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Results tagged with qa.quantum-algebra
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user 703
Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory
3
votes
Deformation quantization of a closed Riemann surface with genus >1
See the paper Quantization of Multiply Connected Manifolds, by Eli Hawkins. arXiv link.
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2
votes
Generators of the Quantum Coordinate Algebras and Quantized Enveloping Algebra Representations
For your first question, the answer is yes, as Casteels pointed out in the comments. The reason is that, for $\mathfrak{sl}_N$, every finite-dimensional irreducible representation appears as a subrep …
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2
votes
Non-Drinfeld–Jimbo deformations and finite quantum groups
I do not know of a general method for quantizing the group algebra of a finite group. However, there is a way to do it for Coxeter groups (finite or not): the result is called an Iwahori-Hecke algebr …
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7
votes
1
answer
287
views
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 antipo …
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7
votes
0
answers
209
views
Does the braid group act faithfully on the quantized enveloping algebra?
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$, where …
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4
votes
Accepted
R-matrices, crystal bases, and the limit as q -> 1
I never found a precise reference for the statement about the R-matrix, so I ended up writing it up myself. The precise statements and proofs can be found in $\S 4.1$ of my paper with Alex Chirvasitu …
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3
votes
Finding the Universal Ideal of a (Covariant) Differential Calculus
I don't know if you still care, but I think I found the answer to your question.
Look at Proposition 1 in Chapter 14 of Quantum Groups and Their Representations by Klimyk and Schmudgen. It shows tha …
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11
votes
1
answer
673
views
Unitary representations of Quantum Groups
Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$; here I am assumin …
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11
votes
Hopf algebras examples
If you're interested in Hopf algebras in categories other than $\mathrm{Vect}$, you can look at the exterior algebra as a Hopf algebra in $\mathrm{SVect}$, the category of super vector spaces with deg …
11
votes
1
answer
761
views
R-matrices, crystal bases, and the limit as q -> 1
I am seeking references for precise statements and rigorous proofs of some facts about the actions of quantum root vectors and $R$-matrices on crystal bases for finite-dimensional representations of q …
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3
votes
An inner product that makes the R-matrix unitary
I'm pretty late to the party here, and Ben, it seems that you already have a satisfactory answer to your question, but I thought for the sake of completeness I would just post this in case anybody stu …
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16
votes
Accepted
Hopf Algebras and Quantum Groups
I don't think that you really need to learn much more algebra before you start on Hopf algebras. As long as you know about groups, rings, etc, you should be fine. An abstract perspective on these th …
13
votes
1
answer
1k
views
2-cocycle twists of braided Hopf algebras
2-cocycle twists of Hopf algebras
Let $H$ be a Hopf algebra over a field $k$. Then a (left, unital) 2-cocycle on $H$ is a map
$$ f: H \otimes H \to k$$
such that
$$ f(x_{(1)},y_{(1)})f(x_{(2)} y_{( …
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4
votes
Accepted
Classification of quantum Lie groups
What Scott's comment is getting at is that you need to have an abstract definition of "quantum Lie group" if you want to have a classification result. As the theory of quantized enveloping algebras a …
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1
vote
Drinfeld's equivalence of quantized function algebras and quantized universal enveloping alg...
I don't have the text of Drinfeld's address near to hand, but the standard way I know to do this is to take the subalgebra of the (finite) dual generated by the matrix coefficients of the irreducible …
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14
votes
0
answers
748
views
Splitting of homomorphism from cactus group to permutation group
We all learned in kindergarten that the category of finite-dimensional (type I, say) $U_q(\mathfrak{g})$-modules is braided monoidal for $\mathfrak{g}$ a complex semisimple Lie algebra. This gives an …
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2
votes
History of the Odd Dimensional Quantum Spheres
(1) I think Podles only introduced the quantum 2-spheres. His paper is linked from MathSciNet, so you should be able to get it if you have access to ams.org. I think the higher-dimensional spheres …
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5
votes
Accepted
Generators of the Odd Dimensional Quantum Spheres
This is shown in the book Quantum Groups and Their Representations, by Klimyk and Schmudgen. The result you ask for is Proposition 63 in Chapter 11. I'd expand more upon this but I have to give a ta …
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14
votes
Accepted
Relationship between "different" quantum deformations
There is certainly a way to quantize the algebra of functions on a Lie group in a way that is compatible with the $q$-deformation of the universal enveloping algebra of its Lie algebra. The standard …
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1
vote
Explicit Coquasi-Triangular Quantised Coordinate Algebra of a Complex Semi-Simple Lie Group?
The coefficients of $R$ are essentially the coefficients of the braiding of the vector representation of $U_q(\mathfrak{g})$. So, more or less, you are asking for a general formula in terms of Cartan …
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1
vote
The relation $S(u^1_i)u^j_1 = q^{-1}u^j_1S(u^1_i)$
For the first question, I would use the dual pairing with $U_q(\mathfrak{sl}_N)$. The $u_i^j$'s are defined to be matrix coefficients of the vector representation of $U_q(\mathfrak{sl}_N)$ with respe …
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7
votes
2
answers
487
views
Idempotency of the q-antisymmetrizer
Background
When constructing the exterior algebra of a (finite-dimensional, complex) vector space $V$, there are two equivalent pictures. The first is the quotient picture. First you define the ten …
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8
votes
Accepted
Reference for the Hecke relation for the universal R-matrix
For the Drinfeld-Jimbo quantum universal enveloping algebras, see Proposition 24 of Chapter 8 in the book Quantum Groups and Their Representations, by Klimyk and Schmudgen. This relation is just in t …
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