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 ...

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1answer
204 views

Dual of a Basis for a Hopf Algebra Conatined in all Dually Paired Hopf Algebras

For an infinite dimensional Hopf algebra $H$, a non-degenerate dually pairing Hopf algebra $H'$, and a choice of basis $e_i$ of $H$, is the dual basis $e^i$ (defined of course by $e^i(e_j) = ...
11
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2answers
496 views

Relationship between “different” quantum deformations

This is a generic question, a good answer to it may be a reference to a corresponding paper\textbook, but any useful comments would be okay too. Let $\mathfrak{g}$ be a (simple) Lie algebra and ...
2
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0answers
137 views

Outer automorphism for $U_q(\mathfrak{su}(2|2))$

It is known that Lie superalgebra $\mathfrak{su}(2|2)$ (and only this one, not arbitrary $\mathfrak{su}(n|n)$) has the nontrivial central extension which forms an $\mathfrak{sl}_2$ triplet, let's call ...
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1answer
466 views

Generators of the Odd Dimensional Quantum Spheres

As is well-known, the $(2N-1)$-quantum sphere $S^{2N-1}_q$ is defined to be the invariant subalgebra of $SU_q(N)$ under the coaction $\Delta_R = (id \otimes \pi) \circ \Delta$, where $\Delta$ is the ...
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1answer
167 views

Generators of the Augmentation Ideal (Counit Kernel)

For the Hopf algebra $SL_q(N)$ it is clear that the kernel of the counit contains the ideal generated by the elements $(u^i_i-1)$ and $u^i_j$, for $i \neq j$. However, I cannot seem to arrive at at ...
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1answer
116 views

Gradings Induced by Coactions?

A "well-known" fact is that a ${C}_q[U(1)]$-coaction on a vector space $V$ induces a $Z$-grading. I don't see why this is so. Clearly, a ${C}_q[U(1)]$-grading will induce a $Z$-grading. But why does ...
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1answer
2k views

Grothendieck and Non-commutative Geometry?

When Grothendieck and his followers were working on their profound progress of algebraic geometry, did they ever consider non-commutative rings? Is there anyway evidence that Grothendieck foresaw the ...
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1answer
371 views

Does There Exists a General Quantum Casimir Extending the $U_q({\mathfrak sl}_2$ Case?

As is well known (see Kassel), when $q$ is not a root of unity, the centre or the quantum enveloping algebra $U_q({\mathfrak sl}_2)$ of ${\mathfrak sl}_2$ is generated by the element $$ C_q = EF + ...
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1answer
672 views

The Major Families of Quantum Groups

If we define a quantum group to be a quasi-triangular or coquasi-triangular Hopf algebra, then what are the major families of quantum groups? Of couse, to start with we have the h-adic completions ...
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1answer
144 views

Explicit Coquasi-Triangular Quantised Coordinate Algebra of a Complex Semi-Simple Lie Group?

Let $SL_q(N)$ be usual quantised coordinate algebra of the special linear group. As is well-known, this is co-quasi-triangular algebra with coquasi-triangular structure given by $$ R(u^i_j \otimes ...
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6answers
1k views

Is there a q-analog to the braid group?

The braid group $B_n$ on $n$ strands fits into a short exact sequence of groups: $$ 1 \longrightarrow P_n \longrightarrow B_n \longrightarrow S_n \longrightarrow 1,$$ where $S_n$ is the symmetric ...
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1answer
457 views

Prove that $U^*U=UU^*=1$ for $U_q(N,C)$

Let $u^i_j$, $i,j = 1, . . . N$, and det$_q^{-1}$ be the standard generators of the quantum group $U_q(N,C)$, and define the matrices $U$ and $U^{\ast}$ by setting $U_{ij} := u^i_j$ and ...
2
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1answer
245 views

Hopf algebra and group structure correspondence for algebraic varieties

Let $V$ be a real algebraic variety and let ${\cal O}(V)$ denote its algebra of regular functions. If we put a group structure on $V$ (not necessarily an algebraic group structure) it will induce a ...
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1answer
185 views

Can we see the geometric realization of $U_q(sl_2)$'s relations as Schubert Conditions?

In Nakajima's Geometric construction of algebras(pages 3-7), he uses the subalgebra of the convolution algebra of $Gr(k^N)\times Gr(k^N)$ invariant under $GL_N$ action to construct $U_q(sl_2)$. To do ...
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1answer
908 views

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 ...
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0answers
209 views

q-deformation of the unitary group integral

There is a well-known orthogonality property of $U(N)$ group characters $$ \int d U \chi_{\mu}(U)\chi_\lambda(U^\dagger V)=\delta_{\mu\lambda}\frac{\chi_\mu(V)}{\dim_\mu} $$ where the integral is ...
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2answers
2k views

When does Lusztig's canonical basis have non-positive structure coefficients?

I've heard asserted in talks quite a few times that Lusztig's canonical basis for irreducible representations is known to not always have positive structure coefficents for the action of $E_i$ and ...
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1answer
419 views

The Killing Form for Co-Quasi-Triangular Hopf Algebras

For a co-quasi-triangular Hopf algebra $H$, with universal $r$-form $r$, there exists an important map $Q$ defined by $$ Q:H \otimes H \to k, ~~~~~~h \otimes g \mapsto r(g_{(1)}\otimes ...
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1answer
122 views

Action of Co-quasi-triangular Universal r-form on $a \otimes 1$

A very easy question I can't seem to answer: For a universal r-form on a co-quasi-triangular Hopf algebra why is $r(a \otimes 1) = r(1 \otimes a) = \epsilon(a)$?
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1answer
187 views

Establishing the Co-Quasi- Triangular Structure of FRT Algebras

Let $V$ be a finite dimensional vector space and let $R$ be a linear invertible mapping from $V \otimes V$ to itself. If we fix a basis $\{e_i\}_{i=1,2, \cdots ,n}$ then we have $N^4$ complex numbers ...
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5answers
469 views

Map constructed from the coquasitriangular structure of SLq(2) which appears not to respect the standard commutation relations

Let $A$ be a Hopf algebra dually paired with a quasi-triangular Hopf algebra $B$. If $x$ is some fixed element of $A$, then we can define a linear map $$ P_x: A \to \mathbb{C} $$ by setting $$ P_x:a ...
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0answers
175 views

What is the q-analogue of the Lefschetz decomposition?

The representation theory behind the Lefschetz decomposition in Kahler geometry was summarised very neatly by Victor Protsak in his answer to 29907 Let $W$ be a $2n$-dimensional symplectic vector ...
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2answers
899 views

Is there a quantum Hermite reciprocity?

It is well known that there is an isomorphism of $SL_2=SL(V)$ representations $$ Sym^n(Sym^m(V))\simeq Sym^m(Sym^n(V)) $$ called Hermite reciprocity (discovered in 1854). My question is: Is there ...
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1answer
1k views

Does the quantum subgroup of quantum su_2 called E_8 have anything at all to do with the Lie algebra E_8?

The ordinary McKay correspondence relates the subgroups of SU(2) to the affine ADE Dynkin diagrams. The correspondence is that the vertices correspond to irreducible representations of the subgroup, ...
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0answers
305 views

Working with quadratic Lie algebras

A quadratic Lie algebra is a Lie algebra with an invariant inner product and the main examples are semisimple Lie algebras. This definition then makes sense in any linear symmetric monoidal category. ...
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1answer
594 views

Weyl Character Formula for Quantum Groups

How much is known about the Weyl character formula for quantum groups? More specifically, has the formula been generalized to the general setting of deformed coordinate algebras $\mathbb{C}[G_q]$ of ...
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0answers
225 views

Explicit Braid Group Reps from quantum SO(N) at roots of unity

This question is related to this one (and indeed the goals are similar). Let $N$ be odd and consider the braided fusion category $\mathcal{C}$ (actually modular) obtained from $U_q\mathfrak{so}_N$ ...
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1answer
151 views

Formula for the Matrix Elements of the Inverse of special linear Universal R-Matrix of Uq(sln)

Motivated by this question, I'm going to ask a much more direct one: Let $R$ be a universal R-matrix for the quasitriangular Hopf algebra ${\cal U}_q ({\mathfrak sl}_N)$, $~$ $R ^{-1}$ its inverse, ...
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1answer
226 views

Reference for the Hecke relation for the universal R-matrix

I've come across a reference in a paper to the Hecke relation for the universal R-matrix of a quasi-triangular Hopf algebra. I've looked around, standard references, online etc, but can't seem ...
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1answer
475 views

The Inverse of a Universal R-Matrix for Quantized Universal Enveloping Algebra of sl2 and the Dual Pairing with SUq(2)

I have recently begun to study quasi-triangular structures and have come across a problem I can't resolve. Let ${\cal U}_q({\mathfrak sl}_N)$ denote the quantised enveloping algebra of ${\mathfrak ...
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2answers
236 views

Group and Hopf Algebra Structures for Projective Varieties

Let $V$ be a projective (or affine) variety. Does there exist a bijective correspondence between group structures on $V$ and Hopf algebra structures on the coordinate ring of $V$?
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3answers
1k views

Quantum group as (relative) Drinfeld double?

The most elementary construction I know of quantum groups associated to a finite dimensional simple Hopf algebra is to construct an algebra with generators $E_i$ and $F_i$ corresponding to the simple ...
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0answers
137 views

The Tangent Spaces of the Bicovariant Calculi over Quantum SU(3)

In Woronowicz's approach to differential calculi on Hopf algebras, a calculus over an algebra $A$ can be specified by a finite dimensional subspace of the dual of $A$. The best known example is his ...
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2answers
566 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 ...
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1answer
477 views

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 ...
5
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2answers
199 views

What is the explanation for the special form of representations of three string braid group constructed using quantum groups information supplied

It is well-known that representations of quantised enveloping algebras give representations of braid groups. For the examples that I know explicitly the representations of the three string braid group ...
3
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0answers
124 views

Restricting Differential Calculi to Quotients

Let $A$ be an algebra, $B \subset A$ a subalgebra, and $(\Omega^1(A),d)$ a first order differential calculus over $A$. Now, as is well known, the subalgebra of $\Omega^1(A)$ consisting of elements of ...
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0answers
192 views

Bicovariant Calculi on the Quantum Unitary Groups

The bicovariant differential calculi on quantum-$SU(n)$ have been classified (by Schmudgen I think) and have been shown to have non-classical dimension. My question is whether or not the bicovariant ...
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3answers
348 views

Group-Adjoint and Hopf-Algebra-Adjoint Maps

I've reading some introductory quantum group material and am trying to understand the algebra-space correspondence in the classical case. One object I'm stuck on is the adjoint coaction $$ Ad_R: a ...
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2answers
987 views

What is the relation between quantum symmetry and quantum groups?

What kind of role do quantum groups play in modern physics ? Do quantum groups naturally arise in quantum mechanics or quantum field theories? What should quantum symmetry refer to ? Can we say that ...
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1answer
2k views

Calculating 6j-symbols (aka Racah-Wigner coefficients) for quantum groups

Which $6j$-symbols for quantised enveloping algebras are known explicitly? The quantum $6j$-symbols for $sl(2)$ are well-known. The references are Masbaum and Vogel and Frenkel and Khovanov. What ...
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2answers
922 views

What do the local systems in Lusztig's perverse sheaves on quiver varieties look like?

In "Quivers, perverse sheaves and quantized enveloping algebras," Lusztig defines a category of perverse sheaves on the moduli stack of representations of a quiver. These perverse sheaves are defined ...
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1answer
339 views

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 ...
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1answer
701 views

Does the canonical basis of a tensor product of quantum group representations span the isotypic components of tilting modules?

It's a theorem of Lusztig that if one takes two (or more) representations of a quantum group at a non-root-of-unity choice of $q$, and look at the canonical basis of their tensor product, then each ...
3
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1answer
227 views

FTR Quantization for any Subalgebra of $GL(n)$?

As is well known, the quantum groups $SU_q(n)$, amongst others, arise from $R$-matrix solutions of the Yang-Baxter Equation. My question is: For any subalgebra of $GL(n)$, does there exist an ...
15
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11answers
4k views

Introduction to deformation theory (of algebras)?

So I know that the idea of deformation theory underlies the concept of quantum groups; I haven't found any single introduction to quantum groups that makes me fully satisfied that I have some kind of ...
5
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1answer
257 views

Are there elements of fixed weight in a crystal not killed by too many Kashiwara operators?

I've come across an annoying lemma trying to finish up an argument, and I was hoping one of you guys knew about it. Question: Given a weight $\lambda$ of a simple Lie algebra $\mathfrak g$, and ...
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1answer
2k views

Which is the correct version of a quantum group at a root of unity?

By this I mean the specialisation of the quantum group Uq(g) with q a root of unity, and the 'correct' meaning of 'correct' (enclosed in quotations since there isn't necessarily a correct answer) is ...
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6answers
944 views

Hopf algebras arising as Group Algebras

Every commutative $C^*$-algebra is isomorphic to the set of continuous functions, that vanish at infinity, of a locally compact Hausdorff space. Every commutative finite dimensional Hopf algebra is ...
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1answer
579 views

Is there a good differential calculus for quantum SU(3)?

For quantum $SU(2)$, Woronowicz gave a well differential calculus. If we denote the generators of quantum $SU(2)$ by $a,b,c,d$, then the ideal of ker($\epsilon)$ corresponding to this calculus is $$ ...