# Questions tagged [kac-moody-algebras]

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### How do you find the generators of $E_{10}$?

$E_{10}$ is a hyperbolic Lie group, i.e. one that not only has infinite generators, but one where the number of generators grows exponentially when you commutate more elements of the algebra. So at ...
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### Functoriality of Feigin–Frenkel duality

For a simple Lie algebra $\mathfrak{g}$, we have the W-algebra of level $k$, denoted by $\mathcal{W}^k(\mathfrak{g})$. Using Wakimoto free field realization and screening operators, Feigin and Frenkel ...
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### Completions of infinite rank Kac-Moody algebras

I am reading Kac's Infinite dimensional Lie algebras, third edition. In section 7.12, Kac discusses completions of Kac-Moody algebras of infinite rank, and define $\bar{\mathfrak{g}}(A)$, for any ...
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### From Cartan-Weyl matrix to structure constants

Assume that some Cartan-Weyl matrix $A_{ij}$ (of a Kac-Moody algebra, not necessarily a semisimple Lie algebra!) is given. Since it completely determines the algebra, it must completely determine the ...
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### Different views on Highest weight irreducible modules of the Virasoro algebra

Every highest weight irreducible representation of the Virasoro algebra can be labelled uniquely by a pair $(c,h)$ of complex numbers . This module can be written as quotient of the unique (up to ...
746 views

### Serre presentations over $\mathbb{Z}$

Given a Cartan matrix $A=(a_{ij})_{i,j\in I}$, a classical result of J.-P. Serre asserts that the complex semisimple Lie algebra $\mathfrak g=\mathfrak g(A)$ corresponding to $A$ admits a presentation ...
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### The use of Schur's lemma for Lie algebras in physics (CFT)

Anytime a one-dimensional central extension appears in the physics literature, immediately they assume that in any irreducible representation the central charge will be a multiple of the identity, ...
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### Physicists misuse the term "Kac Moody algebra". Does that bring problems?

In physics textbooks one frequently sees the name (affine) Kac Moody algebra used to describe the universal (one dimensional) central extension of the loop algebra of a semisimple algebra. But this is ...
182 views

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

Let $\Phi_+$ be the set of all positive roots for a Kac-Moody algebra. Denote by $\alpha_i$ the simple root associated with node $i$ by for $i \in \{1, \ldots, n-1\}$ and by $\beta$ the simple root ...
105 views

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

Let $\mathfrak g$ be a Kac-Moody algebra (symmetric, or hyperbolic, or whatever other assumptions you need) with simple roots $\alpha_i$. For $\alpha$ a root, write $\alpha$ in the basis of simple ...
I have seen several definitions of integrable representation $V$ of Kac-Moody algebra $\mathfrak{g}$ online. Which one is the standard one? Are they actually equivalent? First one is $e_i, f_i$ acts ...