# Questions tagged [kac-moody-algebras]

The kac-moody-algebras tag has no usage guidance.

98
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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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### Convergence in the Weyl-Kac denominator formula

The following is the Weyl-Kac denominator formula:
$\prod_{\alpha \in \Phi^+} (1 - e_{-\alpha})^{m_\alpha}
= \sum_{w \in W} \varepsilon (w) e_{w(\rho) - \rho}$
Now each $e_\lambda : H^* \rightarrow \...

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### Difference between two definitions of affine Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, we have the notion of affinization of $\mathfrak{g}$, which is the central extension of the corresponding loop algebra.
...

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### Identifying the adjoint representation of an affine Kac-Moody algebra

We work over $\mathbb{C}$. Let $\mathfrak{g}$ be an affine Kac-Moody Lie algebra (the question is still relevant for the non-affine case, but I'm specifically interested in the affine case). Suppose ...

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### Proofs of the Frobenius characteristic map

Let $\mathfrak{S}_n$ be the symmetric group on $n$ letters, $\mathsf{Rep}(\mathfrak{S}_n)$ be the abelian category of finite dimensional complex representations of $\mathfrak{S}_n$. A classical result ...

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### Real roots along root strings

Let $A$ be a Cartan matrix, i.e. a $n\times n$ matrix with integer entries such that $A_{ii}=2$ and $A_{ij}\leq0$ for $i\neq j$. Then the corresponding Kac-Moody Lie algebra has a Cartan subalgebra $\...

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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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### Is there a derived version of affine Schur-Weyl duality?

One version of affine Schur-Weyl duality states that there is a fully faithful functor from representation of $A_r$ affine Hecke algebra to the representation of $A_n$ affine Lie group assuming $r<...

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### Action of $\widehat{\mathfrak{sl}_2}$ on symmetric functions with $\mathbb{Z}_{(2)}$ coefficients

It is known that there is a representation of the affine Lie algebra $\widehat{\mathfrak{sl}_q}$ (over $\mathbb{Z}$) on the algebra of symmetric functions, where the action of the Chevalley generators ...

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

The paper "Regular Functions on Certain Infinite-dimensional Groups" by Kac and Peterson describes the construction of a group associated to the datum of a Kac-Moody algebra in a way I haven'...

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

The online database [1] provides a list of some modular tensor categories classified by rank. Let us consider the two modular categories denoted kmA1_$\ell$ and kmC$\ell$_1 (i.e. Kac Moody $A_1$ level ...

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### Representation theory of Kac-Moody algebras in positive characteristic

I have been trying to learn about the representation theory of Kac-Moody algebras in positive characteristic. However, my usual reference for the subject (Infinite dimensional Lie algebras by Kac) ...

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### The Weyl group of Kac-Moody algebra and Coxeter group

Let $\mathfrak{g}$ be a Kac-Moody algebra, $W$ be its Weyl group, generate by fundmental reflections {$r_1,...,r_n$}. For $i,j$, we have relation $(r_ir_j)^{m_{ij}}=1$, $1\le i,j\le n$. Let $W^{'}$ be ...

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### Algebraic ode of exponential generating series

Let $G(z)$ be a rational function. So if we have a series
$$S(x):=\sum_{n}a_n x^n $$ where
$$ a_n = \prod_{i=1}^{n}G((i-1)h) $$
We can conclude that the series satisfies a Linear differential ...

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### Classification of connected finite affine type A crystals

In the survey https://www.aimath.org/WWN/kostka/crysdumb.pdf the following statement is stated as a Conjecture 4.5 (due to Kashiwara): "Every connected affine crystal graph is isomorphic to a ...

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### Which infinite-dimensional Lie algebras have realizations as algebras of global sections of vector bundles with special structure?

I would not be surprised by downvotes since this question is at the same time very naïve, very vague and might be asking about things well known for decades. Specifically vagueness comes from the word ...

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### Action of the Casimir on highest weight modules for Kac-Moody algebra

Let $g$ be a Kac-Moody algebra with a symmetrizable Cartan matrix, and let $\{u_j\}$ and $\{u^j\}$ be bases of $g$ dual with respect to a nondegenerate invariant bilinear form $(\cdot|\cdot)$ on $g$, ...

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### Word/Loop in $L(\Lambda)$

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra, with Chevalley generators $e_i,f_i$ ($i=1,...,n$).
Let $L(\Lambda)$ denote the irreducible module with highest weight $\Lambda$. Let $v$ denote ...

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### Multiplicity relation between highest weight modules, Demazure modules, and crystals

Let $\mathfrak{g}$ be a symmetrizable Kac--Moody algebra, and let $\lambda$ be an associated dominant integral weight. Then two different objects we can relate to this data is $V(\lambda)$, the ...

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

In my research I encounter some elements in a root lattice and I would like to verify that these elements are imaginary roots. Consider the root system $J_{6, 11}$ with the following Dynkin diagram:
\...

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### Macdonald's notes on Kac Moody algebras

Macdonald had given some lectures on Kac-Moody algebras in 1983. The notes are typed here by Arun Ram. However, the website seems to be old and the notes are somewhat not readable because of the ...

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### Affine Kac-Moody algebra from quantum group exchange algebra

In `Hidden Quantum Groups Inside Kac-Moody Algebra', by Alekseev, Faddeev, and Semenov-Tian-Shansky, a relationship between quantum groups and affine Kac-Moody algebras is shown for the WZW model.
...

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### Demazure modules and dimension of weight spaces

Let $\mathfrak{g}$ be a symmetrizable Kac–Moody algebra, $w \in W$ an element of the Weyl group, and $\lambda$ an integral dominant weight with $V(\lambda)$ the associated irreducible highest weight ...

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

I am reading the book: Infinite-Dimensional Lie Algebras (Kac, third edition) and the article: Affine Lie algebras and the Virasoro algebras I (Wakimoto, link). The formulas they wrote for the Lie ...

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### Relationship between crystal root operators and usual $e_i, f_i$?

Suppose I am working in a symmetrizable Kac–Moody Lie algebra $\mathfrak{g}$. Let $e_1,\dotsc,e_n,f_1,\dotsc,f_n$ denote the usual Chevalley generators of $\mathfrak{g}$. Let $V$ be a highest weight ...

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### Finite order automorpisms of affine Kac-Moody Lie algebras

It is known that for a finite order automorphism $\phi$ of a complex semisimple Lie algebra $L$, the fixed point subalgebra $L^{\phi}$ is a reductive Lie algebra and the centralizer of a Cartan ...

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### What is the name of the following root system?

The Dynkin diagram of the root system of affine $D_4$ is
$$
\circ \quad \circ \quad \circ \quad \circ \\
\circ
$$
where all of the four vertices in the first row connects to the vertex in the second ...

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### What is wrong with $A^{(2)}_{2n}$?

When dealing with affine Kac-Moody groups, especially geometrically (e.g. by examining their affine flag varieties or affine Grassmannians) I've been taught that time and time again, issues arise in ...

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### Hopf algebras structure and quantum affine algebras

I'm looking for some information about the Hopf algebras structure and the quantum groups.
In particularly I was wondering if (and eventually where) is defined in the case of quantum affine algebras ...

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### Kac-Moody Lie algebra as derivations of associative algebras

The set of derivations of an algebra $\Bbb A$ forms a Lie algebra. This is one aspect of why Lie algebras are interesting. When $\Bbb A$ is polynomial algebra in $n$ variable then $\text{Der } \Bbb A$ ...

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

I am reading the book "Infinite-dimensional Lie algebras" by Victor G Kac. This is a long question regarding my understanding of the following theorem.
In Theorem 10.7 Kac proves the ...

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### Reference request: Category of finite dimensional representations of loop algebra is not semisimple

For $\mathfrak{g}$ a semisimple Lie algebra, we may define its (untwisted) loop algebra as $L(\mathfrak{g}) = \mathfrak{g} \otimes \mathbb{C}\lbrack t,t^{-1} \rbrack$. Let $\mathcal{F}$ be the ...

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### category O is semisimple

I have been reading the book "Automorphic forms and Lie superalgebras". In Section 2.6, Definition 2.6.15 we have the definition of Category O for BKM Lie superalgebras (I have also checked the book ...

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### Embedding of Verma modules in Kac-Moody Lie algebras

Let $\mathfrak{g}(A)$ be a symmetrizable Kac-Moody Lie algebra over $\mathbb{C}$ and ($\mathfrak{h}$, $\Pi, \Pi^\vee)$ be a realization of the GCM $A$. Assume that $$\mathfrak{g}(A)=\mathfrak{h} \...

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### 2-quotient of integer partition

This question is mostly about understanding the notation used in the following article:
Alex Eskin, Andrei Okounkov, Pillowcases and quasimodular forms, in: Victor Ginzburg (ed.), Algebraic Geometry ...

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### Reference about the root systems $E_{n}$, $n \ge 10$

I am trying to understand the root systems $E_{n}$, $n \ge 10$. In particular, I would like to find some references which describe the number of real roots and imaginary roots of a given degree.
...

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### Imaginary roots in $\widetilde{E}_8$

Consider the root system of 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 associated with $n$.
...

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### Submodules of $V\otimes V^*$

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$ and let $U_q(\hat{\mathfrak{g}})$ be the corresponding quantum affine algebra (here $q$ is not a root of unity). We know ...

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### Equivalence of categories between the loop algebra of $sl_{n+1}$ and the affine Weyl group of $GL_\ell(C)$

In this paper here, Theorem 4.9 page 18, Charri and Pressley are claiming that there exists an equivalence of Categories between certain categories of the Lie algebra $\tilde{ \mathfrak g}=\mathfrak{...

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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 [1]. This module can be written as quotient of the unique (up to ...

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### 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 ...

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### 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 ...

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### 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 ...

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### Definition of integrable representation of Kac-Moody algebra

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 ...

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### Description of real roots of Kac—Moody algebra

Let $\Delta$ be a root system associated to a generalized Cartan matrix, $\alpha_1,\ldots,\alpha_n$ its simple roots.
It is known that if $\Delta$ is of finite, affine or hyperbolic type, $\alpha=\...

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### Kazhdan-Lusztig equivalence for Lie super-algebras

Let $\mathfrak g$ be a semi-simple Lie algebra. Kazhdan and Lusztig studied the category of representations of the corresponding affine Lie algebra (the central extension of $\mathfrak g((t))$) which ...