All Questions
15 questions
33
votes
2
answers
2k
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 $\mathbb{C}^n$. ...
- 1,690
17
votes
0
answers
704
views
When is the determinant an $8$-th power?
I am working over $\mathbb{R}$ (though most of the story goes over any field). I am looking for linear spaces of matrices such that the restriction of the determinant to this spaces can be written (...
- 7,300
13
votes
1
answer
698
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 $p_n(...
- 5,654
11
votes
0
answers
436
views
A rather strange algebra
Let $k$ be an algebraic closed field of zero characteristic and $X$ an affine smooth variety, with $A=\mathcal{O}(X)$ the algebra of regular functions and $\mathcal{V}$ the Lie algebra of vector ...
- 3,318
8
votes
2
answers
577
views
Faithful flatness and non-commutative algebras
$\DeclareMathOperator\Spec{Spec}$When dealing with commutative algebras, a usefull criterion for faithful flatness is the following:
Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then $...
- 541
6
votes
1
answer
570
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 $F_p[G]$?
- 61
5
votes
0
answers
226
views
Cohomology of representation varieties and the Hochschild cohomology
Let $k$ be a field, $A$ a $k$-algebra, and $V$ a $k$-vector space. Then we can consider the representation varieties of $A$ on $V$: $\mathrm{Hom}_{k\textrm{-alg}}(A, \mathfrak{gl}(V))$ and $\mathrm{...
- 985
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
$$
L^...
4
votes
3
answers
1k
views
Fractional Quantum Hall Effect - Mathematics
Just to include something that starts to answer my own question Topological Quantum Computation Lecture notes covers a lot of the Mathematics of the Fractional Quantum Hall effect, or topological ...
- 379
3
votes
1
answer
310
views
Description of the algebra of $G$-invariant polynomials by generators and relations
Fix $n > 1$ and let $\zeta \in \mathbb{C}$ be a primitive $n$-th root of unity. Let $G \subset \text{SL}_2(\mathbb{C})$ be a cyclic subgroup of order $n$ generated by the diagonal matrix $g = \text{...
- 45
2
votes
1
answer
98
views
A weaker version of strongly graded algebras
Let $A = \oplus_{i \in \mathbb{Z}} A_i$ be a graded algebra. We say that it is strongly graded if $A_i.A_j = A_{i+j}$, for all $i,j \in \mathbb{Z}$. Can there be existing a graded algebra such that
$$...
2
votes
0
answers
268
views
Understanding a proof of a result of Schofield
I'm reading a paper of Aidan Schofield- "General Representations of Quivers" and I'm trying to understand the proof of Theorem 3.3. I'm having trouble understanding the argument that's ...
- 839
1
vote
0
answers
119
views
Germs of holomorphic functions and invariant functions
Consider a complex vector space $V \cong {\mathbb C}^n$. Consider the ring of germs of holomorphic functions ${\mathcal O}_0 (V)$ at $0\in V$. We know that this ring is Noetherian.
Now consider a ...
- 803
1
vote
0
answers
128
views
Example of a brick-infinite, tame, triangular algebra of global dimension$\geq 3$
I'm trying to compute some examples and I'm unable to come up with a following example:
What is(are) the example(s) of an acyclic quiver $Q$ with relations such that the 2-Kronecker quiver is NOT a ...
- 839
1
vote
0
answers
115
views
Prove that $B$ is a directing module
Let $A\cong\mathbb{K}Q/I$ be a finite dimensional, associative, basic $\mathbb{K}$ algebra, where $\mathbb{K}$ is algebraically closed and $Q$ is a finite Gabriel quiver on $n$ vertices and $I\...
- 839