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Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory

6
votes
0answers
88 views
+150

Functoriality of the Hopf dual

Given Hopf $\mathbb{C}$-algebra $H$, it's Hopf dual $H^o$ is the largest Hopf algebra contained in $H^*$, the $\mathbb{C}$-linear dual of $H$. (This is well known to be well-defined, see for example ...
3
votes
0answers
40 views

Quotient of quasi-isomorphic nonpositively graded cdga's

I'm looking for a theorem about quotient of quasi-isomorphic cdga's: Let $A, B$ be two cdga's (commutative differential $\mathbb Z$-graded algebra) concentrated in nonpositive degree, and $\mathfrak ...
4
votes
1answer
113 views

Quotient of quasi-isomorphic cdga's

I'm looking for a theorem about quotient of quasi-isomorphic cdga's: Let $A, B$ be two cdga's (commutative differential $\mathbb Z$-graded algebra) of nonpositive degrees, and $\mathfrak m \subset A, ...
4
votes
2answers
187 views

How to compute the inverse of a quantum determinant?

Let $X=(x_{ij})_{mn}$ be a quantum matrix with the commutation relations between entries: \begin{alignat*}{2} & x_{ij} x_{il} = q x_{il} x_{ij}, && j < l, \\ & x_{ij} x_{kj} = q ...
11
votes
3answers
247 views

Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules

In the first answer for this question is writen, about the braided category of representation of the enveloping algebra $U(\frak{g})$, for $\frak{g}$ a semisimple Lie algebra: The space of ...
4
votes
0answers
75 views

Yangians as unique deformation

In Drinfeld's paper "Hopf algebras and the quantum Yang-Baxter equation" there is a statement (Theorem 2) that Yangian is a unique quantization of the corresponding Lie bialgebra. My question is ...
5
votes
0answers
113 views

Quantum groups at $q=-1$

For a Drinfeld--Jimbo quantized enveloping algebra $U_q(\frak{g})$, it is standard knowledge that the categories of modules are very different in the $q$ a root of unity, and $q$ not a root of unity ...
7
votes
0answers
124 views

Universal Enveloping algebra of a L$_\infty$ algebra

In their paper Strongly homotopy Lie algebras, Lada and Markl first show, that there is a symmetrization functor $(-)_L:\mathcal{A}(m)\rightarrow \mathcal{L}(m)$ from the category of $A(m)$-algebras ...
4
votes
1answer
253 views

Is there another quantum deformation of sl(2)?

By looking at defining relations of standard deformation of $\mathfrak{sl}_2$, which are: $$ [E,F] = \frac{q^{H}-q^{-H}}{q-q^{-1}}, \quad [H,E] = 2E, \quad \text{ and } \quad [H,F] = -2F, $$ some ...
8
votes
1answer
143 views

Central extensions of loop groups

Let $LG=\operatorname{Maps}(S^1,G)$ be the loop group of a compact Lie group $G$. I should add some adjectives to $G$, but for sake of simplicity let's just take $G=SU(2)$. There is a central ...
4
votes
0answers
51 views

Do global bases exist for quantum enveloping algebras at $q$ nonroot of unity?

Take $\Bbbk$ to be a field, $q \in \Bbbk$ a nonroot of unity, and $U = U_q(\mathfrak g)$ the quantized enveloping algebra of a complex finite dimensional simple Lie algebra, and write $U^-$ for its ...
16
votes
0answers
163 views

Limiting representation theory of quantum groups at roots of unity and SL(2,C)

Let $V_N$ denote the $N$-dimensional representation of the quantum group $U_q(\mathfrak s\mathfrak l_2)$. I am told that in the limit $N\to\infty$ with $q=e^{2\pi i/n}$ and $N/n\to\alpha\in(0,1)$, ...
5
votes
0answers
173 views

Constructing a noncommutative algebra from a commutative algebra

I was told at a conference that one way to construct a noncommutative algebra from a commutative one is to "replace the product of finite spaces (which on the level of continuous functions corresponds ...
9
votes
1answer
126 views

Separability of compact quantum groups

In the theory of compact quantum groups due Woronowicz, we assume usually that the C*-algebra of the compact quantum group is separable. Is the assumption essential in the theory? Will it eventually ...
7
votes
0answers
121 views

IH-moves on trivalent graphs, and a complex that might be known to low-dimensional topologists

Here is a combinatorial problem which is hard to Google but seems like it might have a solution well known to people who study finite type invariants etc. Let $G_{g,b}$ denote the set of finite ...
9
votes
2answers
207 views

Monoidal Equivalence for Drinfeld--Jimbo Quantum Groups

For $U_q(\frak{g})$ the Drinfeld--Jimbo quantum group, its category of representations is equivalent to the category of representations of $U(\frak{g})$, or equivalently the category of Lie algebra ...
5
votes
0answers
85 views

Relation between different versions of Bar-Natan homology

In Bar-Natan's paper: Khovanov’s homology for tangles and cobordisms, he defined a deformation of Khovanov homology. Namely, for any $m\geq 0$, Bar-Natan's homology $BN^{m}(K)$ is obtained by ...
3
votes
0answers
38 views

Adjoint action of Lusztig's integral form preserves the De Concini-Kac integral form: reference?

Let $U_q(\mathfrak{g})$ be the quantized enveloping algebra of a complex semisimple Lie algebra. I have seen in several papers the following fact stated: the adjoint action of Lusztig's integral form (...
10
votes
2answers
626 views

$(\infty,1)$ 2d TFTs

2d topological field theories $Z : \mathrm{Cob}(2) \to \mathrm{Vect}$ are classified by commutative Frobenius algebras. What can be said about $(\infty,1)$ 2d TFTs $Z: \mathrm{Cob}(2) \to \mathcal{S}$...
13
votes
3answers
541 views

Hopf dual of the Hopf dual

Given any Hopf algebra $A$ over a field $k$, one can also define the Hopf dual $A^*$ of as follows: Let $A^∗$ be the subspace of the full linear dual of $A$ consisting of elements that vanish on some ...
2
votes
0answers
79 views

On the set of indices of irreducible depth 3 subfactors

Let $I_n$ be the set of indices of (finite index) irreducible depth $n$ subfactors. Then $I_2 = \mathbb{Z}_{>0}$. Question 1: Is it true that $I_3$ has no accumulation point? If so: ...
28
votes
1answer
1k views

On a quantum Riemann Hypothesis

Robin's theorem (1984) states that $$ \sigma(n) < e^\gamma n \log \log n$$ for all $n > 5040$ if and only if the Riemann hypothesis is true. Recall that $γ$ is the Euler–Mascheroni ...
2
votes
0answers
76 views

Indecomposable modules for the big quantum group

I am study the representation theory of the big quantum group at a root of unity, and I am wonder if it is known a complete classification of the indecomposable modules for it. To be more specific, ...
3
votes
1answer
190 views

Framing dependence of HOMFLY polynomial

I want to understand the framing dependence of the Khovanov-Rozansky homology, and as its first step, I am trying to understand the framing dependence of the HOMFLY polynomial (i.e. quantum $sl(n)$ ...
4
votes
1answer
111 views

2-morphisms for Bord(n)

I am currently reading in Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors, and have a (I guess) pretty basic question for my understanding of the (...
5
votes
0answers
102 views

Are the integral forms of quantized coordinate algebras always Noetherian?

Let $U_q(\mathfrak{g})$ be the quantized enveloping algebra of a complex semisimple Lie algebra and let $\mathcal{O}_q(G)$ be the quantized coordinate algebra of the corresponding simply-connected ...
5
votes
0answers
72 views

What quantum groups admit quantum topography space structure?

Quantum topography space is a pair $(A,M)$ consisting of a $C^*$-algebra $A$ and an abelian sub algebra $M\subset A$ with approximate identity. The intuition is to take $M$ be the smallest abelian ...
14
votes
1answer
246 views

How are MTCs permuted by the Galois action on the little disk operad?

There is a well-studied action of $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$ on (some version of) the $E_2$ operad; see for example this MO question. Modular tensor categories are examples of $...
1
vote
0answers
41 views

Quantum Lie algebra formalism that doesn't violate P symmetry

begin tl;dr: I just read this paper which gives the equations for the structure constants, braiding operators etc. for a generic quantum Lie algebra. I always found it very annoying that in the ...
9
votes
1answer
201 views

Is the instanton homology for webs and foams a categorified Chern-Simons?

In their paper, Kronheimer and Mrowka constructed an instanton homology $J^{\#}$ for webs and foams and conjectured that for planar webs, $\dim J^{\#}=\#\text{ of Tait colorings}$. According to my ...
7
votes
0answers
186 views

The Fock Space vs the Hilbert space in the context of Quantum Field Theory

Mathematically the definitions are as follows : if $H_n$ is a $n-$dimensional complex Hilbert space then its two different corresponding ``Fock space"(s) are often denoted as $F_{1}$ and $F_{-1}$ ...
7
votes
1answer
158 views

commutative “weakly” Frobenius algebras and 2d TQFT

Fix a field $k$. A classic result written up carefully by Abrams in the article "Two-Dimensional Topological Quantum Field Theories and Frobenius Algebras" says that there is a bijective ...
5
votes
0answers
130 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 ...
5
votes
1answer
194 views

Knot Factorization Homology inputs

Following the paper by Ayala, Francis, and Tanaka: https://arxiv.org/pdf/1409.0848.pdf If we are talking about knots we are talking about framed 3-manifolds with a framed 1-dimensional sub-manifold ...
4
votes
1answer
121 views

Presentations of the monoidal categories of virtual tangles and of welded tangles by generators and relations

Reidemeister theorem implies, without too much fuss, that the monoidal categories of tangles, and of oriented tangles, can be presented by generators and relations. This is done for example in a) ...
7
votes
2answers
281 views

Hopf Subalgebras of Quantized Algebras

As is well known, quantized enveloping algebras $U_q(\frak{g})$ admit far fewer sub-Hopf algebras than classical enveloping algebras $U(\frak{g})$. As one can check directly, for appropriate subsets ...
4
votes
0answers
184 views

Reference Request: Vertex Algebras

I am currently a graduate student in mathematics with an interest in vertex algebras. I am comfortable with the algebraic aspects and would like to learn more about the geometric aspects. The issue is ...
6
votes
1answer
110 views

Jones-Wenzl-type projectors for Brauer algebras

Jones-Wenzl projectors are an explicit combinatorial description of the central projectors of the Temperley-Lieb algebra. They also describe very explicitly the failure of certain representations to ...
0
votes
0answers
89 views

Presentations of $\mathbb{C}_q[G]$

Let $G$ be a semisimple Lie group and $q \in \mathbb{C}^{\times}$, note a root of unity. I am trying to understand the presentations of the quantum group $\mathbb{C}_q[G]$. In the paper FACTORIZABLE ...
6
votes
1answer
241 views

Quadratic algebras, quadratic operads, quadratic categories and quantum cohomology

Motivated by the quantisation of the symmetric laws in physics, the category of quadratic algebras has been endowed with two tensor products by Manin in his Montreal lectures notes. These products ...
2
votes
0answers
108 views

Module algebras and comodule algebras

Let $H$ be a Hopf algebra and $A$ an algebra. Let $H^*$ be the dual Hopf algebra of $H$. Then by Proposition 1.6.11 in the book Foundations of Quantum Group Theory by Shahn Majid, $A$ is a right $H$-...
7
votes
1answer
113 views

Exceptional Quantum Groups as FRT-Algebras

Let $\frak{g}$ be a simple Lie algebra of A,B,C,or D series type. Moreover, let $U_q(\frak{g})$ be its Drinfeld-Jimbo quantized enveloping algebra, and $G_q$ the quantized enveloping algebra. As is ...
1
vote
0answers
152 views

Are creation and annihilation operators closed?

I'm trying to understand some rigorous results in the basic formalism of second quantization and following some lectures I found on the internet I ran into a little trouble in defining creation and ...
4
votes
1answer
135 views

Schur's Lemma for Quantized Universal Enveloping Algebra

Let $U_q(\mathfrak{g})$ (defined over $\mathbb{C}(q)$) be the quantized universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$. Let $M$ a finite-dimensional simple left $U_q(\mathfrak{g})$...
2
votes
0answers
44 views

For any finite-dimensional Hopf C*-algebra, can one make the multiplication and co-multiplication cyclically symmetric simultanously?

For any finite-dimensional *-algebra, one can choose a basis such that the coefficients tensor of the anti-linear map $(a,b)\rightarrow (ab)^*$ becomes cyclically symmetric. (Any *-algebra is ...
4
votes
0answers
39 views

Characteristic classes of invariant star products

Let $\mathfrak{g}$ a Lie algebra, can one compute the $\mathfrak{g}$-invariant Deligne class of an invariant star product by using some kind of invariant local $\nu$-Euler derivation?
5
votes
0answers
45 views

Non-semisimple representations of the braid group in a semisimple braided category

Suppose $\mathcal{C}$ is a semisimple braided tensor category (over $\mathbb{C}$, with finite dimensional hom spaces) and $X$ an object in $\mathcal{C}$. Then for each n > 0 the braid group $B_n$ ...
5
votes
2answers
161 views

Are there examples of finite-dimensional weak Hopf C*-algebras with non-involutive antipode?

For finite-dimensional (non-weak) Hopf C*-algebras it is known that the antipode is always involutive, as claimed e.g. in https://arxiv.org/pdf/1007.5283.pdf. I couldn't find the same statement for ...
14
votes
0answers
338 views

Do TQFTs give a complete set of invariants of manifolds?

An $n$-dimensional TQFT is a representation of the category $n$Cob of $n$-dimensional cobordisms. TQFTs are important sources of invariants of manifolds, and such invariants are highly computable by ...
30
votes
2answers
4k views

What is quantum algebra?

This might be a very naive question. But what is quantum algebra, really? Wikipedia defines quantum algebra as "one of the top-level mathematics categories used by the arXiv". Surely this cannot be a ...