All Questions
Tagged with rt.representation-theory ra.rings-and-algebras
424 questions
4
votes
1
answer
684
views
Quadratic algebras and Koszul algebras
Let $A$ be a quadratic algebra and $B$ the Ext-algebra of $A$.
In case $A$ is a Koszul algebra, we should have that the global dimension of $A$ plus one is equal to the Loewy length of $B$ (is there a ...
- 26.5k
12
votes
1
answer
564
views
Representations of degenerate Clifford algebras
Given a real finite-dimensional vector space $V$ with a symmetric bilinear form $b$, we define the Clifford algebra $Cl(V,b)$ as the quotient of the tensor algebra $\bigotimes V$ by the two sided ...
- 33.3k
3
votes
0
answers
49
views
$c$-matrix reduction in hereditary algebras
Let $k$ be an algebraically closed field, $Q$ be a finite connected quiver and $Q'$ a subquiver of $Q$. Let $C$ be the $c$-matrix of a chamber of the scattering diagram/semi-invariant picture of $kQ$. ...
- 417
3
votes
1
answer
175
views
If the sum of a right (principal) ideal with a left one contains an invertible element and the product is zero then do they contain idempotents?
I am trying to solve a problem on additive categories, that gives the following question on (non-commutative unital associative) rings: if for elements $a$ and $b$ of a ring $R$ we have $ab=0$ and $a+...
- 16.9k
3
votes
0
answers
71
views
$\Omega^2(S) \cong \tau(S)$ for simple modules
Let $A$ be an Artin algebra with $\Omega^2(S) \cong \tau(S)$ for each simple module $S$. Is $A$ a quasi-Frobenius (=selfinjective) algebra?
Here $\tau$ denotes the Auslander-Reiten translate, which is ...
- 26.5k
1
vote
0
answers
146
views
Characterisation of projective modules over tensor products of fields
Let $k$ be a field and $L_1$ and $L_2$ finite field extensions of $k$.
Let $A:=L_1 \otimes_k L_2$ as an algebra.
Question:
Given a finitely generated $A$-module $M$, do we have that $M$ is ...
- 26.5k
4
votes
2
answers
337
views
When is $\Omega^1$ an equivalence?
Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules.
When is the functor $\Omega^1 : \underline{...
- 26.5k
9
votes
0
answers
371
views
Duality between coalgebras and (pseudocompact) algebras - uniqueness
The following result is well-known. It can for example be found in [Iovanov: The representation theory of profinite algebras, Theorem 1.0.2]. For definitions, see below.
Let $k$ be a field. The ...
- 2,450
2
votes
0
answers
201
views
Homological conjecture for finite dimensional algebras
In the theory of finite dimensional algebras there are many homological conjectures. When working over an algebraically closed field it is well known that any such algebra is Morita equivalent to a ...
- 26.5k
3
votes
0
answers
97
views
Infinite-dimensional wild commutative algebras with non-trivial units
Let $A$ be an arbitrary (not necessarily finite-dimensional) associative algebra over an algebraically closed field $K$, and let $\mathrm{fin\,}A$ denote the category of finite-dimensional $A$-modules....
- 482
4
votes
1
answer
159
views
Question on $n$-regular modules
Let $A$ be finite dimensional connected algebra. A simple module $S$ is called $n$-regular in case $pd(S)=n$, $Ext_A^i(S,A)=0$ for $i=0,1,...,n-1$ and $Ext_A^n(S,A)$ being a simple $A$-left module. ...
- 26.5k
5
votes
1
answer
353
views
Existence of non-trivial reflexive modules
Recall that a module $M$ over a ring $R$ is reflexive in case the natural evaluation map $f_M:M \rightarrow M^{**}$ (where $M^{*}=Hom_R(M,R)$) is an isomorphism, where $f_M(m)=g$ with $g(h)=h(m)$, ...
- 26.5k
3
votes
1
answer
211
views
Coinduced modules in the BGG category $\mathcal O$ over complex semisimple Lie algebras
For a given finite-dimensional complex semisimple Lie algebera $\mathfrak g$, we fix Cartan $\mathfrak h$ and Borel subalgebras $\mathfrak b$, then we have the BGG category $\mathcal O$. As usual, we ...
- 55
38
votes
0
answers
1k
views
Groups whose complex irreducible representations are finite dimensional
By a complex irreducible representation of a group $G$, I mean a simple $\mathbb CG$-module. So my representations need not be unitary and we are working in the purely algebraic setting.
It is easy ...
- 38.6k
5
votes
2
answers
332
views
Questions on weakly symmetric algebras
A finite dimensional algebra $A$ over a field $K$ is called weakly symmetric in case $soc(P)=top(P)$ for every indecomposable projective module $P$ and it is called symmetric in case $D(A) \cong A$ as ...
- 26.5k
2
votes
0
answers
74
views
Units in the (stable) center of a Frobenius algebra [duplicate]
Let $A$ be a Frobenius algebra with center $Z(A)$ and $I\subset Z(A)$ the ideal of elements in the image of some $A$-bimodule map $A\rightarrow A\otimes A\rightarrow A$, where the second map is ...
- 15.2k
2
votes
1
answer
203
views
$Ext_A^1(J,J)$ for the Jacobson radical $J$ of an algebra $A$
Let algebras be finite dimensional over a field $K$ and let $J$ denote the Jacobson radical (this is the intersection of all maximal right ideals) of an algebra. Being hereditary means that the ...
- 26.5k
4
votes
0
answers
218
views
Conjugacy class representatives for the automorphism group of a finite abelian group
Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$.
In fact, it's not important that I have exactly one representative from ...
- 286
7
votes
1
answer
239
views
Is being a Frobenius algebra a rare condition for local algebras?
Let $U_{r,l,q}$ be the set of finite dimensional local algebras $A$ over a finite field with $q$ elements such that $J/J^2$ is $r$-dimensional for a number $r \geq 2$ and such that $J^l=0$ for the ...
- 26.5k
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
5
votes
1
answer
271
views
Do all finite-dimensional division algebras appear as Wedderburn factors of rational group rings?
Suppose that $D$ is a division algebra that is finite-dimensional over $\Bbb Q$, does there exist a finite group $G$ such that one of the factors in the Wedderburn decomposition of $\Bbb Q[G]$ is a ...
- 804
1
vote
0
answers
80
views
When is a stable endomorphism ring selfinjective?
Let $A$ be a local symmetric finite dimensional algebra and $M$ an $A$-module with at least two non-isomorphic indecomposable non-projective summands.
In case $\Omega^1(M) \cong M$ in the stable ...
- 26.5k
2
votes
0
answers
84
views
Super global dimension
Let $R$ be a ring of finite global dimension. Define the small super global dimension as $sgl(R):= \sup \{ pd(X)+id(X) | X \in mod-R$ and indecomposable $\}$.
Here $id(X)$ stands for the injective ...
- 26.5k
1
vote
0
answers
64
views
Questions on holonomic modules
An Auslander-Gorenstein ring is a noetherian ring R that has finite left and right selfinjective dimension and such that $fd(I_i) \leq i$ for all $i \geq 0$ for an injective coresolution of the ...
- 26.5k
3
votes
0
answers
169
views
Hochschild homology and Chern character quiver with potential
I am a beginner in quiver theory so this question might not be suitable for mathoverflow.
Let $(Q,W)$ be a quiver with potential and let $\Gamma$ be the Ginzburg DG-algebra associated to $(Q,W)$. Is ...
- 7,300
3
votes
0
answers
54
views
Ext for simple modules in selfinjective algebras
Let $A$ be a finite dimensional symmetric algebra (given by a connected quiver and non-semisimple) and $S$ a simple $A$-module. (I am more interested in symmetric algebras, but selfinjective examples ...
- 26.5k
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
3
votes
0
answers
208
views
A new characterisation of hereditary algebras?
Let $A$ be a quiver algebra with global dimension at most two (or more generally finite global dimension) and $A^e=A^{op} \otimes_K A$ be its enveloping algebra.
Guess:Is $A$ hereditary if and only ...
- 26.5k
1
vote
1
answer
140
views
Is the Cartan permanent odd for finite global dimension?
Define the Cartan permanent of a finite dimensional algebra as the permanent of the Cartan matrix.
Is the Cartan permanent of a finite dimensional algebra with finite global dimension always an odd ...
- 26.5k
5
votes
1
answer
1k
views
Periods in the trivial extension algebra of the incidence algebra of the divisor lattice
Definition of $C_L$ for people who like number theory:
Let $m$ be a number with prime factorisation $m=p_1^{n_1} ... p_r^{n_r}$ with $n_i>0$. Define $I_m$ to be the incidence algebra of the ...
- 26.5k
9
votes
0
answers
259
views
Name for rings with $R \cong R^{\mathrm{op}}$
Is there a name for rings that are isomorphic to their opposite ring $R^{\mathrm{op}}$ in the literature? I'm especially interested for the class of Artin algebras.
- 26.5k
1
vote
0
answers
48
views
Is simreflexive left-right symmetric?
Call an Artin algebra (we can assume that it is basic) $A$ simreflexive in case every simple A-module is reflexive. Is A simreflexive iff the opposite algebra $A^{op}$ is simreflexive? Or equivalently ...
- 26.5k
0
votes
0
answers
66
views
Tameness of the trivial extension of a finite dimensional algebra
The trivial extension T(A) of a (finite dimensional) algebra A is representation-finite if and only if the algebra A is iterated tilted of Dynkin type.
Questions:
Is there a similar classification ...
- 26.5k
2
votes
0
answers
127
views
Multiplicative subgroups of $GL(V)$ which are almost additively closed
Edit:
According to comments of YCor and Vincent, I revise the question.I appreciate their comments:
Let $G$ be a group.
We say that $G$ is a special group if it is isomorphic to a subgroup $H$ ...
- 356
6
votes
0
answers
79
views
Canonical forms for representations of tame algebras
Recently I have been studying the representation type of algebras (tame-wild dichotomy). A question that I feel is probably well known but that I cannot answer is the following:
Can I define tame ...
- 61
3
votes
0
answers
427
views
When is the stable category abelian
For which Artin algebras $A$ is the stable module (that is the module category modulo projectives) category of finitely generated modules abelian?
If you like you may take rings that are not Artin ...
- 26.5k
4
votes
0
answers
228
views
Question on $n$-torsionless modules
Let $A$ be a finite dimensional algebra. Recall that a module $M$ is called $n$-torsionfree in case $Ext_A^i(D(A),\tau(M))=0$ for all $i=1,...,n$ when $\tau$ denotes the Auslander-Reiten translate. ...
- 26.5k
7
votes
1
answer
730
views
Where is my mistake in calculating duals?
I'm confused and probably have a thinking error. Exercise 6 on page 420 of Lam's Lectures on Modules and Rings says essentially:
Let $R$ be a ring and $C$ a cyclic right $R$-module: $C=R/A$ with ...
- 26.5k
2
votes
0
answers
75
views
Double dual of the simple module in local algebras
Let $A$ be a local Artin algebra that is not selfinjective with simple module $S$.
Questions:
Can $S^{**}$ be indecomposable?
$S^{**}$ be somehow generally be described (for example as a ...
- 26.5k
1
vote
1
answer
174
views
Reference for a result of Auslander about the global dimension
One of Auslanders famous theorems is that he proved that the global dimension of a semiprimary ring is equal to the maximum of the projective dimensions of the simple modules of the ring. This result ...
- 26.5k
6
votes
1
answer
290
views
(Non-)formality for ADE preprojective algebras
Given a quiver $Q$, I can associate to $Q$ a certain 2-Calabi-Yau (dg-)algebra $\Gamma_Q$ by a 2-dimensional version of the "Ginzburg dga" construction: i.e., start by doubling $Q$, and then impose ...
- 423
5
votes
2
answers
226
views
Algebras with all simples reflexive
Let $A$ be a ring. Recall that a module $M$ is called reflexive in case the canonical evaluation map $f_M : M \rightarrow M^{**}$ is an isomorphism. Here $(-)^{*}$ denotes the functor $Hom_A(-,A)$.
In ...
- 26.5k
2
votes
1
answer
98
views
Reflexive modules up to multiplicity
Call an indecomposable module $M$ over a ring $A$ (restrict to finite dimensional algebras if you like or if it helps) $n$-almost reflexive in case $M^{**} \cong nM$, when $(-)^{*}=Hom_A(-,A)$ and $nM$...
- 26.5k
3
votes
0
answers
169
views
Characterisation of reflexive modules for general rings
A module $M$ over a general ring A is called reflexive in case the canonical evaluation map $e_M: M \rightarrow M^{**}$ is an isomorphism, when $(-)^{*}:=Hom_A(-,A)$. Is $M$ reflexive iff $M \cong M^{*...
- 26.5k
4
votes
1
answer
615
views
Characterisation of reflexive modules
Let $A$ be a semiperfect noetherian ring.
A module $M$ is called reflexive in case the canonical map $f_M: M^{**} \cong M$ is an isomorphism, when $(-)^{*}:=Hom_A(-,A)$. This is equivalent to say that ...
- 26.5k
3
votes
1
answer
319
views
Functions on a field representable by Hahn series?
It is well known (see here for example) that a function over $\mathbb{R}$ is representable by a power series iff its analytic continuation to $\mathbb{C}$ is holomorphic on some open subset of $\...
- 10.1k
1
vote
0
answers
155
views
Deduce Schur-Zassenhaus theorem from Wedderburn-Malcev theorem
Is it possible to derive Schur-Zassenhaus theorem from Wedderburn-Malcev theorem? For the existance part I have the following idea: Let $G$ be a finite Group and $N$ an abelian Hall $p$-subgroup. Let $...
- 581
5
votes
2
answers
360
views
Injective dimension of the Jacobson radical and global dimension
Given a finite dimensional algebra $A$ with Jacobson radical $J$.
Is the global dimension of $A$ equal to the injective dimension of $J$?
Together with Xiao-Wu Chen and Srikanth Iyengar we proved this ...
- 26.5k
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
13
votes
0
answers
355
views
Analog of Haar element in an algebra
In a Hopf algebra $H $ (over some field $ k $), there is the notion of a Haar element $ h \in H$. This is an element of the algebra which has the property that if $ V $ is a representation of $ H $, ...
- 4,612