All Questions
Tagged with qa.quantum-algebra ag.algebraic-geometry
31 questions
3
votes
0
answers
97
views
What algebras generate polynomial count varieties as their representations spaces ? Is it preserved by the Koszul duality, Manin's endomorphisms?
Consider for example commutative polynomial algebra $C[x,y]: xy=yx$, look on $F_p$-matrices satisfying that relation - the number over $F_p$ will be given by polynomial in $p$ (classical result due to ...
- 24.9k
21
votes
3
answers
808
views
Examples when quantum $q$ equals to arithmetic $q$
First, as a disclaimer, I should say that this post is not about any specific propositions, but is more of some philosophical flavor.
In the world of quantum mathematics, the letter $q$ is a standard ...
- 1,391
1
vote
0
answers
161
views
Points and algebraic geometry on the quantum plane
The "quantum plane" is the "space" of the algebra $A=\Bbbk\langle X,Y\rangle/(YX-qXY)$, for a scalar $q$ (e.g. $\Bbbk=\mathbb C(q)$). I would like to know how much algebraic ...
- 2,519
2
votes
1
answer
287
views
On the definition of the Cherednik algebra of a variety with a finite group action
Let $X$ be a connected complex smooth affine variety, acted on by a finite group $G$. We define a reflection hypersurface $(Y,g)$ as a smooth codimension one subvariety $Y\subset X$ which is fixed by $...
- 541
7
votes
2
answers
484
views
Tsuchiya-Ueno-Yamada's proof that sheaves of conformal blocks are locally free
I'm referring to Tsuchiya-Ueno-Yamada's (TUY hereafter) celebrated paper Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries. One of the main goals of their paper is to ...
- 585
18
votes
1
answer
3k
views
Conjectures of Peter Scholze about q-de Rham complex: examples
Peter Scholze formulated several conjectures about $q$-de Rham complex in the paper
Canonical $q$-deformations in arithmetic geometry, Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 5, pp 1163–...
- 4,316
5
votes
0
answers
158
views
Can we see the symmetry of the quantum Schubert polynomial of a point
Let $X=G/B$ be a homogeneous space and consider the quantization map
$$
S_W\otimes\mathbb{C}[q]\to(S(\mathfrak{h})\otimes\mathbb{C}[q])/I_W^q\,,
$$
where
$S_W$ is the coinvariant algebra of the Weyl ...
- 1,066
2
votes
1
answer
254
views
Finitely Generated Commutative Hopf $*$-Algebras
As is well known, using the Hilbert Nullstellensatz (and a more recent result of Cartier) one can show that commutative finitely generated Hopf algebras over $\mathbb{C}$ are equivalent to algebraic ...
7
votes
0
answers
323
views
Flat connection from gauged WZW model
$\newcommand{\g}{\mathfrak g}$
$\newcommand{\h}{\mathfrak h}$
In short my question is :
Has someone worked out the flat connection that one should get from the gauged WZW model in genus 0 ?
Some ...
- 8,524
4
votes
1
answer
623
views
Verlinde Formula and Theta Function Identities
The paper Fusion rules and modular transformations in 2D conformal field theory by Erik Verlinde mentions a simple case of rational conformal field theory, where the fusion algebra is just $(\mathbb{Z}...
- 22.8k
8
votes
0
answers
463
views
On the cohomology of Kontsevich graph complex
Kontsevich's formality theorem asserts that a certain quasi-isomorphism of chain complexes between the graded Lie algebra of polyvector fields on $\mathbb{R}^n$ and the dg Lie algebra of ...
- 1,609
10
votes
1
answer
1k
views
Are there workable algebraic geometry approaches for the pentagon equation?
A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring.
A fusion ring is given by a finite set of integer ...
18
votes
2
answers
984
views
A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?
In a 1986 paper, Harer and Zagier proved the recursion:
$$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$
where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...
- 1,303
0
votes
0
answers
354
views
abstract algebra for component wise operations on "vectors" or what it might be called
I have a quite tough problem to solve and need an algebra that allows to "vectors" following operations:
- multiplication between two vectors are componentwise that means v=(v1, v2, v3,...) multiplied ...
- 254
4
votes
2
answers
615
views
When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau?
I would like to ask a simple question. Let $A=\mathbb{C}\langle x_{1},\dots,x_{n} \rangle/I$, where $I$ is the two-sided ideal generated by $x_{i}x_{j}=a_{ij}x_{j}x_{i}$ for $1\le i,j\le n$. We say a $...
- 63
4
votes
1
answer
393
views
braids and dynamics of roots of a polynomial
The 2-variable polynomial equation $f(z,t) = 0$ with $z = \mathbb{C}, \\,t \in \mathbb{S^1}$ has $n = \mathrm{deg}_z f$ solutions each fixed $t$. I wanted to follow the roots as they travel with time ...
- 22.8k
12
votes
1
answer
946
views
Compatibility of the KZ connection with operadic composition
In what sense is the Knizhnik-Zamolodchikov connection compatible with the operadic composition in the little discs operad and/or the operad of $\overline{M}_{0,n}$'s?
Here are (some) details, ...
- 407
5
votes
0
answers
303
views
Quantum dynamics on varieties: asymptotic quantum trace distance on SHL varieties
The Question Asked
Definition: the Second-Hand Lion trace distance $D_k$
Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the ...
- 1,389
-3
votes
1
answer
2k
views
Quantum dynamics on varieties and Salmon Prizes
Concluding Progressive Remarks
A new finding is Bates and Oeding's preprint "Toward a salmon conjecture" (arXiv:1009.6181), with its reference to the Salmon Prize.
The Salmon Prize (photo of the ...
5
votes
1
answer
2k
views
Homotopic morphisms between curved A-infinity algebras
I know how to think about (curved) $A_\infty$-algebras 'geometrically', i.e. via formal non-commutative geometry in the sense of Kontsevich etc. I also know how to think about $A_\infty$-morphisms in ...
- 460
5
votes
1
answer
694
views
Log structure and degeneration
I am interested in compactification of the moduli space of elliptic curves, and I heard that Log geometry is very important for the problem.
I am developping the same technique for quantum geometry.
...
- 53
7
votes
0
answers
460
views
Quantum polynomial rings and singularities
Something I've been thinking about lately has led me to wonder about the following. Consider the quantum polynomial ring $ Q= \mathbb{C}_{-1}[x_1,...x_n]$ generated as a graded ring in degree 1 with ...
- 3,758
16
votes
2
answers
2k
views
Deformation quantization and quantum cohomology (or Fukaya category) -- are they related?
Good afternoon.
Let $M$ be, say, a compact symplectic manifold. Both deformation quantization (as in Kontsevich) and quantum cohomology yield "deformations" (in the appropriate respective senses) of "...
- 21k
24
votes
6
answers
4k
views
A few questions about Kontsevich formality
[K] refers to Kontsevich's paper "Deformation quantization of Poisson manifolds, I".
Background
Let $X$ be a smooth affine variety (over $\mathbb{C}$ or maybe a field of characteristic zero) ...
- 21k
23
votes
4
answers
3k
views
Deformations of Nakajima quiver varieties
Are deformations of Nakajima quiver varieties also Nakajima quiver varieties ?
In case the answer to this is (don't k)no(w), here are some simpler things to ask for.
(If you're a differential ...
- 1,191
6
votes
1
answer
1k
views
Weyl Character Formula for Quantum Groups
How much is known about the Weyl character formula for quantum groups? More specifically, has the formula been generalized to the general setting of deformed coordinate algebras $\mathbb{C}[G_q]$ of ...
- 1,462
43
votes
6
answers
9k
views
The 'real' use of Quantum Algebra, Non-commutative Geometry, Representation Theory, and Algebraic Geometry to Physics
In this question, Orbicular made the following comment to Feb7 and my own answers;
Please keep in mind that - even though it is stated very often - noncommutative geometry does not give "real" ...
- 4,842
9
votes
1
answer
820
views
Quantum equivariant $K$-theory and DAHA.
Theorem 3.2 of the paper "Quantum cohomology of the Springer resolution" by Braverman, Maulik and Okounkov relates equivariant quantum cohomology of the cotangent bundle of $G/B$ to the trigonometric ...
8
votes
3
answers
1k
views
Is there an analogue Beilinson-Bernstein localization for quantized enveloping algebra
I am completely a beginner in this field. I wonder know whether there is appropriate notion for quantum flag variety of finite dimensional Lie algebra. If so, what is the correspondent notion for &...
- 81
13
votes
4
answers
2k
views
Why would I want to know (equivariant) quantum cohomology?
Let's say that I have a variety I think is interesting, and based on some papers I don't fully understand, I can compute quite explicitly its equivariant quantum cohomology in terms of explicit ...
- 44.7k
19
votes
4
answers
2k
views
What are the points of Spec(Vassiliev Invariants)?
Background
Recall that a (oriented) knot is a smoothly embedded circle $S^1$ in $\mathbb R^3$, up to some natural equivalence relation (which is not quite trivial to write down). The collection of ...
- 54.6k