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22
votes
3answers
783 views

What's the state of affairs concerning the identification between quantum group reps at root of unity, and positive energy affine Lie algebra reps?

In his paper [1], Finkelberg used Kazhdan-Lusztig's massive work [4,5,6,7,8] to prove that $Rep^{ss}(U_q\mathfrak g)$ (the semisimplification of the category of finite dimensional reps of ...
11
votes
1answer
366 views

What are the simple Lie superalgebras of type E?

Background Simple finite dimensional Lie superalgebras over $\Bbb C$ have been classified. There are the Cartan type superalgebras (algebras of purely odd vector fields), two strange families P(n) ...
10
votes
1answer
268 views

Cohomological Proof of Serre Relations for a Symmetrizable Kac-Moody Algebra

In 'Infinite Dimensional Lie Algebras, 3rd edition', Kac mentions at the top of p. 170 that 'a simple cohomological proof of Theorem 9.11 was found by O. Mathieu (unpublished)'. Does anyone know how ...
7
votes
1answer
260 views

Restriction of highest-weight representations to Heisenberg subalgebras

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $\tilde{\mathfrak{g}}=\mathfrak{g}((t))\oplus \mathbf{C}K\oplus \mathbf{C}d$ its Kac-Moody extension ($K$ is the level and $d$ ...
7
votes
1answer
176 views

minimal energy of affine Lie algebra reps

Let $\mathfrak g$ be a simple Lie algebra. Let $\widetilde{L\mathfrak g}$ be the universal central extension of $L\mathfrak g:=\mathfrak g[t,t^{-1}]$. Let $V_\lambda$ be a positive energy ...
6
votes
4answers
487 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 ...
6
votes
0answers
110 views

Are Schubert varieties for Kac-Moody groups cut out by linear equations?

Let $G$ be a reductive group, and let $X$ be a partial flag variety for $G$. Then it is known that for any projective embedding of $X$, that the equations scheme-theoretically cutting out a Schubert ...
5
votes
1answer
319 views

Is the centralizer of a torus in a Kac-Moody algebra always a Borcherds algebra?

If one has a finite dimension simple Lie algebra, one can easily calculate that taking the centralizer of a torus (or toral subalgebra), that is, summing the weight spaces that lie in some proper ...
5
votes
1answer
435 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?
5
votes
0answers
203 views

What are global sections of the determinant bundle on the Beilinson-Drinfeld Grassmannian?

Let $X$ be a smooth proper algebraic curve over $\mathbb{C}$, and let $G$ be a reductive group over $\mathbb{C}$. Let $Gr_{X,n}$ be the Beilinson-Drinfeld Grassmannian (for n points in $X$), which ...
4
votes
2answers
300 views

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?
3
votes
1answer
669 views

A possible mistake in Kac's “Infinite Dimensional Lie Algebras”

I have a paperback 3rd edition and on page 65 you can find Proposition 5.8. My question is about part (c): If $A$ is of indefinite type, then $$ \overline{X} = \{ h \in \{ \frak h_{\mathbb{R}} ...
3
votes
1answer
314 views

Kac Moody algebra defintion

Why is the dimension of the cartan subalgebra $2n-\text{rank}(A)$ in the defintion from Kumar's book. From a few examples I can see why the defintion is the way it is, but, I would like a better ...
2
votes
1answer
123 views

Is is possible to lift an equivariant map of Loop lie algebras to an equivariant map of Loop groups?

For brevity, let $LG=\mathbb{T}\ltimes \tilde{L}G$, the affine loop group and let $G$ be a simple simply conneceted Lie group. I have a map $\phi:L\mathfrak{g} \to L\mathfrak{g}$ that is equivariant. ...
1
vote
0answers
118 views

Are Generalized Verma modules natural w.r.t isometries?

Let $H$ be a subgroup of $G$ with Lie algebras $\mathfrak{h}$ and $\mathfrak{g}$ respectively. If I have 2 representations $V, W$ of $\mathfrak{h}$ equipped with a $\mathfrak{h}$ invariant inner ...
1
vote
0answers
228 views

Does Iwahori subalgebra correspond to any Cartan decomposition for affine Kac-Moody algebra?

Let $\hat{\mathfrak{g}}$ be an affine Kac-Moody algebra which is the central extension of $\mathfrak{g}[t,t^{-1}]$(polynomial version). Consider Iwahori subalgebra $I$. My question is whether $I$ ...
0
votes
0answers
18 views

How to find the character of $\mathfrak{U}\left(\mathfrak{n}_{-}\right)$? [migrated]

Let $\mathfrak g$ be a Kac-Moody algebra. Then $$ \mathfrak{n}_{-}=\oplus_{\alpha\in\varPhi_{+}}\mathfrak{g}_{-\alpha} $$ and for $\mathfrak{U}\left(\mathfrak{n}_{-}\right)$ the ...
0
votes
0answers
68 views

Equivalence of Kahler structures of based loop group and its Grassmannian model

In Pressley-Segal's Loop Groups, we have the following spaces equipped with Kahler structures. Let $G$ be a compact, connected, (simply connected) group with Lie algebra $\mathfrak g$. Let ...