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Questions tagged [qa.quantum-algebra]

Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory

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7 votes
1 answer
196 views

What is a good modern reference (or maybe this is known to someone here and lacks a good reference) which explains the relation between the Temperley-Lieb algebra and representations of $U_q(\mathfrak{...
11 votes
2 answers
202 views

Consider the quantum group $U_q(\mathfrak{sl}_2)$ for generic $q,ドル which is a $\mathbb{C}$-algebra. There are two "completions" of this quantum group that one can consider to accommodate a ...
8 votes
0 answers
324 views

Up to what order are finite-dimensional Hopf $\mathbb{C}$-algebras classified? Is there a table of this classification available somewhere?
10 votes
1 answer
302 views

Let $H_n$ be the Hecke algebra of type $A_{n-1}$ over $\mathbb{Z}[v,v^{-1}]$. The standard trace $\tau:H_n \to \mathbb{Z}[v,v^{-1}]$ is given on the standard basis $\{T_w\}_{w \in S_{n}}$ by ...
3 votes
0 answers
88 views

Let $\mathbf{U}$ be a quantized enveloping algebra defined by a root datum as in Lusztig's book. Even in finite type, this is slightly more general than something like $\mathbf{U}(\mathfrak{g}),ドル ...
9 votes
2 answers
416 views

Let $G$ be a reductive connected algebraic group over the field $\mathbb{C}$ of complex numbers (say $G = SL_{2}$ for instance). The Peter-Weyl theorem is the isomorphism of $G \times G$-modules: $$ \...
6 votes
1 answer
331 views

Let $ M $ be a submonoid of a finite group $ G $. Then $ M $ is itself a finite group. Indeed, since $ M \subset G $ is finite, for any $ g \in M ,ドル there exist integers $ n < m $ such that $ g^n = ...
5 votes
2 answers
353 views

As mentioned on pages 8-9 of notes on q-Schur algebra, the $q$-Schur algebra $S_q(n,r)$ can be realized as a quotient of the quantized universal enveloping algebra $U_q(\mathfrak{g}),ドル which is ...
2 votes
0 answers
78 views

In Jiang-Hua Lu's paper "Moment Maps at the Quantum Level" (Comm. Math. Phys. 157, 1993), Theorem 4.4 states that the smash product algebra $$ \mathcal{O}_\hbar(P) \# U_\hbar(\mathfrak{g}) $$...
3 votes
0 answers
92 views

(crossposted from mse, where it didn't get much traction) It's a famous theorem of Ringel (later extended by Green and others) that the hall algebra of the $\mathbb{F}_q$-valued representations of an ...
3 votes
1 answer
261 views

I've read (for example here on mathoverflow) that finite-dimensional cocommutative Hopf algebras over $\mathbb C$ are always isomorphic to group Hopf algebras. If the algebra is also commutative, the ...
2 votes
0 answers
72 views

Let $\mathfrak{b}_+$ be the Borel subalgebra of $\mathfrak{sl}_2$. It is equipped with a Lie bialgebra structure given by $$[H, E] = 2E, \quad \delta(E) = E \otimes H - H \otimes E, \quad \delta(H) = ...
2 votes
1 answer
225 views

I am trying to understand how one can perform second quantization in the case of the Hamiltonian of the hydrogen atom, i.e. when the one particle Hamiltonian acquires an external coulomb potential. ...
4 votes
1 answer
202 views

I am looking for a proof of Proposition 10.2 in Etingof and Schiffmann’s Lectures on Quantum Groups. In particular, I would like to understand the proof of the following statement in an elementary way:...
3 votes
0 answers
137 views

First of all, my question is motivated by the following phenomenon. Suppose that $Y$ is a quantisation of a Lie bialgebra (say $y$), and suppose that $Y' \subset Y$ be a coideal subalgebra. In the ...

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