All Questions
Tagged with qa.quantum-algebra quantum-groups
323 questions
10
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5
answers
3k
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Solutions of the Quantum Yang-Baxter Equation
I am interested in finding non-constant solutions to the following Yang Baxter equation
$$R_{12}(x/y) R_{13}(x/z) R_{23}(y/z) = R_{23}(y/z)
R_{13}(x/z) R_{12}(x/y)$$
where $R(x)$ is an endomorphism ...
- 8,866
10
votes
2
answers
836
views
Computing in quantum groups
I'd be interested in doing some computations in quantum groups $ U_q(\mathfrak g)$ that are conceptually simple (``is this element 0"?, and $\mathfrak g = sl_5$), but are somewhat lengthy to do by ...
- 2,869
10
votes
2
answers
808
views
What is the discrete quantum group associated to a compact group?
Let $G$ be a compact topological group. Then $G$ is a CQG with function algebra $C(G)$ and the usual comultiplication on $C(G)$. Is there an easy description of the dual discrete quantum group $\...
- 175
10
votes
1
answer
806
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DW, state sum models, and fully extended TQFTs
I am interested in state sum models and their relations with some other of TQFTs, especially the fully extended TQFTs and the Dijkgraaf-Witten TQFTs (generalized, in the sense that finite-group-...
- 5,230
10
votes
2
answers
1k
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Quantization of conjugacy classes in a Lie group
Let $G$ be a Lie group (and to be safe, let's assume it is semisimple). Consider the action of $G$ on itself by conjugation, and form the GIT (algebro-geometric) quotient $G/\!/G$. Then let $\pi:G\...
- 18.7k
10
votes
2
answers
363
views
Symmetries of module categories over the category of representations of quantum $sl(2)$
The category $\mathcal{C}_l$ of tilting modules of the quantum group $U_q(sl_2)$ quotiented out by the modules of zero quantum dimension has a natural structure as a semisimple monoidal category when ...
- 103
10
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1
answer
324
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Is the quantum $\mathfrak{sl}_3$ invariant stronger than the quantum $\mathfrak{sl}_2$ invariant?
Both the $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ quantum framed link invariants can be computed using linear skeins. The first being computed using the Kauffman bracket and the second using a similar ...
- 103
10
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1
answer
871
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Is there a good reference for the relationship between the Yangian and formal based loop group?
For every finite dimensional semi-simple Lie group $\mathfrak{g}$, we have a loop algebra $\mathfrak{g}[t,t^{-1}]$. This loop algebra has a natural invariant inner product by taking the residue at ...
- 44.7k
10
votes
1
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518
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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 ...
- 835
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1
answer
191
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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 ...
- 459
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0
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287
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What's the relation between half-twists, star structures and bar involutions on Hopf algebras?
A star structure on a Hopf algebra is an antilinear antiautomorphism squaring to 1 and satisfying some further axioms. A Hopf algebra with a star structure is then a star algebra and a Hopf algebra in ...
- 5,565
9
votes
2
answers
479
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Is there a nice q-analogue of the Jacobi identity in a quantized enveloping algebra?
In a Lie algebra $\mathfrak{g}$ the Jacobi identity $\newcommand{\bracket}[2]{\left[#1\,#2\right]} \bracket{x}{\bracket{y}{z}} + \bracket{z}{\bracket{x}{y}} + \bracket{y}{\bracket{z}{x}} = 0$ holds. ...
- 1,161
9
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2
answers
362
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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 ...
- 219
9
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1
answer
586
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Finite compact quantum groups
Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz). It is called finite if $A$ is a finite-dimensional $C^*$-algebra. By elementary $C^*$-algebra theory, we ...
user167952
9
votes
1
answer
207
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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 ...
- 151
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1
answer
766
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The difference between $q$-deformations and $h$-deformations
What is the difference between $q$-deformations and $h$-deformations of universal enveloping algebras?
In chapter XVI of Quantum groups by Kassel, a very precise definition of a quantum enveloping ...
- 395
9
votes
1
answer
332
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The double cover in the classical limit of $U_q(\mathfrak{sl}_2)$
I am trying to learn about Drinfeld–Jimbo quantum groups and I am having trouble with the classical limit of $U_q(\mathfrak{sl}_2)$. When properly expressed the limit makes sense as $q\to 1$ — see for ...
- 141
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1
answer
814
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Yetter--Drinfeld Modules and Braidings
Let $H$ be a Hopf algebra, $M$ a right $H$-comodule for a coaction $\Delta_R$, and $\triangleleft$ a right $H$-action on $M$ such that $M$ is a Yetter--Drindfeld module. We know from general theory ...
- 1,462
9
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0
answers
381
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Finite-dimensional representations of quantum $SU(2)$
The most famous of all the quantum groups is $SU_q(2)$ - the Quantum special unitary group. The irreducible comodules of this quantum group are very well understood - they are labelled by integers (or ...
- 1,144
8
votes
3
answers
528
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Classification of $\operatorname{Rep} D(G)$
Let $G$ be a finite group and $D(G)$ its quantum double. Its finite dimensional complex representations are classified in this Dijkgraaf et al. Quasi-Quantum Groups Related To Orbifold Models. However,...
- 5,230
8
votes
2
answers
819
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Are there interesting monoidal structures on representations of quantum affine algebras?
Is there a good monoidal structure on a category of integrable representations of a quantum affine algebra? In the ordinary affine Kac-Moody case, there is the usual tensor product (symmetric, adds ...
- 45.7k
8
votes
1
answer
288
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Name for the action of a bialgebra on an algebra
Give an algebra $A$, a bialgebra $B$, and an action, that is, a bilinear map $\triangleright: B \times A \to A$ such that
$$
(b_1b_2) \triangleright a = b_1\triangleright(b_2 \triangleright a).
$$
...
- 357
8
votes
2
answers
931
views
Drinfeld's equivalence of quantized function algebras and quantized universal enveloping algebras
In his 1986 ICM address, Drinfeld discusses a way of producing a quantized function algebra (or more precisely a quantized formal series Hopf algebra) from a quantized universal enveloping algebra -- ...
- 4,612
8
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1
answer
266
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Classifying Low Dimensional Solutions of the Yang--Baxter Equation
What is the present situation with classifying solutions of the Yang--Baxter equation in low dimensions?
To make my question more specific, have all solutions for dimension $2$ and $3$ been ...
- 101
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1
answer
2k
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Quantum Group Calculations in Mathematica
I'm trying to learn how to do algebraic manipulations in Mathematica but not finding the help very helpful. I'm going to ask about a specific quantum group example related to a previous question of ...
- 1,462
8
votes
1
answer
578
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Transmutation versus operads
A while ago, I was reading Majid's book Foundations of quantum group theory, and Section 9.4 has a rather fascinating description of a Tannaka-Krein reconstruction result for quantum groups. In ...
- 45.7k
8
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1
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390
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What is the ring structure on Lusztig's integral form of quantum $\mathfrak{sl}(2)$?
Consider the quantum group $U_q(\mathfrak{sl}_2)$, with generators $E,F,K$ such that $[E,F]=\frac{K-K^{-1}}{q-q^{-1}}$. Write $[n]=\frac{q^n-q^{-n}}{q-q^{-1}}$, and $[n]!=[n][n-1]\dotsm[1]$.
In ...
- 492
8
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1
answer
394
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Image of Comultiplication on Finite Quantum Groups/Hopf Algebras
Let $A=:F(G)$ be the algebra of functions on a finite quantum groups aka a finite dimensional C*-Hopf Algebra.
Suppose that $F(G)$ is neither commutative nor cocommutative.
In their 1966 paper Kac and ...
- 1,037
8
votes
1
answer
358
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Abel's five terms relation from Yang-Baxter equation?
Can the famous Abel's five terms relation satisfied by the dilogarithm be derived from (a particular case of) the theory of Yang-Baxter equations?
If yes, how?
Thanks for any help.
- 838
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1
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355
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What are the primitives of Lusztig's twisted bialgebra $\mathbf{f}$?
I'm trying to understand the coproduct on Lusztig's $\mathbf{f}$ and, apart from the Chevalley generators, I don't know of any more primitive elements. In the $q=1$ case, the Weyl group acts as a Hopf ...
- 131
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0
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233
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$U_q(\mathfrak{g})$ is to knot theory as $U_q(\hat{\mathfrak{g}})$ is to $?$
Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over the complex numbers, e.g. $\mathfrak{sl}_n$.
Then every representation $\DeclareMathOperator\Rep{Rep}V\in \Rep U_q(\mathfrak{g})$ ...
- 5,701
7
votes
4
answers
1k
views
Compact Quantum Groups from Hopf Algebras
For a compact quantum group $C_q[G]$, it was shown by Woronowicz that $C_q[G]$ contains a dense Hopf algebra generalising the algebra of representations of $G$. I am interested in the other way around,...
- 1,462
7
votes
2
answers
469
views
Low dimensional noncommutative non-cocommutative Hopf algebras
Sweedler's Hopf algebra (see here) is the lowest dimesnional ($4$-dimensional) Hopf algebra that is noncommutative and non-cocommutative. What are the next examples? Are there noncommutative, ...
- 141
7
votes
2
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654
views
What is the state in the WRT TQFT associated to a handlebody?
Let $Y^3$ be a handlebody with boundary $\Sigma$. By definition, there is some associated vector $v_{WRT}(Y^3)\in Z(\Sigma)$, the (finite dimensional) Hilbert space associated to $\Sigma$ by the ...
- 18.7k
7
votes
1
answer
853
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Trace identities and the Kauffman Bracket skein module
Let's consider $K_t(M)$, the Kauffman bracket skein module (see this and this papers) of a three-manifold $M$. When $t=-1$, $K_t(M)$ is easily seen to be isomorphic to the ring of functions on the ...
- 18.7k
7
votes
2
answers
201
views
Does the canonical element associated to a finite dimensional $\mathbb{C}^* $-Hopf algebra always have finite order?
Let $\mathcal{A}$ be a finite dimensional $\mathbb{C}^* $-Hopf algebra. Let $B(\mathcal{A})$ be a basis of $\mathcal{A}$ and let
$B(\mathcal{A}^* )=\{ \delta_x\in\mathcal{A}^* | x\in B(\mathcal{A}) \}$...
- 727
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2
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378
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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 ...
- 219
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1
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986
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Understanding "Decategorified" symplectic Khovanov homology
In http://arxiv.org/abs/math/0405089 Seidel and Smith constructed a link invariant using Lagrangian Floer theory that was conjectured to be equivalent to Khovanov homology. The equivalence was ...
- 73
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2
answers
623
views
How to make a premodular category a modular tensor category?
A premodular category (also called ribbon fusion category) is roughly speaking a tensor category where fusion and braiding of the objects are defined. With an extra nondegeneracy condition for the ...
- 875
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2
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631
views
Abelian category from the category of Hopf algebras
The kernel of a Hopf algebra map $\phi:H_1 \to H_2$ is in general not a Hopf
sub-algebra of $H_1$. Is there some replacement or alteration of the notion
of a kernel in the Hopf algebra setting. Same ...
- 1,144
7
votes
1
answer
386
views
Inner automorphisms of Hopf algebras
Is there a reasonable notion of an inner automorphism of a Hopf algebra $H$ which in the case of a group ring $H=\mathbb KG$ for a group $G$ reduces to a conjugation by $g\in G$?
- 2,390
7
votes
1
answer
556
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Quantum coordinate ring at root of unity
Noah Snyder gave a great answer to this question about different versions of a quantum group $U_q(\mathfrak g)$ when $q$ is a root of unity. I want to ask about forms of the deformed coordinate ring $...
- 18.7k
7
votes
2
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1k
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Alternative Definition of the Quantum Determinant?
Let $M_q(n)$ be the standard quantum matrices (over the complex numbers) with generators $u^i_j$ for $i,j = 1, \ldots ,N$, and reations
$$
u^i_ju^k_j = qu^k_ju^i_j, \text{ for } i < k, ~~~~~~~ u^...
- 637
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1
answer
672
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The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group
I asked this question on Math.Stack but have not had any answers.
Question
What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group, $A$?
The trivial ...
- 1,037
7
votes
1
answer
345
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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 (...
- 1,586
7
votes
3
answers
650
views
Representations 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 $\...
- 5,230
7
votes
2
answers
1k
views
Reference for the existence of a Shapovalov-type form on the tensor product of integrable modules
Shapovalov and Jantzen showed us how to construct a nice inner product on finite dimensional representations of a semi-simple Lie algebra, by simply giving the highest weight vector inner product 1 ...
- 44.7k
7
votes
1
answer
556
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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 ...
- 3,652
7
votes
1
answer
321
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Real forms of Drinfeld-Jimbo quantum groups
A real form of a Hopf algebra $H$ over $\mathbb{C}$ is defined to be a $\ast$-structure on $H$ which is compatible with the coproduct. Compatibility of the $\ast$-structure with the counit and ...
- 8,559
7
votes
1
answer
751
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Is there a good differential calculus for quantum SU(3)?
For quantum $\operatorname{SU}(2)$, Woronowicz gave a well differential calculus. If we denote the generators of quantum $\operatorname{SU}(2)$ by $a$, $b$, $c$, $d$, then the ideal of $\ker(\epsilon)$...
- 1,776