# Tagged Questions

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### Why should affine lie algebras and quantum groups have equivalent representation theories?

Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$ and let $\hat{\mathfrak{g}}$ be the Kac-Moody algebra obtained as the canonical central extension of the algebraic loop algebra ...
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### PBW basis and canonical basis

Consider the example of $\mathfrak{g} = sl_3$. Then $$\mathfrak{g} = \mathfrak{n} \oplus \mathfrak{h} \oplus \mathfrak{n}^{-},$$ where $\mathfrak{n}$ is generated by $E_{12}, E_{13}, E_{23}$, ...
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### “Quantum Littlewood-Richardson” Rule?

Let $\frak{g}$ be a complex semi-simple Lie algebra, and $\lambda,\mu \in P^+$ two positive dominant weights with corresponding irreducible representations $V(\lambda)$ and $V(\mu)$. The tensor ...
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### lie algebras, Kac Moody, and quantum mechanics book

Hi all, I've just finished a graduated course on Kac-Moody algebras, and I'm really looking for some reading in regard to their applications to Quantum Mechanics. Can you help?
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### The Killing form on quantized enveloping algebras and reduction to the classical case

Let $U_q$ be the quantized enveloping algebra associated to a semisimple Lie algebra $\mathfrak g$. It is a result due to Tanisaki (see here; also see Chapter 6 of Jantzen's book Lectures on Quantum ...
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### Modules which are direct sum of weight spaces.

For a semisimple Lie algebra $\mathfrak{g}$, a highest weight module $V(\lambda)$ with highest weight weight $\lambda$ has the property that every submodule $W$ of $V(\lambda)$ is a direct sum of the ...
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### Does there exist a canonical “degree” filtration on quantum groups?

For any lie algebra $\mathfrak g$, there is a natural filtration on $U(\mathfrak g)$ by "degree": the filtered piece $U^{\leq n}(\mathfrak g)$ is just the image in $U(\mathfrak g)$ of ...
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### Fixed points of quantised enveloping algebra for affine $\mathfrak{sl}_n$

Consider the automorphism of the algebra $U_q(\widehat{\mathfrak{sl}}_n)$ induced by the obvious diagram automorphism of the extended type A Dynkin diagram. More precisely, if the vertices of the ...
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### 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 ...
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### 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, ...