Questions tagged [qa.quantum-algebra]

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

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Show that $(\iota\otimes \hat{\Delta})(W) = W_{12}W_{13}$

Let $\mathbb{G}$ be a CQG (in the sense of Woronowicz) with associated dense Hopf $^*$-subalgebra $\mathbb{C}[\mathbb{G}]$. Endow the (algebraic) dual space $\mathbb{C}[\mathbb{G}]'= \mathscr{U}(\...
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Reference request : table of quantum Clebsch-Gordan coefficient

From a quick Google search, one can find a table of the first Clebsch-Gordan coefficient. For example this table. Those are used to pass between the tensor product bases and the bases as sum of ...
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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 $...
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4 votes
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Strongly simple fusion categories: the known examples?

A fusion category is called simple if its fusion subcategories are just $Vec$ and itself. Let us call a fusion category strongly simple if every fusion category Morita equivalent to it (i.e. same ...
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2 votes
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Relation between the modular categories SU(2)_n and Sp(n)_1

The online database [1] provides a list of some modular tensor categories classified by rank. Let us consider the two modular categories denoted kmA1_$\ell$ and kmC$\ell$_1 (i.e. Kac Moody $A_1$ level ...
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Associativity of Quantum Double

Here is the statement about the associativity of the quantum double of bialgebras in Klimyk-Schmudgen "Quantum Groups ..." (Sec 8.2.1) Can anyone help me derive the formula of on bottom of ...
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Quantum double construction of a quantum group

Let $\mathfrak g$ be a semisimple Lie algebra over $\mathbb C$ with a Cartan subalgebra $\mathfrak t$ and Borel subalgebras $\mathfrak b_\pm \subset \mathfrak g$. (You can assume $\mathfrak g=sl(2)$ ...
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Is there a geometric interpretation of the second derivative of the Alexander polynomial at $1$?

For an (oriented) knot in $S^3$ the number $\Gamma(K) := \Delta_K’’(1)$ shows up in a number of places in knot theory, for example the Casson-Walker-Lescop invariant. Here $\Delta_K(t)$ is the ...
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Nullstellensatz for maximal left ideals of quantum plane

Let $R=\mathbb{C}\langle x,y\rangle/\langle xy=qyx\rangle$ be the quantum plane algebra. Does some sort of Nullstellensatz holds for the maximal left ideals of $R$? By this we mean all maximal left ...
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Relating different definitions of dual of a compact quantum group

Let $\mathbb{G}$ be a compact quantum group in the sense of Woronowicz. We can look at its associated dense Hopf$^*$-subalgebra $\mathbb{C}[\mathbb{G}]$. Hence, in the framework of multiplier Hopf $*$-...
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Motivating quantum groups from knot invariants

Quantum groups are useful for making knot/link invariants: for example, $U_q(\mathfrak{sl}_2$) you get the Jones polynomial. This boils down to the fact that $\mathcal C = \operatorname{rep }U_q(\...
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Lusztig's root datum

In his "Introduction to quantum groups", Lusztig first defines a Cartan datum $(I,\cdot)$ and then a root datum of $(I,\cdot)$-type, which suggests (to me) that a root datum is not ...
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PBW basis for the quantized enveloping Lie algebra of $\mathfrak{g}_2$

I would like to know if you have any reference where I can find the canonical PBW basis for $U_q(\mathfrak{g}_2),$ computed using the action of the braid group as defined by Luzstig. Alternatively I ...
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Dimensions of centralizers in quantum Lie algebras associated to $\mathfrak{sl}_n$

Following ideas of Woronowicz, Lyubaschenko and Sudbery defined in Quantum Lie algebras of type $A_n$ the notion of a quantum Lie algebra $\mathfrak{sl}_n$. Let me focus on the case where $q$ is not a ...
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Hopf algebra with a non-invertible antipode

What is an example of a Hopf algebra with a non-invertible antipode?
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Product of $U^+_q(\mathfrak{sl}_2)_i$ in $U_q(\mathfrak{g})$ according to some reduced expression

Let $\mathfrak{g}$ be some simple Lie algebra, $\alpha_1,\alpha_2,\cdots,\alpha_n$ be its simple roots. Let $U^+_q(\mathfrak{sl}_2)_i$ be the subalgebra of $U_q(\mathfrak{g})$ generated by $E_i,K_i$. ...
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Interpretation of superfactorial in terms of plane partitions

Recently I got interested in plane partitions and the following formula by MacMahon, which counts the number of plane partitions $\pi \in B(r,s,t)$ fitting in an $(r,s,t)$-box: $$ \binom{r+s+t}{r,s,t}...
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2 votes
1 answer
198 views

Calculating fusion rules for $\operatorname{Rep}(G)$ and $G_{k}$ [reference request]

Could somebody please direct me to textbooks / literature that (perhaps lay the foundations for and) detail the method for determining the fusion rules for categories such as $\operatorname{Rep}(G)$ ...
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1 answer
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Norm of contragredient of unitary representations of compact quantum groups

Maybe the answer to the following question is known but I am unable to find it in the literature. Anyway, let me begin my question by fixing some notations and terms. Let $G = (A, \Delta)$ be a ...
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How to understand a definition in KLR algebra in the setting of quantum affine algebras?

I am trying to figure out what should the following definition correspond to in the setting of quantum affine algebra: $$ X \circ Y = Ind_{\beta, \gamma}^{\beta+\gamma} X \boxtimes Y \quad (1) $$ This ...
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Closed-form expressions for the Kashaev invariant via surgery

For a knot $K$, let $J_N(K)$ denote the $N$th Kashaev invariant of $K$. This is the same as the $N$th colored Jones polynomial evaluated at an $N$th root of unity (or $2N$th depending on your ...
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4 votes
2 answers
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In what precise sense is quantum (i.e., non-commutative) probability not expressable in terms of classical probability?

The quantum set-up has many settings, so let's fix some definitions. I will be taking the Hilbert space approach with a minor modification that I will make explicit. We begin with a Hilbert space $\...
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107 views

Representation of quantum groups

Let $\mathbb{G}=(A,\Delta_A)$ be a C*-quantum group and $\mathbb{H}=(B,\Delta_B)$ be a closed quantum subgroup of $\mathbb{G}$. We say that $\mathbb{H}$ is a closed quantum subgroup of $\mathbb{G}$ if ...
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Is the comultiplication of a compact quantum group always injective?

Let $(A, \Delta)$ be a compact quantum group in the sense of Woronowicz. Is it true that the comultiplication $\Delta : A \to A \otimes A$ always injective? This is true for both the universal (...
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4 votes
1 answer
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Exponential map and Lie correspondence within a Hopf algebra setting

The Cartier-Konstant-Milnor-Moore (et al.) theorem for Hopf algebras states that a cocommutative Hopf algebra over $\mathbb{C}$ is isomorphic to a smash product of a universal enveloping algebra of a ...
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How to interpret compositional diagrams for quantum sets algebraically

$\newcommand{\id}{\mathrm{id}}$My reference for this post is Musto, Reutter and Verdon's A compositional approach to quantum functions, arXiv:1711.07945. Questions are in bold below. Allow me to begin ...
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3 votes
1 answer
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Norm antipode on a Kac-type compact quantum group

Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group. Consider the associated dense Hopf$^*$-subalgebra $\mathcal{O}(\mathbb{G})$ and let $S: \mathcal{O}(\mathbb{G})\to \mathcal{O}(\mathbb{G})$ ...
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Woronowicz characters are multiplicative

I'm reading the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset. Let $G$ be a $C^*$-algebraic compact quantum group with function algebra $(C(G), \Delta)...
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Quantum groups at small roots of 1

I wonder if there is any literature about representations of quantum groups at a root of 1 of small order. For example, I would like to understand the case of $\mathrm{SL}(2)$ and $q=-1$ (in the ...
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4 votes
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135 views

Tensor product of representations on a compact quantum group

Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz) with function algebra $(C(\mathbb{G}), \Delta)$. Let $X \in M(B_0(H)\otimes C(\mathbb{G}))$ and $Y \in M(B_0(K)\...
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Drinfel'd polynomials for evaluation representations of $\mathbf{U}_q(\mathbf{L}\mathfrak{g})$?

We know for type $A$, there is an evaluation homomorphism from quantum affine algebra to quantum algebra, $$\operatorname{ev}_a:\mathbf{U}_q(\mathbf{L}\mathfrak{g})\to \mathbf{U}_q(\mathfrak{g})$$ for ...
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Subrepresentations of C*-algebraic compact quantum groups

Let $\mathbb{G}$ be a compact quantum group with function algebra $(C(\mathbb{G}), \Delta)$ (in the sense of Woronowicz). Let $X \in M(B_0(H) \otimes C(\mathbb{G}))$ be a (possibly infinite-...
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  • 273
2 votes
1 answer
121 views

Timmerman's "An invitation to quantum groups and duality" corollary 3.2.8

Consider the following propositions of Timmerman's book "An invitation to quantum groups and duality": I do not understand the equivalence $$h(\mathfrak{C}(\delta_V)S(\mathfrak{C}(\delta_W))...
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3 votes
0 answers
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Relative strength of Jones and colored Jones polynomials

this is my first post here. I've been studying some Knot Theory and I came to a question concerning invariants. We know that the Jones polynomial is related to the RT-invariant associated to the two-...
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12 votes
2 answers
317 views

Reference for free symmetric monoidal categories with duals on symmetric monoidal categories

The inclusion $U$ of the (2,1)-category of symmetric monoidal categories with duals into the (2,1)-category of symmetric monoidal categories admits a left 2-adjoint functor $L$ for formal reasons. In ...
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3 votes
1 answer
189 views

Characterising algebraic compact quantum groups among Hopf $^*$-algebras

Let $(A, \Delta)$ be a Hopf $^*$-algebra. Assume that $\{u^\alpha\}_{\alpha \in I}$ is a maximal collection of pairwise inequivalent irreducible unitary corepresentation matrices. I want to show that $...
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1 vote
1 answer
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Orthogonality relations for Haar state and antipode (Timmerman)

Consider the following proposition from Timmerman's "An invitation to quantum groups and duality": I am having trouble seeing why the boxed equations are true (Note that on the left the ...
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3 votes
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Explicit construction of the Drinfeld double for quasi-bialgebras

Let $A$ be a quasi-bialgebra over a field $k$ and $A$-$\mathrm{mod}$ be the category of finite dimensional left $A$-modules. Since $A$-$\mathrm{mod}$ is a monoidal category, its Drinfel’d center $\...
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3 votes
1 answer
216 views

Is there a definition of Heisenberg double for quasi-Hopf algebras?

$\newcommand{\dmod}{\text{-}\mathrm{mod}}$Let $A$ be a finite-dimensional quasi-Hopf algebra over a field $k$ and $A\dmod$ be the category of finite dimensional left $A$-modules. Since $A\dmod$ is a ...
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2 votes
1 answer
148 views

A twisted Haagerup category without pivotal structure

Let $G$ be a finite group, $\tau$ a group automorphism of $G$ of period two and $m$ a natural number. Following [1, Definition 2.1], a complex fusion category $\mathcal{C}$ is called a quadratic ...
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Indecomposable comodules

For a Hopf algebra $A$, we say that a comodule $V$ is indecomposable if it is not equivalent to a direct sum of irreducible comodules. $\bullet$ What is an example of a finite dimensional ...
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2 votes
1 answer
485 views

What is a coalgebra?

A coalgebra is a triple $(A,\Delta,\epsilon)$ consisting of a vector space, a coproduct, and a counit. Now as we all know, just like the unit in an algebra, the counit of a coalgebra is unique, i.e. ...
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2 votes
0 answers
150 views

Categorical dimension and formal codegrees

Let $\mathcal{C}$ be a complex fusion category. If it admits a pivotal structure $a$ then by [1, Proposition 4.7.12], $\dim_a$ induces a character $\chi$ on the Grothendieck ring $Gr(\mathcal{C})$, of ...
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5 votes
2 answers
177 views

Drinfeld center of a Deligne tensor product

Let $\mathcal{C}$ and $\mathcal{D}$ be two tensor categories (if necessary, assume they are fusion categories). Is the canonical braided monoidal functor $$\mathcal{Z}(\mathcal{C})\boxtimes\mathcal{Z}(...
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3 votes
1 answer
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Is there a fusion subcategory in sphericalization tensor equivalent to the original one?

Let $C$ be a fusion category. Then $C$ is not necessary spherical. But its sphericalization $\tilde{C}$ has a canonical spherical structure $i:Id\to **$. The simple objects of $\tilde{C}$ are pairs $(...
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4 votes
1 answer
103 views

Link invariants from modular categories (strictification and computation)

By the theory of Reshetikhin and Turaev, a modular tensor category $C$ gives rise to a link invariant. While $C$ is strict as a monoidal category (e.g. $\mathbb{Fib}$), calculating the link can be ...
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5 votes
2 answers
299 views

Classifying Hopf algebras that admit a single irreducible comodule

Is it possible to classify Hopf algebras $H$, over a field $k$, which admit a unique (up to isomorphism) irreducible comodule, namely the trivial $1$-dim comodule $$ k \to k \otimes H, ~~ k \mapsto k ...
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4 votes
0 answers
243 views

The fusion categories with a strict skeleton

We refer to the book Tensor Categories (by Etingof-Gelaki-Nikshych-Ostrik) for all the notions mentioned in this post. A fusion category is skeletal if two isomorphic objects are always equal. Every ...
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3 votes
1 answer
94 views

Nonstandard Podles spheres as $U_c(\frak{h})$ invariants

In this paper Podles introduced a $2$-parameter family of $q$-deformed spheres $S_{q,c}$ that are now called the "Podles spheres". The case of $c=0$ is very special and is known as the "...
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3 votes
1 answer
206 views

The adjoint representation of $U_q({\frak sl}_2)$ on itself

Let $U_q(\frak{sl}_2)$ denote the quantum universal enveloping algebra of $\frak{sl}_2$, and consider the adjoint action $$ \mathrm{ad}_X: U_q({\frak sl}_2) \to U_q({\frak sl}_2), ~~ Y \mapsto S(X_{(...
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