9
votes
0answers
198 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edition', Kac mentions at the top of p. 170 that 'a simple cohomological proof of Theorem 9.11 was found by O. Mathieu (unpublished)'.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 ...
1
vote
0answers
108 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 ...
2
votes
1answer
287 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 ...
7
votes
1answer
235 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 $d$ ...
4
votes
2answers
282 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?
10
votes
1answer
346 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) ...
3
votes
1answer
643 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}} ...
5
votes
1answer
304 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 ...
6
votes
4answers
446 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
vote
0answers
216 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 $I$ ...