10 votes
Accepted

Realisation of Kac-Moody Lie algebras

An equivalent definition of a realisation of a GCM $A=(a_{ij})_{1\leq i,j\leq n}$ of rank $\ell$ is as follows: it is a triple $(\mathfrak h, \Pi, \Pi^{\vee})$ where $\mathfrak h$ is a complex vector ...
9 votes

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

In a recent paper https://arxiv.org/abs/2003.12930 the space of global sections on Schubert subvarietis of Beilinson-Drinfeld Grassmanian was computed. It turns out to be global Demazure module over ...
9 votes
Accepted

Number of real roots in type $\tilde{E}_8$

I don't believe there is a reference for this, for this follows immediately from the description of real roots in affine root systems. Namely, by Proposition 6.3(a) in "Infinite dimensional Lie ...
8 votes

Physicists misuse the term "Kac Moody algebra". Does that bring problems?

I can't address all uses by all physicists, but in many contexts, they consider only representations at a fixed level that admit a well-behaved energy grading. That is, sometimes an energy grading is ...
  • 43.4k
6 votes
Accepted

The use of Schur's lemma for Lie algebras in physics (CFT)

Let $\mathfrak{g}$ be a complex Lie algebra with a distinguished nonzero central element $x$, and let $V$ be an irreducible representation of $\mathfrak{g}$. The usual proof of Schur's lemma can be ...
  • 43.4k
6 votes
Accepted

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

The proof is written in Mathieu, Olivier. Formules de caractères pour les algèbres de Kac-Moody générales. (French) [Character formulas for general Kac-Moody algebras] Astérisque No. 159-160 (1988), ...
5 votes
Accepted

Highest weight representations of Kac—Moody algebras: what is inside the weight spaces?

The key word you're looking for is Littelmann path model, which gives a nice geometric description for the basis of a weight space of a highest-weight representation of a symmetrisable Kac-Moody ...
  • 4,108
4 votes

Reference on Highest Weight Module of Kac-Moody Algebra

I highly recommend the book Affine Lie Algebras, Weight Multiplicities, and Branching Rules by Kass, Moody, Patera, and Slansky. https://www.ucpress.edu/op.php?isbn=9780520067684
4 votes
Accepted

Does the Weyl group preserve coprimality in Kac-Moody algebras?

Note that, in your notation, if $d_i \in \mathbb Z$ are such that $\sum_i d_i a_i = \gcd_i a_i$, then also $\sum_{i \ne j} (d_i + \langle\alpha_j^\vee, \alpha_i\rangle d_j)a_i + d_j(a_j - \sum_i \...
  • 9,067
4 votes

Kazhdan-Lusztig equivalence for Lie super-algebras

In: Kazhdan-Lustzig polynomials and character formulae for the Lie superalgebra $gl(m|n)$, J. Amer. Math. Soc. 16 (2002), 185–231, J. Brundan develops a conjecture on the characters for the ...
3 votes
Accepted

Constructing a Kac-Moody group as a quotient of the free product of its root subgroups

Note first that the derived Kac-Moody algebra $\mathfrak g'$ is the Kac-Moody algebra $\mathfrak g_{\mathcal D}$ associated to the Kac-Moody root datum $\mathcal D$ of simply connected type (see [1, ...
3 votes
Accepted

Graph of a Lie super algebra

See §15 in Dictionary on Lie Superalgebras, by L. Frappat, A. Sciarrino, P. Sorba. They define the Dynkin diagram of basic Lie superalgebras (i.e., those with an even nondegenerate invariant bilinear ...
3 votes
Accepted

Is every weight of an integrable highest weight module in the Tits cone?

Yes. The Tits cone, as the name implies, is a cone: in particular, it's convex. Any weight for a highest weight module is an affine linear combination of finitely many extremal weights (I'll leave ...
  • 42k
3 votes
Accepted

Quantum dimension in SU(N) level k Kac-Moody algebra

I found a nice database on Web. For example https://www.math.ksu.edu/~gerald/voas/mtc/kmA3_3.html list the data for SU(4) level 3 case.
3 votes
Accepted

How does an element $T\left(z\right)$ act on a $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)\left[\left[z\right]\right]$-module?

For the meaning of $T(z)$, see the paper of Frenkel and Hernandez (http://arxiv.org/abs/1308.3444, version 4) Example 5.6. The eigenvalues of $T(z)$ are related to the so-called Frenkel-Reshetikhin q-...
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2 votes

exceptional cases in Kazhdan-Lusztig

If I remember correctly, Kazhdan and Lusztig first pick an appropriate irreducible rigid object/module ${\bf V}^{\kappa}_{\lambda}$ in the tensor category $\tilde{\mathcal{O}}_{\kappa}$ of $\tilde{\...
  • 176
2 votes

The normalised form for the twisted Kac-Moody algebra

Right - I think I see now... you consider the two twisted loop algebras as subalgebras of $\mathfrak{sl}_n({\mathbb C}((t)))$, in fact as fixed points for the involutions $\sigma$, $\tau$ on this ...
  • 1,164
2 votes

Normalized invariant form on a Kac-Moody Algebra

There are textbook discussions which deal with such questions in detail: see for example Chapter 16 in the book Lie Algebras of Finite and Affine Type by Roger Carter (Cambridge Univ. Press, 2005). ...
2 votes

Graph of a Lie super algebra

A good review article on Lie superalgebras by V.G. Kac can be found here: https://www.sciencedirect.com/science/article/pii/0001870877900172 Certain subtle points about Cartan matrices and Dynkin ...
2 votes
Accepted

2-quotient of integer partition

I suspect $\alpha_i$ and $\beta_i$ refer to heights in the Russian way to describe Young diagrams (rotate the English notation 135 degrees), this makes it into a piecewise linear function, and this ...
2 votes

Twisted affine Lie algebras, Lie bracket and normalized standard invariant form

I think that there is just a little mess between things that are denoted $K$, $K'$ in the book, as well as $d$, $d'$. For that, let us examine these formulas carefully. Using the first formula for the ...
2 votes

Finite order automorpisms of affine Kac-Moody Lie algebras

Heintze, Ernst; Groß, Christian. Finite order automorphisms and real forms of affine Kac-Moody algebras in the smooth and algebraic category. (English summary) Mem. Amer. Math. Soc. 219 (2012), no. ...
1 vote

Complete reducibility of integrable modules over symmetrizable Kac-Moody Lie algebras

I managed to prove (3) : Claim: The Casimir element $\Omega$ preserves the weight spaces of $V$. Proof: Let $v \in V_{\lambda}$. Since $\Omega$ commutes with the action of $\operatorname{lie} g$ ...
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1 vote
Accepted

Relation between the modular categories SU(2)_n and Sp(n)_1

Below is an email of Andrew Schopieray answering positively this question. I randomly noticed a question of yours on mathoverflow about the fusion rules of some quantum group categories. Feel free to ...
1 vote
Accepted

How to verify that an element in the root lattice is an imaginary root of a non-hyperbolic root system?

Denote $$K = \{ \alpha\in Q_+\setminus\{0\} \mid \langle \alpha,\alpha_i^\vee \rangle \leqslant 0 \text{ for all $i$ and $\operatorname{supp}(\alpha)$ is connected} \}.$$ Here $Q_+$ is the positive ...
1 vote

Affine Kac-Moody algebra from quantum group exchange algebra

$\let\eps\varepsilon$ I believe there is an error in the expression for the derivative of $R^{-1}$, it should be $$ \frac{d}{dx}R(x-y)^{-1}=-2\textrm{ln}(q)P\delta(x-y)R(x-y)^{-1} $$ Accordingly, one ...
1 vote

Graph of a Lie super algebra

Classical, Simple, Complex, Lie superalgebras and Complex, Affine, Kac-Moody algebras and Complex, Kac-Moody Lie superalgebras have an associated graph -up to isomorphism- in the sense of a ...
1 vote
Accepted

unitary representations of Kac-Moody algebras

Chapter III of the lecture notes on Kac-Moody and Virasoro algebras by Antony Wasserman (2010) might well be what you are looking for: This course develops the representation theory of affine Kac–...
1 vote

presentation for a nilpotent group associated to the square of a coxeter element

This is not an answer to your main question but response to your post script remark. In general it is not the case that $N \big(c^2 \big) \cong N(c) \rtimes N(c)$. Consider the case of $\vec{\Gamma} = ...
  • 1,685
1 vote

inductive construction of unipotent radicals

To A. Leverkühn, This is not a general answer to your question but rather a nice confirmation in the case of $\vec{\Gamma} = B_n $. The dynkin diagram $B_n$ has two leaves --- one joined by a type $...
  • 1,685

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