All Questions
Tagged with rt.representation-theory ra.rings-and-algebras
424 questions
4
votes
1
answer
135
views
Global splitting field for algebras
Let $A$ be a finite dimensional algebra.
A field $K$ is a splitting field for an indecomposable $A$-module $M$ in case the local algebra $End_A(M)/(rad(End_A(M))$ is 1-dimensional.
$K$ is called a ...
- 26.5k
6
votes
1
answer
203
views
Question on a subcategory being extension-closed
In the article "Homological theory of noetherian rings" by Idun Reiten from 1996, it was stated that it seems to be not known whether the subcategory $\operatorname{Tr}(\Omega^i(\mathrm{mod}\...
- 26.5k
6
votes
0
answers
182
views
On properties of an algebra as a bimodule
Let $A$ be a two-sided artinian ring.
Recall that a module $M$ is said to have dominant dimension at least $n$ in case the terms $I_i$ in the minimal injective coresolution of $M$ are projective for $...
- 26.5k
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 ...
4
votes
0
answers
141
views
Frobenius algebras of small dimensions
In Classification of commutative Frobenius algebras , Jeremy Rickard showed that there are infinitely many commutative (local without loss of generality) Frobenius algebras of vector space dimension ...
- 26.5k
3
votes
1
answer
104
views
Irreducibility of product bicomodules
Let $H$ be a Hopf algebra, and $V$ and $W$ a left, and a right,
$H$-comodule respectively. The tensor product
$$
V \otimes W
$$
has an obvious $H$-$H$-bicomodule structure.
If $V$ and $W$ are ...
- 1,144
2
votes
0
answers
110
views
Generalising injective modules
Free modules over a ring generalise to projective modules over a ring, which generalise to flat modules, which generalise to torsion free modules:
$$
\textrm{free} \to
\textrm{projective}
\to
\textrm{...
3
votes
0
answers
56
views
Weakly symmetric rings and derived equivalences
A ring $R$ with Jacobson radical $J$ is called Frobenius in case $R/J \cong soc(R)$ as left and right $R$-modules and weakly symmetric in case we even have $R/J \cong soc(R)$ as $R$-bimodules.
...
- 26.5k
3
votes
0
answers
39
views
Positive roots of the Tits unit form and dimension vectors
Let $A$ be a finite dimensional quiver algebra such that two indecomposable modules are isomorphic iff their dimension vectors are the same. Let $T_A$ be the tits unit form of $A$ and $r_A$ the set of ...
- 26.5k
3
votes
1
answer
98
views
On algebras where indecomposable modules are determined by their dimension vectors
Let $A$ be a finite dimensional quiver algebra such that any two indecomposable modules with the same dimension vector are isomorphic.
Question: In case $A$ has $n$ simple modules and finite global ...
- 26.5k
1
vote
1
answer
178
views
Representation of algebras as bounded nilpotent operators
Let $A$ be a real/complex algebra (just a real/complex vector space with a multiplication; none PI's are required). Let $\mathcal{H}$ be a real/complex Hilbert space. Let $\operatorname{B}(\mathcal{H})...
3
votes
0
answers
173
views
Comparing zeroth Hochschild homology and cohomology for algebras
Let $A$ be a finite dimensional algebra and $[A,A]$ its commutator and $Z(A)$ its center.
Question 1: Do we have $dim(A/[A,A]) \geq dim(Z(A))$?
Note that A/[A,A] is the zeroth Hochschild homology ...
- 26.5k
6
votes
0
answers
57
views
A characterisation of weakly symmetric algebras
A finite dimensional algebra $A$ over a field $K$ is called a Frobenius algebra in case there exists a $K$-linear map $f: A \rightarrow K$ such that $ker(f)$ contains no non-zero right ideal of $A$ (...
- 26.5k
2
votes
1
answer
196
views
Weight space dimension of the fundamental representation $\pi_n$ for type $C_n$
Will the fundamental representation $\pi_n$ of type $C_n$, for $n > 3$, have weight spaces of dimension greater than $1$? Is there some online resource where weight space multiplicities can be ...
3
votes
0
answers
138
views
Meaning of an algebra having "sufficiently many primitive idempotents"?
This is a phrase Ringel uses a few times in his writing, and I'm not sure exactly what he means by it. The context is that we have a quiver $Q$ with path algebra $\mathbf{k}Q$. If $Q$ is not a finite ...
- 1,161
2
votes
1
answer
250
views
Example of a projective bimodule with isomorphic left and right duals
What is an example of a non-free finitely generated $R$-bimodule $M$ satisfying
i) $M$ is projective as both a left and right $R$-module
ii) the right dual $\mathrm{Hom}_R(M,R)$ and the left dual ...
3
votes
1
answer
244
views
Left module which cannot be made into a bimodule?
Let $A$ be a noncommutative unital algebra, defined over $\mathbb{C}$ say. What is an example of a left $A$-module $M$ that does not admit a right $A$-module structure giving $M$ the structure of a ...
3
votes
2
answers
1k
views
Dual of a projective module
Let $R$ be a noncommutative ring with unit, let $P$ be a projective left $R$-module, and denote $^{\vee}\!P := \,_R\mathrm{Hom}(P,R)$. One often sees it written that projectivity implies an ...
5
votes
1
answer
224
views
Convolution algebra associated to a finite dimensional algebra
Given a finite dimensional $k$-algebra $A$ (we can assume it is given by a connected quiver with relations). One can form its trivial extension $T(A)$ (see for example https://math.stackexchange.com/...
- 26.5k
5
votes
1
answer
704
views
Representations of tensor products of algebras
For two associative unital algebras $A$ and $B$, defined over $\mathbb{K} = \mathbb{R}, \mathbb{C}$, is it possible to have an irreducible representation of $A \otimes_{\mathbb{K}}B$ which is not of ...
- 993
6
votes
1
answer
339
views
Monoidal categories from the projective modules of a ring
Let $R$ be a not necessarily commutative ring, and denote by $_R\mathrm{lp}_R$ the category of $R$-bimodules, which are finitely generated projective as left modules, with morphism $R$-bimodule maps, ...
- 993
2
votes
1
answer
98
views
A weaker version of strongly graded algebras
Let $A = \oplus_{i \in \mathbb{Z}} A_i$ be a graded algebra. We say that it is strongly graded if $A_i.A_j = A_{i+j}$, for all $i,j \in \mathbb{Z}$. Can there be existing a graded algebra such that
$$...
5
votes
2
answers
680
views
Characters on Hopf algebras
For any algebra $A$, a character for $A$ is a non-zero algebra map $c:A \to \mathbb{C}$. For $H$ be a Hopf algebra, a character is given by $\epsilon:H \to \mathbb{C}$ the counit of $H$. I am looking ...
4
votes
1
answer
375
views
Invertible bimodules and projectivity
Let $A$ be a noncommutative algebra over a field, say $\mathbb{C}$ or $\mathbb{R}$, and let $L$ be a bimodule over $A$. If $L$ is invertible, that is, if the dual right $A$-module $L^*$ satisfies
$$
L^...
3
votes
1
answer
313
views
indecomposable modules of gentle algebras
Let $A = \mathcal{k}Q/I$ be a gentle algebra (where $\mathcal{k}$ is algebraically closed). In the paper Auslander-Reiten Sequences with Few Middle Terms and Application to String Algebras, Butler and ...
- 31
8
votes
2
answers
498
views
Left-right non-bimodule examples
Let $A$ be a unital algebra, defined over the complex numbers. Any bimodule $M$ over $A$ must, by definition, be a left, and right, module satisfing
$$
a.(m.b) = (a.m).b, ~~~~~~~ \textrm{ for all } a,...
4
votes
1
answer
175
views
Group representation with algebra structure
I haven't seen this question in standard textbooks, so I decide to give it a try here. It might relate to deeper structures of certain TQFTs, but I'm not sure.
Let $G$ be a finite group. Its finite-...
- 5,230
2
votes
0
answers
72
views
injective map between tensor products of two irreducible modules of simple Lie algebra sl_{n+1}
Let $1 \leq i_1 < i_2 < i_3 \leq n$. I know that there is an injective map from $V(\omega_{i_1}+\omega_{{i_2} -1})\otimes V(\omega_{{i_3}+1})$ to $V(\omega_{i_1}+\omega_{i_2})\otimes V(\omega_{...
- 137
6
votes
2
answers
358
views
Duals of the spinor representations of $\frak{so}_{2n}$
For the $D_n$-series simple Lie algebra $\frak{so}_{2n}$
a curious phenomenon occurs for the fundamental representations corresponding to the spinor nodes of the Dynkin diagram, which is to say the ...
2
votes
1
answer
211
views
Choice of a ground ring for cluster algebras
In order to define cluster algebra one needs to define its ground ring. In most cases, we take a group $P$ (often called a coefficient group) which is taken to be an abelian multiplicative group . ...
- 225
4
votes
2
answers
360
views
Double centralizer in special linear algebra
It is well known that for a matrix $A$ in $\mathfrak{sl}_n(\mathbb{C})$, we have the following equivalence:
$$\dim Z(A) \text{ is minimal} \leftrightarrow A \text{ is cyclic}$$
where $Z(A)$ is the ...
- 617
6
votes
0
answers
259
views
Diameter of finite rational matrix groups
Suppose $G$ is a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$.
For a set $\mathcal{M} \subseteq G$ that generates $G$, define the $\mathcal{M}$-diameter $\mathit{diam}(G, \mathcal{M})$ of $G$ to be ...
- 165
6
votes
2
answers
501
views
Complete reducibility and field extension
Let $\pi$ be a representation of a Lie algebra $L$ in a finite-dimensional linear space $V$ over the field $F$. Let $K$ be a field extension of $F$. Let $\pi_K=\pi\otimes K$ be the corresponding ...
- 71
4
votes
1
answer
520
views
List of Casimir elements of low dimensional Lie algebras
I am interested in explicit formulae for the Casimir elements (or "Casimir operators") of low-dimensional, real, non-Abelian Lie algebras (d=2,3, and possibly 4). I am wondering if there is any ...
5
votes
0
answers
140
views
Open problems about Morita and derived invariants
Are there properties of rings of which one does not know whether they are Morita or derived invariances?
For a recent such example for Morita invariance, see https://www.sciencedirect.com/science/...
- 26.5k
13
votes
3
answers
2k
views
Classification of commutative Frobenius algebras
Are there attempts to classify commutative finite dimensional Frobenius algebras? They appear often in mathematics, such as in algebraic geometry and the famous category equivalence between ...
- 26.5k
5
votes
1
answer
175
views
Ideals of commutative Frobenius algebras
Given a finite dimensional commutative (connected=local) Frobenius algebra $A$ over a field $K$.
Question 1: Does $A$ have only finitely many ideals? (the answer should be no in the non-commutative ...
- 26.5k
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 ...
5
votes
1
answer
256
views
Definition of a Dirac operator
So it seems that a Dirac operator acting on spinors on $\psi=\psi(\mathfrak{su}(2),\mathbb{C}^2)$ can be written in this case simply as:
$D=\sum_{i,j} E_{ij}\otimes e_{ji}$, where $E_{ij}$ are ...
- 327
7
votes
1
answer
370
views
Gorenstein symmetric conjecture for arbitrary rings
The Gorenstein symmetric conjecture states that for Artin algebras $A$ one has the the regular module has finite injective dimension as a right module if and only if it has finite injective dimension ...
- 26.5k
3
votes
1
answer
102
views
Algorithms for the explicit matrix isomorphism problem over $\mathbb{C}$
Suppose that $A$ is a $d^2$ dimensional algebra over $\mathbb{C}$ and we know the multiplication tensor $c_{ij}^k$ and the unit $u^k$ in some basis. If $A$ is semi-simple and has a single simple ...
- 3,830
4
votes
0
answers
153
views
Recovering the bimodule from the trivial extension
Given a ring $S$ with a non-zero $S$-bimodule $M$, the trivial extension of $(S,M)$ is defined as the ring $R:=T_M(S)$ with $R= S \oplus M$ with multiplication $(s,m)(s',m')=(s s', sm' +m s')$.
We ...
- 26.5k
3
votes
0
answers
134
views
Proving that the exterior algebra is symmetric via the polynomial ring
Recall that a finite dimensional algebra $A$ over a field $K$ is called Frobenius in case $A \cong D(A)$ as right modules, and it is called symmetric in case $A \cong D(A)$ as bimodules (where $D=...
- 26.5k
12
votes
1
answer
922
views
Does this algebra have finite global dimension ? (Human vs computer)
Usually computers can calculate the global dimension of a finite dimensional quiver algebra much faster than humans. But in this case a high end computer (calculating for 3 weeks) was not able to ...
- 26.5k
4
votes
1
answer
215
views
Explicit examples of finite dimensional, involutive Hopf algebras with traceless antipode?
$\require{AMScd}$
In the paper [1], it is shown that there exist finite dimensional, semisimple Hopf algebras $H$ where the antipode $S:H \to H$ is traceless.
Unfortunately, the method of proof in [...
- 1,231
5
votes
1
answer
1k
views
$Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C}) ?$
$G$ is an p-adic group, and $\pi$ is an irreducible representation of $G$, then do we naturally have
$Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C})$? I think it is true, but ...
- 467
1
vote
0
answers
71
views
tensor product of two compact induced representations
Suppose $A \subset A^{\prime}$ and $B \subset B^{\prime}$ are all p-adic groups, and $V_{\pi}$ is a representation of $A$; $V_{\rho}$ is a representation of $B$.
Define
$ind_{A}^{A^{\prime}}V_{\pi}...
- 467
7
votes
2
answers
450
views
Ideals invariant under ring automorphisms
I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:
$I$ is generated by two homogeneous elements;
$I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}...
- 11.1k
4
votes
2
answers
453
views
Global dimension of the tensor algebra
Let $R$ be a semisimple ring with a non-zero $R$-bimodule V. Let $T_R(V):= \bigoplus\limits_{k=0}^{\infty}{V^{\otimes_k}}$ be the tensor algebra of $V$.
Question 1: Is there a simple proof that $...
- 26.5k
4
votes
1
answer
463
views
Global dimension of a graded algebra
Let $A= \bigoplus\limits_{n=0}^{\infty}{A_n}$ be an $\mathbb{N}$-graded algebra with semisimple $A_0$.
Question: Do we have that the global dimension of $A$ is equal to $\sup \{i \geq 0 | Ext_A^i(...
- 26.5k