# Tag Info

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### 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 ...

### 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 ...
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### 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 ...
• 2,866

### 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
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### 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
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### 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), ...
• 2,998
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### 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

### 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
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• 176

### 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

### 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). ...
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### 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 ...
• 15.5k
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### 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 ...
• 14.8k

### 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 ...
• 15.1k

### 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. ...
• 4,586
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 $\operatorname{lie} g$ ...
• 1,179
1 vote
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### 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 ...
• 23.9k
1 vote
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### 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 ...
• 2,866
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 ...
• 16.4k
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### 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
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### 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–...
• 155k
1 vote

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$has two leaves --- one joined by a type$...
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