Simple highest weight modules of quantum affine algebras
Let $U_q(\hat{\mathfrak g})$ be a quantum affine algebra, and let $L$ be an integrable simple highest weight module of $U_q(\hat{\mathfrak g})$. In [Lu], Lusztig proved that the limit of $L$ when $q\to...
View ArticleDoes Manin's construction of non-commutative endomorphism algebra...
$\newcommand{\dual}{\mathrm{dual}}\DeclareMathOperator\End{End}\DeclareMathOperator\Fun{Fun}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\GL{GL}$Around 1986–7 Yu.Manin proposed natural and simple...
View ArticleHow to get $U(N)_k$ Kac-Moody modules and characters from $N \cdot k$ Dirac...
It is known that the $U(N)_k$ Kac-Moody algebra can be written as the coset $U(N)_k = U(N \cdot k)_1 / SU(k)_N$. (This fact is related to the level-rank duality of $U(N)_k \leftrightarrow U(k)_N$.) A...
View ArticleIs there a non-pointed simple integral modular fusion category?
The weakly group-theoretical conjecture, which suggests a negative response to [ENO11, Question 2], is formulated as follows:Statement 1 (open): Every integral fusion category is weakly...
View ArticleBraidings on Temperley-Lieb Category
Let $k$ be a field, and let $q\in k^{\times}$. We can then consider the Temperley-Lieb category $TL(q)$. The objects of $TL(q)$ are the non-negative integers, and morphisms are roughly isotopy classes...
View ArticleIs there a non-split super-modular positive integral fusion category?
We reference [EGNO] for the concept of a braided fusion category. Following the conventions in [JFR], let $\mathcal{C}$ denote a braided fusion category equipped with a braiding $\beta$, and let...
View ArticleAlternative to Kontsevich formality
Has anyone considered an alternative approach to Kontsevich formality in which the DGLA of poly-vector fields is deformed to an $L_\infty$-algebra?Some vocabulary:DGLA = Differential Graded Lie...
View ArticleMonoidal class vs gauge class vs Grothendieck class
In the comments under this post, three notions of equivalence classes of unitary modular tensor categories are brought up. They are monoidal classes, gauge classes, and Grothendieck class. Could...
View ArticleIs the Drinfeld element of a semisimple quasitriangular Hopf algebra...
Let $A$ be a finite dimensional semisimple quasitriangular Hopf algebra over $\mathbb{C}$ with universal $R$-matrix denoted by $\mathcal{R}\in A\otimes A$. The Drinfeld element of $A$ is defined...
View Articlequantum invariants, ribbon Tannakian duality and classification of ribbon...
In a nutshell, my question is:Q0: is there a classification of invariant of (framed) tangles arising from the Reshetikhin–Turaev construction?I will now make it more precise. One could define a...
View ArticleWhat algebras generate polynomial count varieties as their representations...
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...
View ArticleIs there an integral fusion category of the Ising type?
In [EGNO, Section 8.27.3], we read:Any braided fusion category ${\mathcal C}$ is obtained from a weakly anisotropiccategory (namely, the core of ${\mathcal C}$) using finite groups (via...
View ArticleProof that every commutative locally compact quantum group arises from a...
It is well-known that there is a bijection (up to isomorphisms) between locally compact quantum groups whose algebra is commutative and classical locally compact groups. I seem to cannot find a proof...
View ArticleVertex operator algebras and modular fusion categories
Let $\mathcal{V}$ be a vertex operator algebra (VOA), and let $\mathcal{C} = \text{Rep}(\mathcal{V})$ be the tensor category of $\mathcal{V}$-modules. It is a conjecture by Vaughan Jones whether every...
View ArticleAffiliating the whole algebra of 'coordinates' with a locally compact quantum...
When constructing a quantum deformation of a classical (matrix) locally compact group, we usually start with the *-Hopf algebra $A$ of matrix entries/coordinates. We then deform this algebra ($A_q$)...
View ArticleRepresentations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$
Let $G$ be a finite group and $D(G)$ its quantum double. As in my previous question, a typical irreducible representation (finite dimensional over $\mathbb{C}$) is labeled by $(\theta,\pi)$, where...
View ArticleNon weakly-group-theoretical integral fusion category
Is there an integral fusion category of rank $7$, FPdim $210$ and type $(1,5,5,5,6,7,7)$ with the following fusion rules (or the little $\color{purple}{\text{variation}}$...
View ArticleProof of redundancy for defining relation in current algebra $J$ presentation
Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$. The $\mathfrak{g}$-current (Lie) algebra is $\mathfrak{g} \otimes \mathbb{C}[t]$, with Lie bracket given by $[a \otimes t^m, b \otimes t^n]...
View ArticleSmallest finite dimensional $\mathbb{C}^*$-Hopf algebra whose Drinfeld double...
In this question, let us call a finite dimensional $\mathbb{C}^*$ Hopf algebra $H$"trivial" if there exists a finite group $G$ such that one of the following three equivalent statements is true:(1)....
View ArticleDoes the canonical element associated to a finite dimensional $\mathbb{C}^*...
Let $\mathcal{A}$ be a finite dimensional $\mathbb{C}^* $-Hopf algebra. Let $B(\mathcal{A})$ be a basis of $\mathcal{A}$ and let$B(\mathcal{A}^* )=\{ \delta_x\in\mathcal{A}^* | x\in B(\mathcal{A})...
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