All Questions
Tagged with qa.quantum-algebra rt.representation-theory
149 questions
7
votes
0
answers
213
views
Decomposition of certain projectives for cyclotomic q-Schur algebras
In representation theory, a very popular set of finite dimensional algebras are the $q$-Schur algebras, which are given by looking at the endomorphisms of $V^{\otimes d}$ where $V$ is the standard ...
- 44.7k
6
votes
4
answers
2k
views
level 2,3 characters of affine su(2)
Does anyone know where I can find an explicit formula to compute the level 2 or level 3
characters of affine $su(2)$? I have found several sources that give a formula to compute the
level 1 ...
- 1,709
6
votes
2
answers
1k
views
Categorifications of the Fibonacci Fusion Ring arising from Conformal Field Theory
I was reading about realizations of the "Fibonacci" fusion ring $X \otimes X = X \oplus 1$ in Fusion Categories of Rank 2 by Victor Ostrik. Apparently, there are two of them and they arise in various ...
- 22.8k
6
votes
1
answer
226
views
Tensor representations of the quantum algebra $U_q(\mathfrak{sl}(2))$ at the roots of unity
I'm trying to understand how the representation theory of $U_q(\mathfrak{sl}(2))$
works and I had a look to some books and lecture notes available on the internet. The case of $q^m\neq1$ is discussed ...
6
votes
1
answer
392
views
Corepresentations of Tensor Products of Hopf Algebras
Given two cosemisimple Hopf algebras $H,G$ over ${\mathbb C}$, denote their usual (not braided) tensor product by $G \otimes H$. What conditions do we need to impose on the Hopf algebras to ensure ...
- 159
6
votes
1
answer
272
views
Bialgebraic structure of Sklyanin algebra
Does Sklyanin algebra (which is an elliptic extension of the quantum group) admit a bialgebra structure or even Hopf algebraic structure? Or is it proved that it is impossible to have such a structure?...
- 367
6
votes
1
answer
634
views
What is the image of the half/full twist in the Hecke algebra, in the Kazhdan-Lusztig basis? What is the corresponding complex of Soergel bimodules?
Let $B_n$ be the braid group on $n$ strands. It has generators $\tau_i$ for $i = 1,\ldots,n-1$ which exchange the $i$th and $(i+1)$st strands, and which satisfy the relations
$\tau_i \tau_j = \tau_j \...
- 8,723
6
votes
1
answer
298
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
...
- 44.7k
6
votes
1
answer
210
views
Positivity of Schur elements in Iwahori-Hecke algebras
I'm interested in finite Iwahori-Hecke algebras.
If $\mathcal{H}$ is such a Hecke algebra, defined over $\mathbb{Z}[q^{\pm 1/2}]$, and $\Lambda$ an irreductible representation, there is the notion of ...
- 617
6
votes
1
answer
243
views
Jones-Wenzl-type projectors for Brauer algebras
Jones-Wenzl projectors are an explicit combinatorial description of the central projectors of the Temperley-Lieb algebra.
They also describe very explicitly the failure of certain representations to ...
- 2,533
6
votes
0
answers
442
views
Conceptual proof of braid group actions on quantum groups
Roughly 1990, Lusztig wrote a series of papers on quantum groups. Perhaps the result that the braid groups acts on $U_q(\mathfrak{g})$ is the proof which is least conceptual.
The original paper ...
- 719
6
votes
0
answers
238
views
Category of modules over a coPoisson-bialgebra
Fix a ground commutative ring $k$. A coPoisson-bialgebra is a bialgebra $H$ equipped with a linear mapping $\pi:H \rightarrow H \otimes_k H$ s.t.
$\pi$ is a coLie bracket
$\pi$ is a coderivation
$\pi(...
- 1,368
5
votes
2
answers
554
views
Jones polynomial of the concatenation of two braids
Let $\sigma_1$ and $\sigma_2$ be two braids with $n$-strings.
Are there any formulas relating $J_{\widehat{\sigma_1\sigma_2}}(q)$, $J_{\hat{\sigma_1}}(q)$, and $J_{\hat{\sigma_2}}(q)$?
Here, $J_L(q)...
- 103
5
votes
4
answers
532
views
Non-Drinfeld–Jimbo deformations and finite quantum groups
As is well-known, for the compact semi-simple Lie groups $G$, there exist non-commutative Hopf algebra deformations ${\cal O}_q[G]$ of their coordinate algebras ${\cal O}[G]$, the so-called Drinfeld--...
- 263
5
votes
3
answers
2k
views
Practical Ways to get Skew-Schur Functions
The Schur polynomials satisfy many, many identities and there is a whole book about them.
I think the easiest way is with the Vandermonde Determinant.
$$s_{3,1,1}(a,b,c) = \frac{\left|\begin{array}{...
- 22.8k
5
votes
1
answer
564
views
Is there a fusion rule in positive characteristic?
Verlinde's fusion gives a certain "tensor product" of representations of loop groups. The category of representations of loop groups has (essentially equivalent) two incarnations. One is analytic, ...
- 18.7k
5
votes
2
answers
403
views
Indecomposable, non-simple, modules of quantum groups at roots of unity
Let us consider the quantum group $U_q(\mathfrak{sl}_2)$ (as defined in Kassel's book on quantum groups), for $q$ being a root of unity of order $d$ (i.e., $d$ is the smallest positive integer for ...
- 7,746
5
votes
1
answer
497
views
Quantum groups at $q=-1$
For a Drinfeld--Jimbo quantized enveloping algebra $U_q(\frak{g})$, it is standard knowledge that the categories of modules are very different in the $q$ a root of unity, and $q$ not a root of unity ...
- 219
5
votes
1
answer
429
views
Crystal basis for quantum groups and Lie algebras
Lie $g$ be a finite dimensional complex simple Lie algebra and $U_q(g)$ the corresponding quantum group, where $q$ is not a root of unity. Every simple finite dimensional $g$-module is of the form $V(\...
- 6,201
5
votes
2
answers
923
views
Status of a conjectural definition of H. Nakajima
In his paper '$t$-analogue of $q$-characters of finite dimensional representations of quantum affine algebras' - http://arxiv.org/abs/math/0009231 - H. Nakajima states a conjectural definition of the $...
- 1,201
5
votes
1
answer
143
views
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 ...
5
votes
3
answers
348
views
Software for Planar Algebras or Group Rings
Does software exists for calculating with planar algebras or group rings? It could be part of Mathematica or be an extension of Python or Java or C. What would go into designing such a data-type ...
- 22.8k
5
votes
1
answer
505
views
Generalized Wigner 3-j symbol and Legendre functions
Let $P_{n}(x)$ the $n-th$ Legendre polynomial. It is well-knonw that $$\int_{-1}^1 P_n(x) P_m(x) P_h(x) \, dx=2\left(\begin{array}{ccc}
n & m & h\\
0 & 0 & 0
\end{array}\right)^{2}\tag{...
- 219
5
votes
2
answers
680
views
Characters on Hopf algebras
For any algebra $A$, a character for $A$ is a non-zero algebra map $c:A \to \mathbb{C}$. For $H$ be a Hopf algebra, a character is given by $\epsilon:H \to \mathbb{C}$ the counit of $H$. I am looking ...
5
votes
1
answer
209
views
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-...
- 175
5
votes
1
answer
181
views
Matrix coefficients of a compact quantum group
Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz).
Definition: A corepresentation matrix of $(A, \Delta)$ is a matrix $a=(a_{i,j}) \in M_n(A)$ such that
$$\...
user167952
5
votes
1
answer
183
views
Schur's Lemma for Quantized Universal Enveloping Algebra
Let $U_q(\mathfrak{g})$ (defined over $\mathbb{C}(q)$) be the quantized universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$. Let $M$ a finite-dimensional simple left $U_q(\mathfrak{g})$...
- 671
5
votes
1
answer
297
views
Quantum 9j symbols?
A formula for (SU2) quantum 6j symbols exists. A formula expressing ordinary (q=1)
9j symbols in terms of 6j symbols is long known. Unfortunately, combining both (I tried it myself) got tricky - the ...
- 4,829
5
votes
1
answer
404
views
What is the "right" hermitian structure on tensor products of quantum group representations?
This is pretty specific, but there are some experts around.
So, in Chari & Pressley, it's explained that in the standard *-structure, every irreducible, finite-dimensional representation of a ...
- 44.7k
5
votes
1
answer
215
views
Classification of $\operatorname{Rep}D(H)$
Question
Let $H$ be a finite dimensional complex Hopf algebra and $D(H)$ its quantum double. Can we classify the simple objects in $\operatorname{Rep}D(H)$ if the representations of $H$ are well-...
- 5,230
5
votes
1
answer
600
views
exceptional cases in Kazhdan-Lusztig
The Kazhdan-Lusztig story doesn't apply to the four exceptional cases $(E_6)_1$, $(E_7)_1$, $(E_8)_1$, $(E_8)_2$ (see this earlier question of mine).
What's special about those cases?
- 43.2k
5
votes
2
answers
384
views
a question about finite dimensional representation of a Hopf algebra
Let $H$ be a Hopf algebra over a field $k$ and $V$ a finite
dimensional left $H$-module. Then $End_{k}(V)$ is a right $H$-module
via $(f\cdot h)(v)=S(h_{1})f(h_{2}\cdot v)$.
We set $Ann(End_{k}(V))$={...
5
votes
1
answer
951
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 ...
- 44.7k
5
votes
0
answers
128
views
Fusion categories with $\mathrm{PSU}(2)_k$ fusion rules
Let $R_k$ be a fusion ring with $\mathrm{SU}(2)_k$ fusion rules (or equivalently $A_{k+1}$ fusion rules).
All fusion categories with such fusion rules have been classified by Frohlich and Kerler in ...
- 293
5
votes
0
answers
207
views
parameter of a quantum group
I am currently learning about quantum groups, and I got a question about how two different ways of thinking the $q$-parameter of quantum groups are related to each other. Here by a quantum group, I ...
- 323
5
votes
0
answers
85
views
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})$$
for ...
- 719
5
votes
0
answers
287
views
Drinfeld Polynomial for Yangian $Y(\mathfrak{sl}_2)$
I am looking for a direct proof that a highest weight representation of $Y(\mathfrak{sl}_2)$ is finite-dimensional if its highest weight is determined by a Drinfeld polynomial.
The results was ...
- 671
5
votes
0
answers
144
views
Are the integral forms of quantized coordinate algebras always Noetherian?
Let $U_q(\mathfrak{g})$ be the quantized enveloping algebra of a complex semisimple Lie algebra and let $\mathcal{O}_q(G)$ be the quantized coordinate algebra of the corresponding simply-connected ...
- 126
5
votes
0
answers
191
views
Modular double of elliptic quantum group
By studying dynamical quantum Yang-Baxter equations and corresponding $RLL$ relations, Felder defined an elliptic version of quantum group $E_{\tau, \eta}(sl_2)$, which can be understood as $\mathfrak{...
- 367
5
votes
0
answers
123
views
Signs associated to self-dual simple objects in a fusion category
Every self-dual simple object $X$ in a fusion category can canonically be assigned a number $a$, from its "snake" associator element:
The square of $a$ equals Muger's "squared dimension" of $X$, an ...
- 1,821
4
votes
1
answer
629
views
Kazhdan Lusztig Map and conjugacy classes of Weyl groups.
The Kazhdan Lusztig map gives a correspondence between conjugacy classes of Weyl groups and nilpotent orbits. So, let $w$ be a conjugacy class and
$N$ be the nilpotent orbit it gets mapped to under ...
- 1,073
4
votes
3
answers
344
views
Coinvariants of tensor products of Hopf algebras
Let $G$ be a Hopf algebra, considered as a right $G$-comodule in the obvious way.
The axioms of Hopf algebras imply that
$$
G^{\operatorname{coinv}(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \...
4
votes
3
answers
540
views
Name for a Hopf algebra admitting no non-trivial 1-dimensional comodule
A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional. See for example this question for a discussion.
Now every Hopf algebra $H$ admits a one-dimensional ...
- 1,144
4
votes
1
answer
192
views
Why are the divided difference operators of the nil Hecke ring only of degree 1?
In the paper "A Diagrammatic Approach to the Categorification of Quantum Groups I" (arXiv, journal, MSN), Khovanov and Lauda put a strand with a single dot in degree 2, and put the crossing operator ...
- 185
4
votes
1
answer
959
views
Jones polynomial of tangles using Temperley-Lieb algbra?
The definition of the Jones polynomial of tangles (a la Reshetikhin and Turaev) uses the space of invariants for $U_q sl_2$ and R-matrices. It seems to me the same thing cane be done in terms of the ...
- 803
4
votes
1
answer
101
views
Non-cosemisimple duals of pointed Hopf algebras
I take the following quote from an answer to this question
A Hopf algebra is called pointed if all its simple left (or right)
comodules are one-dimensional. The quantized enveloping algebras and
...
- 145
4
votes
1
answer
167
views
Reference request: decomposability of $\mathbb{G}$-Hilbert modules
Let $\mathbb{G}$ be a compact quantum group, $B$ be a $C^*$-algebra together with a right action
$$\beta: B \to B\otimes C(\mathbb{G})$$ which is a non-degenerate $*$-homomorphism satisfying $(\beta \...
- 525
4
votes
1
answer
170
views
Quantum Hamiltonian reduction and tensor products
Let $k$ be a field of characteristic zero, $\mathfrak{g}$ a finite-dimensional Lie algebra over $k$, and let $A,B$ associative $k$-algebras.
Suppose that $\mathfrak{g}$ acts on $A$ and $B$, and ...
- 1,077
4
votes
1
answer
445
views
A submodule of a tensor product of $U_q^{\prime}(\mathfrak{g})$-modules
Does anyone have a proof for the following Lemma?
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$ and $U_q^{\prime}(\mathfrak{g})$ be the quantum affine algebra over $\...
- 353
4
votes
1
answer
147
views
Equivalence of star products on two differents Poisson algebras?
Let $A$, $B$ be two commutative and associative $\mathbb k$-algebras and let $A_\hbar:=A[[\hbar]]$, $B_\hbar:=B[[\hbar]]$ be the corresponding ring of formal series. Of sense [Deformation theory and ...
- 53