1
vote
0answers
102 views

Cycles in Quivers and Path Algebras

I cannot find anything giving the algebra of a quiver with a single cycle on three or more vertices. In other words if your quiver consists of n vertices (n>2), and e_i is connected to e_{i+1} (taking ...
5
votes
1answer
163 views

Equivariant Formality

Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$. Assume further there exists a ...
2
votes
1answer
125 views

Jacobson radical and group rings/subalgebras

Let $G$ be a finite group and $N\le G$ be a subgroup. Consider the group algebra $kN$ as a subalgebra of $kG$ over an algebraically closed field $k$ of positive characteristic. What can we deduce ...
7
votes
0answers
126 views

What happens to simple modules under Ringel duality?

If $A$ is a quasi-hereditary algebra then its Ringel dual $A'=End_A(T)$ is the endomorphism algebra of a (minimal) full tilting module $T$ for $A$. The algebra $A'$ is again quasi-hereditary, although ...
2
votes
0answers
57 views

Isomorphic bound quiver algebras for different admissible Ideals

We know that for a path algebra KQ, whether or not KQ is finite dimensional (namely, Q may or may not have oriented cycles), there might be different admissible ideals I and J of KQ for which the ...
13
votes
2answers
362 views

Does base extension reflect the property of being isomorphic?

Let L/K be a (separable?) field extension, let A be a finite dimensional algebra over K, and let M and N be two A-modules. Let $A' = L \otimes_K A$ be the algebra given by extension of scalars, and ...
4
votes
1answer
340 views

Homotopy of quivers

The matrix ring $k^{n\times n}$ can be realized in many ways as a quotient of a path algebra: For example choose the quiver $1\leftrightarrows 2 \leftrightarrows \cdots \leftrightarrows ...
5
votes
1answer
283 views

Does $\text{Mat}_n(k)$ have some universal properties similar to its universal enveloping algebra?

Let $k$ be a field and $\text{Mat}_n(k)$ be $n \times n$ matrices over $k$. Let's consider $\text{Mat}_n(k)$ as an associative algebra and denote $gl_n(k)$ be the same $k$-linear space as ...
1
vote
1answer
113 views

Modules for an idempotented algebra

Recall that an associative algebra $A$ is called idempotented provided that is the filtered union of subalgebras $eAe$ for $e \in A$ idempotent. I think sometimes people say that $A$ has approximate ...
9
votes
1answer
374 views

Counting representations of $k[x,y]$ when $k$ is finite

$\newcommand{\GFq}{\mathbf F_q}$ Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial ...
3
votes
2answers
251 views

Do morphisms of finitely-decomposable Quiver representations map indecomposables nicely?

Consider two quivers $Q$ and $Q'$ of type $A_n$, laid out horizontally like so: Given representations of $Q$ and $Q'$, Gabriel's theorem guarantees the existence of finitely many indecomposables ...
4
votes
1answer
374 views

“as close to being semisimple as it can possibly be.”

I had originally asked this question on math stack exchange but I think maybe it's more appropriate to ask it here. In the paper of Beilinson, Ginzburg and Soergel entitled "Koszul Duality ...
10
votes
2answers
274 views

Temperley-Lieb algebras for other Weyl groups?

The Temperley-Lieb algebra has the same generators as the $S_n$ group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen ...
2
votes
1answer
245 views

finite dimensional irreducible representation of finite dimensional nilpotent Lie algebra

Let $k$ be a field, $L$ be a finite dimensional nilpotent Lie algebra over $k$ and $M$ be a finite dimensional irreducible representation of $L$. Assume that there is a linear function $\rho : ...
1
vote
0answers
108 views

Cotorsion theory and its relative homology

Let (F(R), Cot(R)) be a cotorsion theory, Such that F(R) is the class of flat R-modules and Cot(R) the cotorsion modules. Why this is true that, For $ N\in Cot(R) $, $ \text{Ext}_{F(R)}^i(M, N)\cong ...
2
votes
0answers
89 views

Tits-Kantor-Koecher construction for Jordan algebra of symmetric bilinear form

Suppose $f$ is a nondegenerate, symmetric bilinear form on a vector space $V$ over a field $F$. Then $J = V + F\cdot 1$ is a unital Jordan algebra (known as the "spin factor" Jordan algebra when $F = ...
17
votes
4answers
911 views

Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?

Let $R$ be a finitely generated ring with identity, $M_n(R)$ the set of $n\times n$ matrices. Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$? This should be an elementary ...
0
votes
0answers
69 views

Degree of a commutator in a hyperalgebra or enveloping algebra

Consider a semisimple algebraic group $G$ over an algebraically closed field of arbitrary characteristic and let $\bar U(G)$ denote its hyperalgebra (ie, the restricted Hopf dual of the coordinate ...
2
votes
0answers
143 views

Minimal prime ideals of a group ring

Let $R$ be a left Noetherian ring (if you prefer you can just think to $R$ as a skew field, I'll be happy with an answer under that hypothesis) and $G$ be a polycyclic-by-finite (or, if you prefer ...
4
votes
2answers
260 views

BGG-like resolutions and translations

This question originated from my confusion after I read the following paragraph (page 31, section 4.8) in Resolutions and Hilbert series of the unitary highest weight modules of the exceptional ...
5
votes
2answers
212 views

Lie algebra embeddings and the center of their enveloping algrabras

Let $\mathfrak{g}_1\subset \mathfrak{g}_2$ be a Lie algebra embedding. Assume both are semisimple. For instance take the standard diagonal embedding $\mathfrak{sl}(2, \mathbb{C})\subset ...
1
vote
1answer
187 views

translation functors in parabolic category $\mathcal{O}$

I'm looking for a reference for a treatment of translation functors (as defined e.g. in this [question][1]) in parabolic versions of BGG category $\mathcal{O}$. I am mainly interested in the ...
1
vote
1answer
249 views

Is the universal enveloping algebra of a finite-dimensional Lie algebra (left) noetherian?

The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is a flat deformation of $S(\mathfrak{g})$, so these algebras should be similar in many ways. Does at least this general similarity ...
3
votes
0answers
236 views

det(A)det(B) = det(AB+correction), Capelli identities, “factorzied” representation of gl_n

Context: some probably know that there are Capelli identities which state det(A)det(B) = det(AB+correction) for some matrices with non-commuting elements, they go back to 19-th century, but also ...
5
votes
0answers
238 views

Homological dimension of completed path algebras.

Let A = c[Q]/I be a finite dimensional quotient of a path algebra over a quiver Q, with I being the ideal of relations. Is it true that the I-adic completion of A has finite homological dimension?
7
votes
3answers
429 views

Why/when classification of simple objects is “simple” ? E.g. (unknown) classification of simple Lie algebras in char =2,3…

Classification of simple finite-dim Lie algebras for char >=5 has been accomplished not so long time ago, and char p=2,3 is open problem. I wonder what is known/expected for char p=2,3 ? More vague ...
4
votes
1answer
186 views

Well defined Tensoring of spectral triples

Hi, I have a misunderstanding that I am hoping is really quite trivial. I will give my question directly and context below for those that need/want it. Question: In connes standard model he takes ...
1
vote
1answer
292 views

Question about the shape of the dual module of a module

Hi, I am reading this paper and have the following question:. On page 5 you can see different types of $A_n$-modules ($\ A_n=k[x,y]/\langle x^2,y^{n+2},xy^{n+1}\rangle\ $)$\ $ and later in the paper ...
28
votes
2answers
938 views

What do cluster algebras tell us about Grassmannians?

One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
0
votes
0answers
97 views

Quaternion algebras over $k[[u,v]]$

Let $R=k[[u,v]]$ be a power series ring over algebraically closed field of characteristic zero. The quaternionic $R$-algebra is $A=R\langle x,y\rangle/I$, where $I=(x^2-a, y^2-b, xy+yx-2c)$ and ...
7
votes
1answer
359 views

Modular representation theory: central idempotents in $\mathbb{Z}_p[G]$

Let $G$ be a finite group and let $p$ be a prime dividing the order of $G$. Let $\chi$ be a $\mathbb{C}_p$-valued irreducible character of $G$ and let $e_{\chi} = |G|^{-1}\chi(1)\sum_{g \in G} ...
6
votes
0answers
122 views

Classifying algebras with two idempotent generators and involution

Say that a finite-dimensional algebra $H$ over a field $K$ is dihedral if $H$ is generated by idempotents $P_1$ and $P_2$ and there is an algebra involution interchanging $P_1$ and $P_2$. For ...
3
votes
1answer
350 views

Relaxing commutativity. For c1,c2 find q1,q2: (1) [c1,c2]=q1c2-q2c1 (2) [q1,q2]=0, (3)…

Consider some elements c1,c2 in some ring. Let me say that they are "relaxed commutative" if there exists two elements q1,q2, such that the following conditions hold: (1) $ [c_1,c_2]=c_1q_2-c_2q_1$ ...
4
votes
0answers
137 views

Endomorphismrings of maximal submodules.

The question I am interested in answering is the following: Suppose that for a pair of $d$-dimensional modules $M$ and $N$ over a $k$-algebra ($k$ a field) $R$ we have that $\dim_k ...
0
votes
0answers
127 views

Splitting of noncommutative finite-dimensional local algebras over the residue field

Let $R$ be a finite-dimensional local (associative, unital, and not necessarily commutative) algebra over a field $k$ (that is, $R$ has a unique maximal two-sided ideal $\mathfrak M$) such that ...
5
votes
1answer
376 views

Generators of the derived category

For a ring $R$, which is a finite-dimensional algebra over a field, the category of finite-dimensional, projective, right $R$-modules, $\mathcal{P}_R$ is generated by the indecomposable projective ...
2
votes
1answer
163 views

Indecomposable extensions of regular simple modules by preprojectives

Given four points in general position on $\mathbb{P}^2$ there exists a projection to $\mathbb{P}^1$ collapsing these four pairwise to two points. Its kernel is some fifth point on $\mathbb{P}^2$. In ...
1
vote
1answer
228 views

Restrictions of Modules and Dimensions

Let K be a finite field and let R,P be groups (with R a subgroup of P). I know that the irreducible KP-modules have dimensions 1,4 and 16 over K. I have a KP-module M, and I know that M has dimension ...
3
votes
1answer
453 views

Reference Request: Basis in terms of ring of symmetric polynomials

As part of the result of solving the problem I am working on, my advisor and I translated the task of finding a basis for $R(T_{sl_{\mathbb{C}}(n)})$ in terms of $R(sl_{\mathbb{C}}(n))$ into the ...
0
votes
1answer
509 views

Hereditary algebras

I have the following problem: If $\Lambda$ is a hereditary, basic and connected algebra and $e$ is an idempotent of $\Lambda$, how can I prove that $e\Lambda e$ is also hereditary?
7
votes
2answers
508 views

“Composition of Morita equivalences” or “Morita equivalence and the Nakayama functor”

This problem occured to me, when trying to find a Morita invariant for finite dimensional algebras. Suppose $\Lambda$ and $\Gamma$ are two self-injective $k$-algebras ($k$ being a field) which are ...
5
votes
1answer
223 views

Radical of group algebra

Let $G$ be a finite group of size $p^a\cdot r$. Does there exist a simple way to calculate radical of the group algebra $F_p[G]$?
2
votes
2answers
460 views

Functoriality of adjugate matrix

i'm interested in geometric interpretations of many linear algebra notions (check also related geometric interpretation of matrix minors). it came to me recently that geometric description of adjugate ...
2
votes
2answers
224 views

Action of centralizer on Borel-Moore homology of Springer Fibers for Affine Hecke Algebra

In Chriss and Ginzburg's "Representation Theory and Complex Geometry", they describe a geometric construction of representations of the affine Hecke algebra, using the Borel-Moore homology of ...
0
votes
0answers
184 views

Does regularity of a D-module for an unusual filtration imply regularity for the usual one?

One definition of regular D-modules on affine space is that a D-module is regular if it has a filtration compatible with the order filtration on differential operators whose associated graded is ...
3
votes
2answers
583 views

Geometric interpretation of matrix minors

i was recently interested in geometric interpretation of various notions showing up in linear algebra because in most cases linear algebra with geometry courses focus too much on linear algebra not ...
3
votes
3answers
730 views

Fractional Quantum Hall Effect - Mathematics

Just to include something that starts to answer my own question Topological Quantum Computation Lecture notes covers a lot of the Mathematics of the Fractional Quantum Hall effect, or topological ...
4
votes
2answers
431 views

Translation functors for category $\mathcal O$

Let $\mathfrak g = \mathfrak n^- \oplus \mathfrak h \oplus \mathfrak n$ be a semisimple Lie algebra over $\mathbb C$ and let $\mathcal U$ be its enveloping algebra. Category $\mathcal O$ is the ...
0
votes
1answer
570 views

Does it make sense that “Representations of groups over finite ring” ?

I am an undergrad student who wants to know about the representation theory over arbitrary finite fields or finite rings of characteristic p (p a prime). (called modular representation theory.) In ...
3
votes
2answers
371 views

An example where finitistic dimension does not equal right global dimension?

The (right) big finitistic dimension of a ring is Findim$(R) =$ sup{proj.dim(M) | $M$ a right $R$-module of finite projective dimension}. The (right) little finitistic dimension findim$(R)$ is the sup ...