigoplus_i U_i$ be
a decomposition of $B$ into ...
1
vote
0
answers
66
views
To find $\dim(D^{\mu})=\dim\frac{S^{\mu}}{S^{\mu}\cap S^{\mu\perp}}$
Let $S_6$ be the symmetric group of degree $6$ and $F$ be any finite field of characteristic $2.$ Then $2$-regular partition of $6$ are $(5,1)$, $(4,2)$ and $(3,2,1)$ . I have to find $$\dim(D^{\mu})=\...
- 141
1
vote
1
answer
153
views
About locally finite condition in category $\mathcal{O}^\mathfrak{p}$
Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$.
Denote by $\Phi$
the root system of $(\mathfrak{g},\mathfrak{h})$ and ...
- 1,875
13
votes
1
answer
748
views
Tilting Objects in BGG Categories $\mathcal{O}$
Let $\mathcal{O}$ be the BGG category $\mathcal{O}$ with respect to a finite dimensional semisimple Lie algebra $\mathfrak{g}$ and its Borel subalgebra $\mathfrak{b}$ (as define in this book by ...
- 295
1
vote
0
answers
77
views
n-Gorenstein algebras and tilting modules
Let $\Lambda$ be an artin algebra over a commutative artinian ring $R$. $\Lambda$ is said to be $n$-Gorenstein, for some natural number $n$, provided it have finite self-injective dimension at most $n$...
- 775
3
votes
1
answer
252
views
Higher Extension Group Question
Suppose we have an associative unital ring $R$, and we have an $R$-module $M$ with a length 3 socle filtration, i.e. write
$$soc(M) \text{ for the socle of } M,$$
$$soc^2(M) \text{ for the preimage ...
- 1,077
5
votes
3
answers
903
views
Irreducible representations and invariant subspaces
Consider two invertible n-by-n matrices, $n>2$, $X$ and $Y$ over a finite field $k$ (say for simplicity $k=\mathbb Z/ \mathbb Z_2$). Is there any reasonable way to check that there is no proper ...
1
vote
0
answers
77
views
Existence of a certain direct summand
Let $P$ be a cyclic group of order $p^n$. Let $M$ be a $\mathbb{Z}_p[P]$-module and $M_p :=\mathbb{Q}_p\otimes M$ be the associated $\mathbb{Q}_p[P]$-module. When will $M_p$ have a direct summand ...
- 1,283
5
votes
1
answer
192
views
Given a representation-infinite algebra, when is every AR component infinite?
Let $A$ be a finite dimensional algebra over an algebraically closed field $K$. The Auslander-Reiten quiver $\Gamma_A$ of $A$ is a means of presenting the category of finitely generated right $A$-...
- 482
7
votes
1
answer
349
views
Is there a converse to the Brauer–Nesbitt theorem?
$\DeclareMathOperator\Tr{Tr}$Say that we have an algebra $R$ over $\mathbb{C}$. If, for two finitely generated (edit: and semisimple) $R$-modules $M, N$ we know that $\Tr_M(r)=\Tr_N(r)$ for all $r\in ...
- 463
6
votes
3
answers
446
views
Is the category of symmetric bimodules over a commutative ring closed under extensions?
Let $A$ be a commutative ring, and consider the category of bimodules over $A$.
An $A$-bimodule $M$ is called symmetric if $a\cdot m = m \cdot a$ for all $a \in A$, $m \in M$.
Is the category of ...
- 61
10
votes
2
answers
503
views
A diagram for understanding action/coaction compatibility in a Yetter-Drinfeld module
For a Hopf algebra $H$ with antipode $S$, let $M$ be a left $H$-module with the action $h \otimes m \mapsto \rho(h,m)$, and also a left $H$-comodule with coaction $\delta \colon m \mapsto m^{(-1)} \...
- 1,161
5
votes
0
answers
856
views
Dual of representation is irreducible implies the representation is irreducible?
Suppose $V$ is an infinite dimensional $\mathbb{Q}_p$-representation of Lie algebra $\mathfrak{g}$ over $\mathbb{Q}_p$. If its dual representation $V^{\prime}$ is irreducible then, is it always true ...
- 685
4
votes
1
answer
419
views
Weights of BGG dual of Verma module
Consider the Verma module $M(\lambda)=\oplus_{\mu \in \mathfrak{h}^{\star}} M(\lambda)_{\mu}$. Denote its BGG dual by $M(\lambda)^{\vee}=\oplus_{\mu \in \mathfrak{h}^{\star}} M(\lambda)^{\star}_{\mu}$ ...
- 685
5
votes
1
answer
243
views
G-modules and ideals of secant varieties
Consider the action of $G = SL(n+1)$ on $\mathbb{P}^N$, and embed $\mathbb{P}^n$ in $\mathbb{P}^N$ via the degree two Veronese embedding. Let $V\subset\mathbb{P}^N$ be the corresponding Veronese ...
user117617
1
vote
1
answer
149
views
Finding modules to check for finite global dimension
Given a finite dimensional algebra $A$ and a generator-cogenerator $M$ and let $B:=End_A(M)$. $B$ has finite global dimension iff every $A$-module has finite $add(M)$-resolution dimension, which is ...
- 26.5k
1
vote
2
answers
337
views
Finding all submodules
Given a finite dimensional local commutative algebra over a finite field $K$ and a finite dimensional module $M$. What is the fastest/best way to obtain all submodule from $M$ using a Computer algebra ...
- 26.5k
3
votes
1
answer
247
views
Kazhdan-Lusztig Theorem for a special class of integral weights
This question may be seen as a follow up of this one.
Let $\mathfrak{g}$ be a rank $r$ simple complex Lie algebra, and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. We call $\alpha_i$ ...
- 457
8
votes
1
answer
704
views
Kazhdan-Lusztig theorem for composition factors of Verma modules
The Kazhdan-Lusztig Conjecture (which is actually a theorem) gives the character of some irreducible modules of a (say) simple complex Lie algebra $\mathfrak{g}$ in terms of characters of Verma ...
- 457
6
votes
1
answer
302
views
Endomorphism ring of bimodules
Given an algebra $A$ with a right $A$-module $M$ with $End_A(M) \cong A$.
Then we can view $M$ as a natural $A$-bimodule. When is $M$ as a bimodule indecomposable and what is its endomorphism ring as ...
- 26.5k
3
votes
1
answer
286
views
Why we study Endo-Trivial Modules?
I made the exact same question on MSE some days ago and there wasn't any response whatsoever so far, so thought probably I have to ask here to get an answer. So the question goes as follows:
Recently ...
- 441
4
votes
2
answers
288
views
The explicit indecomposable representations of (any) Euclidean quiver of type E
It is known that for any quiver $Q$ that is an orientation of $\tilde{\mathbb{E}}_8$, the hereditary path algebra $KQ$ ($K$ being an algebraically closed field) is tame (but not finite). That is, in ...
- 482
1
vote
0
answers
360
views
Property of the syzygy functor of $\operatorname{\underline{mod}} A$
Let $A$ be an artin algebra. We denote by $\operatorname{mod}A$ the category of finitely generated left $A$-modules. We denote by $[A]$ the ideal of morphisms between $A$-modules which factor through ...
- 775
3
votes
1
answer
361
views
Whether hom functor and socle (radical) are commutative?
Let $A$ be an algebra. We know that $Soc M =\oplus _A Soc M_{\alpha}$ and $Rad M =\oplus _A Rad M_{\alpha}$ if $M= \oplus _A M_{\alpha}$ as $A$-modules.
Now let $M,N$ be $A$-modules, $\Lambda:=End_A(...
- 775
2
votes
1
answer
119
views
How to get that $\Omega^2_{\Lambda}(N) \cong \textrm{Hom}_A(M,Y)$?
Let $A$ be an algebra over a field k. A module $_AM$ is called a generator if $\textrm{add}(A) \subseteq \textrm{add}(M)$, a cogenerator if $\textrm{add}\big(D(A)\big) \subseteq \textrm{add}(M)$. $M$ ...
- 775
4
votes
1
answer
382
views
Non-existence of projective covers
I didn't get the argument of Example 7.3.11, page 123, from the representation theory book of Peter Webb, available also online at:
http://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf
In ...
- 137
4
votes
1
answer
241
views
Questions in the paper "The category of good modules over a quasi-hereditary algebra has almost split sequences"
I am reading the paper "The category of good modules over a quasi-hereditary algebra has almost split sequences", the link is here:https://pub.uni-bielefeld.de/publication/1780235.
In the paper, $A$ ...
- 775
3
votes
1
answer
777
views
Long exact sequence of free modules and rank inequality
Given a long exact sequence of free (left) modules $M_i$ of finite rank $k_i$ over possibly non-commutative ring $R$:
$\dots \to M_{i-1}\to M_i \to M_{i+1}\to \dots$
What is the condition on $R$ ...
- 31
8
votes
1
answer
351
views
Irreducible $S_n$-modules and $S_n$-actions on projective spaces
Let $V$ be an $(N+1)$-dimensional vector space with an action of the symmetric group $S_n$, such that $V$ is an irreducible $S_n$ -module.
Let $\{p_1,...,p_h\}\in \mathbb{P}(V)$ be $h\geq N+2$ ...
user91659
0
votes
1
answer
235
views
Questions about Lowey length
Let $\Lambda$ be an artin algebra.
If $M$ is a finitely generated $\Lambda$-module with Loewy length 2 and finite projective dimension. How to get the exact sequence $$0 \rightarrow A \rightarrow P/...
- 775
6
votes
1
answer
361
views
How to check whether a module is an n-th syzygy
Given a finite dimensional algebra $A$, define $\Omega^{n}(mod-A)$ (modules here are always finite dimensional) to be the full subcategory of projective modules or modules $M$ such that $M \cong \...
- 26.5k
1
vote
0
answers
33
views
How to get that $add(E)=add(\nu(E))$ equivalent to $add(soc(E))\cong add(top(E))$?
Let $A$ be a finite dimensional algebra over a field K. $E$ is an injective $A$ module. $\nu=DHom_A(-,A)$ is the Nakayama functor. $add(E)$ is the full subcategory of $A$-module consisting of all ...
- 775
2
votes
0
answers
105
views
Why $T'$ dosen't have projective direct summand?
Let $A$ be a k-algebra, where k is a fixed field. Let $S$ be a simple, non-injective $A$-module such that $Ext^{i}_{A}(S,S)=0$ for $1 \leq i \leq n$. Let $P(S)$ be the projective cover of $S$, and let ...
- 775
2
votes
1
answer
263
views
Nonnegatively graded algebra $A$ finitely generated as $k$-algebra iff $A_0$ finitely generated, $A_{>0}$ finitely generated as $A$-module?
This is related to my question here. My question is as follows. How do I see that a nonnegatively graded algebra $A$ is finitely generated as a $k$-algebra if and only if $A_0$ is finitely generated ...
- 349
3
votes
1
answer
173
views
$M$ is finitely generated as $A$-module iff $M/A_{>0}M$ is finitely generated as $A$ module?
Let $A$ be a nonnegatively graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A_{>0}M$ is a graded $A$-submodule of $M$.
(Here, given a nonnegatively graded algebra $A$, we've ...
- 349
0
votes
0
answers
68
views
Is there a simple module $S$ satisfies the following conditions?
Let $A$ be a k-algebra,where k is a fixed field. {$x_1,x_2, \cdots,x_n $} is a complete set of primitive orthogonal idempotents of $A$. $M$ is a left $A$-module such that $x_iM=0$ for $i=1,2,\cdots,n-...
- 775
2
votes
1
answer
450
views
The definitions of a generator module?
Recently I have seen two definition of a generator module:
1) A generator for a category $C$ is an object $G$ such that for any two parallel morphisms $f,g:X \rightarrow Y$ with $f \neq g$, then ...
- 775
5
votes
1
answer
394
views
Classification of indecomposable modules in tame hereditary algebras
An algebra $A$ is said to be tame if the isomorphism classes of indecomposable $A$-modules in each dimension occur in a finite number of 1-parameter families. $A$ is said to be of finite ...
- 482
1
vote
1
answer
126
views
Action is determined by a braiding
Let $H$ be a bialgebra over $\mathbb{C}$. Suppose that $V$ is a left $H$-comodule and $W$ is an $H$-module. Then we can defined map $\Psi$ by
\begin{align}
& \Psi: V \otimes W \to W \otimes V, \\
&...
- 6,201
1
vote
1
answer
271
views
Rank of a locally free $\mathbb Z[G]$-module
This is a basic question. Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$, i.e. is locally free.
...
- 255
3
votes
1
answer
151
views
Unclear asymmetry in Lie-algebra module structure on space of linear transformations Hom(V,W)
Let $L$ be a (finite dimensional) Lie-algebra. Let $V, W$ be finite-dimensional vector spaces. If $V,\; W$ are in addition $L$-modules (see, e.g., 6.1 in Humphreys Introduction to Lie Algebras), then ...
- 583
7
votes
1
answer
455
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 ...
- 11.2k
2
votes
0
answers
202
views
Could Partial Tiltings be studied as Almost Complete Tiltings?
The first part of what follows is a brief recap of the definitions, setting and motivations for my questions. Experts can find the questions at the end.
Here $k$ denotes an algebraically closed field,...
- 493
9
votes
1
answer
472
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 ...
1
vote
0
answers
120
views
Irreducible representations of quantum affine algebras
The finite dimensional simple modules of quantum affine algebras are parameterized by Drinfeld polynomials. Some other modules of quantum affine algebras are studied in the paper. Some infinite ...
- 6,201
8
votes
1
answer
387
views
Jordan-Hölder-like statements for modules with $\Delta$-filtrations over a quasihereditary algebra
Definitions
Let $A$ be an Artin algebra (for instance, take $A$ to be a finite dimensional algebra over some field) and label the isomorphism classes of simple $A$-modules by the elements of a ...
- 425
2
votes
1
answer
268
views
Constructing a simple $A$-module
Let $n \ge 2$, and let $A$ be the (unital and associative, but noncommutative) $\mathbb{C}$-algebra with generators $x_1, \dots, x_n$ and relations $x_ix_j + x_j x_i = 2\delta_{ij}$. What is a ...
2
votes
1
answer
395
views
Projectivity of torsion-free modules over integral group rings
Let $G$ be a torsion-free group and assume that the integral group ring $\mathbb{Z}G$ is torsion-free as well. Let $M$ be a torsion-free, finitely generated module over $\mathbb{Z}G$.
If we assume ...
- 2,998