Questions tagged [quantum-groups]
Questions about algebraic structures known as quantum groups, and their categories of representations. Quasitriangular Hopf algebras and their Drinfel'd twists, triangular Hopf algebras, $C^\star$ quantum groups, h-adic quantum groups, various semisimplified categories at roots of unity which are called "quantum groups", bicrossproduct quantum groups, and quantum groups coming from braided tensor categories.
192 questions with no upvoted or accepted answers
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How to understand extremal vector?
Extremal vectors are defined in Kashiwara's paper. The definition is as follows.
Simple reflections in the Weyl group of $\mathfrak{g}$ acts on the crystal basis of integrable $U_q(\mathfrak{g})$-...
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156
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Questions about ``$p$-canonical basis" for $\widehat{\mathfrak{sl}_n}$ module (wedge power of natural representation)
Let $p$ be a prime number. Consider the natural representation of the affine Lie algebra $\widehat{\mathfrak{sl}_p}$, defined as follows.
$$A = \bigoplus_{i=1}^N \mathbb{C}a_i; \qquad \text{nat}_p = A ...
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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 ...
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$q$-characters of quantum affine algebras
The theory of $q$-characters for quantum affine algebras are studied in The q-characters of representations of quantum affine algebras and deformations of W-algebras, Combinatorics of q-characters of ...
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234
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Quantum Schur-Weyl duality for quantum affine algebras of other types
In the paper by Chari and Pressley, it is proved that the there is functor from the category $C_m$ of finite dimensional representations of the affine Hecke algebra of $GL(m)$ to the category $D_n$ of ...
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Partially permutative matrices
Let $V$ be a finite dimensional vector space over a field $K$. Then a map
$L:V\otimes V\rightarrow V\otimes V$ is said to satisfy the Yang-Baxter equation if $(L\otimes I)(I\otimes L)(L\otimes I)=(I\...
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Antipode action on quantum minors
Let ${\cal O}(SU_q(n))$ be the standard $q$-deformed coordinate algebra of $SU(n)$, with the canonical generators $x_{i,j}$. For $I = \{i_1,\ldots, i_r\}, J=\{j_1,\ldots,j_r\}\subseteq \{1, \ldots,n\}$...
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149
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Symmetries of modular categories coming from quantum groups
This is a request for references about the computation of the braided autoequivalences of fusion categories coming from a quantum group. I could not find even the description of braided ...
- 1,529
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Simple modules of quantum toroidal algebras
Many properties of quantum toroidal algebras are similar to quantum affine algebras. Every simple module of a quantum affine algebra of rank $n$ corresponds to an $n$-tuple of Drinfeld polynomials.
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Quantum Groups and quantum spaces - From algebra to Analysis
My question will be about the non-standard quantum projective space $\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. I want to see this algebra now on a von Neumann algebraic ...
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Center of $U_q(sl_3)$ and $U_q(sl_4)$
In the book a guide to quantum groups, page 285, the center of $U_q(g)$ is described in Theorem 9.1.6. The center of $U_q(sl_2)$ is computed explicitly in Example 9.1.7. I tried to compute the center ...
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The convolution on a semisimple finite quantum groupoid
Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to \hat{\mathbb{...
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(Non trivial) coidempotents(Co-$K$-theory)
I was interested to know about coalgebraic version of "Idempotents".
So I seached the web and I found the following interesting post :
https://math.stackexchange.com/questions/689322/co-idempotents-...
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Canonical basis of quantum groups
I am trying to understand the canonical basis of quantum groups and different ways to construct the canonical basis of quantum groups. In the comments of Lusztig's papers, the paper [92], CANONICAL ...
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quantum deformations of tensor category
I was told that, if I understand correctly, that the enveloping algebra of semisimple Lie algebra admits one family of quantum deformation as Hopf algebra, which was proved by Drinfeld. Anyone can ...
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Quantum Coordinate Algebras at Roots of Unity and Non-Standard Irrep Types
Let $\frak{g}$ be a complex semi-simple Lie algebra of rank $n$, and $U_q(\frak{g})$ the corresponding Drinfeld-Jimbo algebra. As is well-known, for $q$ not a root of unity, the irreducible finite-...
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Hopf Algebra Pairings and Module-Comodule-Equivalences
Let $\left< , \right> : G \times H \to {\bf C}$ be a dually pairing for two complex Hopf algebras $G$ and $H$. For any (left)-$G$-comodule $(V,\Delta_R)$, we can give $V$ the structure of a left ...
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Haar Functionals and Coquasi-triangular Structures
In this question it is mentioned that the coordinate algebra $C_q[G]$ Drinfeld--Jimbo algebras, for $G$ a compact semi-simple Lie group, admit a unique positive definite Haar functional. I was ...
- 1,453
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A question about q-binomials at roots of unity
I have a question about a lemma $9.3.6$ in the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. This question comes from page 301, "The restricted specialization&...
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Action of Lusztig braid group operators on locally finite part
Let $\mathbf{U}_q(\mathfrak{g})$
be a Drinfel'd-Jimbo quantum group. The quantum group $\mathbf{U}_q(\mathfrak{g})$ acts on itself by the left adjoint action $ad(u)(x)=u_{(1)}u S(u_{(2)})$, where we ...
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Quantum Schubert cell algebra and quantum odd-dimensional euclidean space
De Concini, Kac, Procesi introduced quantum Schubert cell algebra associated to a complex Lie algebra $\mathfrak{g}$ which is denoted by $\mathcal{U}^{w}_{\epsilon}$ where $w$ is an element of Weyl ...
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Is there an explicit description of a gauge transformation $F$ such that $U_{\hbar}(\mathfrak{g})$ and $(U(\mathfrak{g})[[\hbar]])_F$ are isomorphic?
Let $\mathfrak{g}$ be a semisimple Lie algebra, let $t$ be its canonical 2-tensor, and let $\Phi_{KZ}$ be a Drinfeld associator.When $R_{KZ}=e^{\hbar t/2}$, $(U(\mathfrak{g})[[\hbar]],\Phi_{KZ},R_{KZ})...
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102
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Category O for (Yangian) toroidal Lie algebras?
Suppose throughout that $g$ is a finite-dimensional simple Lie algebra over $\mathbb{C}$ and let us denote:
$$g_{[2]} := g \otimes_{\mathbb{C}} \mathbb{C}[v^{\pm 1}, t^{\pm 1}]$$
$$g_{[2]}^+ := g \...
- 1,516
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Examples of (weak-)bialgebras/Hopf algebras with a finite dimensional unitary representation and corepresentation and polynomial growth rate
I need examples (the more the better, even better if there is a systematic way of construction) of (weak-)bialgebras or (weak-)Hopf algebras $H$ with a finite dimensional representation $\rho$ and a ...
- 727
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Product of matrix entry and representation
Consider a semisimple lie algebra $\mathfrak{g}$ and the corresponding quantum group $\mathcal U_q(\mathfrak{g})$ over $\mathbf{Q}(q)$. Consider two dominant weights $\lambda,\mu\in P^+$, a matrix ...
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Invariant weights associated to algebraic quantum groups
Consider an algebraic quantum group $(A, \Delta)$ in the sense of Van Daele, i.e. a regular multiplier Hopf $^*$-algebra with a positive left integral $\varphi$ and a positive right integral $\psi$.
...
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Morphism of discrete quantum groups
In the paper Kazhdan's Property T for Discrete Quantum Groups
, we read the following fragment:
First, note that I think there is a typo and that codomain and domain of the dual maps have to be ...
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90
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Quantum groups as bialgebra cohomology classes
My question below is about how to view the quantum group $U_q(\mathfrak{g})$ as a bialgebra cohomology class.
Background: If $A$ is a bialgebra, Gerstenhaber and Schack in Bialgebra cohomology, ...
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The dual of elements $E$, $F$, and $H$ of $U_h(\mathfrak{sl}_2)$ corresponds to which element of $F_h(\mathrm{SL}_2)$ by isomorphism?
$\newcommand{\sl}{\mathfrak{sl}}\DeclareMathOperator\SL{SL}$Let $U_h(\sl_2)$ be the quantized universal enveloping algebra of $\sl_2(\mathbb{C})$ and $F_h(\SL_2)$ be the quantized function algebra of $...
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What is meant by saying that monoidal category of $U_h (sl_2 (\mathbb C))$ is different from that of $U(sl_2 (\mathbb C))[[h]]\ $?
I am following the notes on Quantum Groups given by our instructor. There I found that Drinfeld-Jimbo quantum group $U_h (sl_2 (\mathbb C))$ corresponding to the Lie algebra $sl_2 (\mathbb C)$ and the ...
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Canonical basis of the invariant part of $O_q(\mathfrak g)^{\otimes N}$
Let $\mathfrak g$ be a semi-simple Lie algebra (We can assume $\mathfrak g=sl(n)$ for simplicity) and let $O_q(\mathfrak g)$ be the corresponding quantum algebra of functions. Then $O_q(\mathfrak g)^{\...
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Anticommutation of convolution products on trace class operators of quantum groups
This question was originally posted to MathStackExchange.
Let $\mathbb{G}$ be a locally compact quantum group and let $W$ and $V$ be the left and right fundamental unitaries, i.e., they implement the ...
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Superfluous axioms for ribbon Hopf algebra
In his book Foundations of quantum group theory, Majid defines (2.1.10) a ribbon Hopf algebra as a quasi-triangular Hopf algebra $(H, R)$ with a special central element $v \in H$ satisfying
(1) $v^2 ...
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Relation Hopf categories and categorified quantum groups
In the paper Hopf Categories Crane and Frenkel gave a definition of a Hopf category, which they considered as a categorification of a quantum group. Categorifications of quantum groups have later been ...
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Frobenius reciprocities
An adjunction of the form $\mathrm{Hom}(A \otimes X, Y) \cong \mathrm{Hom}(X, A^* \otimes Y)$ in a rigid monoidal category is sometimes called Frobenius reciprocity. Is there a result that unifies ...
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Kontsevich integral on tangles and Fubini
I am reading about the Kontsevich integral, following this text : https://pdfs.semanticscholar.org/635b/c6370e8aba381724eaaa36abefba7f7a5bec.pdf
At some point (page 10 to be exact) the author claims ...
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Embedding problems on quantum groups？
We work over the field of complex numbers.
We have known that Lie algebra of type $A_2 $is a subalgebra of type $G_2$. However, when we consider their quantum groups, is this true i.e. does there ...
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almost magic unitary
A magic unitary is a unitary matrix $u=(p_{ij})_{ij}$ whose entries are all projections (in some Hilbert space) and in each row they sum to the identity and same holds for each column $(\sum_i p_{ij}=...
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Why are the quantum Fock spaces in FLOTW the same as Uglov's?
Theorem 2.5 in the well-known FLOTW paper [1] and Theorem 2.1 in Uglov's paper [2] both refer to the original JMMO paper [3]
to define quantum Fock spaces, i.e. Fock spaces for $U_q(\widehat{\...
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Branching rule for degenerate cyclotomic Hecke algebras
Grojnowski and Vazirani showed that there is a crystal isomorphism between the crystal defined by modular branching of simple modules of Ariki-Koike algebras and that of an integral highest weight ...
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How to compute the upper global basis (dual canonical basis) of an irreducible $U_q(\mathfrak{g})$-module?
Let $\mathfrak{g}$ be a semisimple Lie algebra and $U_q(\mathfrak{g})$ the corresponding quantum group. Are there some general method to compute the upper global basis (dual canonical basis) of an ...
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What is the intuition of lower global bases?
In the paper: Crystallizing the Q-analogue of Universal Enveloping Algebras, Kashiwara introduced the upper global bases. In the paper: https://projecteuclid.org/euclid.dmj/1077295931, Kashiwara ...
- 6,201
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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 $H$-...
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Quantum invariant: The canonical $2$-tensor
In Chapter XVI Kassel introduces a proper definition of a quantum universal enveloping algebra of a Lie algebra $\mathfrak{g}$. (See definition XVI.5.1). Notice that a quantum enveloping algebra has a ...
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Partial Flag Varieties and Quotients of Symmetric Polynomials
$\def\Q{\mathbf Q}\DeclareMathOperator{\Gr}{Gr}$First, consider a Grassmannian $\Gr(k, N)$ of $k$-dimensional subspaces in an $N$-dimensional space. It is known that its cohomology ring is
$$H_k=\Q[...
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Reference of a result about R-matrices
Are there some references about the result Lemma 1.2 in BRAIDED SYMMETRIC AND EXTERIOR ALGEBRAS by Arkady Berenstein and Sebastian Zwicknagl?
I type Lemma 1.2 in the following.
Let $C \in Z(\widehat{...
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104
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Comodule structure on finite dimensional Hopf algebra
Actually I am trying to establish that the following are equivalent for $f\in H^*$:
(i) $f\in \pi(H^*)$. where $\pi(H^*)$ is the vector subspace of $H^*$ (the subspace of coinvariants).
(ii) $f:H \...
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71
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Comodules of the $B,C$ and $D$ series quantum groups
In Section 11.5 of Klimyk and Schmudgen's book on quantum groups, explicit presentations of the isomorphism classes of comodules of ${\cal O}(GL_q(N))$ are given in terms of its "quantum minors". In ...
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Constant solution of the CYBE
I am learing how to solve the system of equations
\begin{align}
r_{12}+r_{21}=t,\ [[r,r]]=0,
\end{align}
where $t$ is the Casimir element of $g\otimes g$ corresponding to a non-degenerate invariant ...
- 348
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How to value $\Omega$ in T-system for twisted quantum affine algebras?
Let us proceed to the unrestricted T-systems. Choose $h\in {\mathbb{C}\backslash 2\pi \sqrt{-1} \mathbb{Q}}$ arbitrarily.
The unrestricted T-system for $U_{q}(X_{N}^{(\mathfrak{k})})$ is the following ...
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