# Questions tagged [kac-moody-algebras]

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81
questions

**9**

votes

**1**answer

591 views

### 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 ...

**2**

votes

**1**answer

116 views

### 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 $g$, ...

**5**

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**0**answers

167 views

### 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 ...

**3**

votes

**0**answers

53 views

### 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 ...

**1**

vote

**1**answer

43 views

### 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:
\...

**3**

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**0**answers

112 views

### 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 ...

**6**

votes

**1**answer

201 views

### 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.
...

**6**

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86 views

### 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 ...

**3**

votes

**1**answer

112 views

### 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 ...

**12**

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162 views

### 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 ...

**5**

votes

**1**answer

204 views

### 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 ...

**0**

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103 views

### 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 ...

**9**

votes

**0**answers

391 views

### 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 ...

**3**

votes

**0**answers

78 views

### 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 ...

**6**

votes

**0**answers

78 views

### 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 $\text{Der } \Bbb A$ ...

**3**

votes

**1**answer

123 views

### 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 ...

**3**

votes

**0**answers

55 views

### 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 ...

**1**

vote

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180 views

### 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 ...

**3**

votes

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56 views

### 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} \...

**2**

votes

**1**answer

90 views

### 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 ...

**5**

votes

**0**answers

154 views

### 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.
...

**2**

votes

**0**answers

95 views

### 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 $n$.
...

**1**

vote

**0**answers

121 views

### 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 ...

**1**

vote

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105 views

### 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{...

**0**

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75 views

### 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 ...

**7**

votes

**0**answers

715 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 ...

**6**

votes

**1**answer

713 views

### 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, ...

**11**

votes

**1**answer

637 views

### 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 ...

**5**

votes

**1**answer

160 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 ...

**3**

votes

**1**answer

91 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 ...

**1**

vote

**0**answers

111 views

### 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 ...

**3**

votes

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97 views

### 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=\...

**10**

votes

**1**answer

457 views

### 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 ...

**1**

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**0**answers

87 views

### Weyl Group action on the complement of the Tits Cone in a Kac-Moody algebra

Given a Kac-Moody algebra $\mathfrak h$ and its Weyl group $W$, the action of $W$ on the Tits cone $X$ is well understood. Decompose $\mathfrak h$ into $X\cup -X\cup L$. Then the action of $W$ on $-...

**10**

votes

**1**answer

456 views

### Realisation of Kac-Moody Lie algebras

I am reading Infinite dimensional lie algebras by Kac. He starts with a $n \times n$ GCM (Generalized Cartan Matrix) $A$ of rank $l$, then he defines the realization associated with the matrix $A$ ...

**2**

votes

**3**answers

358 views

### Graph of a Lie super algebra

Let $A$ be a generalized Cartan matrix and let $\mathfrak{g}$ be the Kac-Moody Lie algebra associated to $A$. There is an associated graph of $\mathfrak{g}$ which is known as the Dynkin diagram of $\...

**5**

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76 views

### Conformal Dimension and Highest Weight States of Coset CFT

I am trying to understand the vertex operator algebras of the following form:
$$\frac{U(M|N)_{k_1}}{U(L)_{k_2}}$$
Where $U(M|N)$ is the unitary supergroup, $U(L)$ is the usual unitary group, and $...

**6**

votes

**1**answer

310 views

### Reference on Highest Weight Module of Kac-Moody Algebra

I am trying to understand this paper. The construction requires the understanding of the following concepts in the representation theory of simple and affine Lie algebras:
The construction of Verma ...

**9**

votes

**1**answer

342 views

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

Let $V(\lambda)$ be the unique irreducible representation of a Kac—Moody algebra $\mathfrak{g}$ with the highest weight $\lambda$. If $\mathfrak{g}$ is not of finite type, then even for $\lambda$ one ...

**3**

votes

**1**answer

107 views

### The normalised form for the twisted Kac-Moody algebra

Consider the affine Kac-Moody algebra $\mathfrak g=\widehat{\mathfrak{sl}}_r(\mathbb C((t)))$ and consider the two involutions $$a(t)\rightarrow \sigma(a(t))=-\,^ta(-t),$$ and when $r$ is even $$a(t)\...

**4**

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102 views

### Gaussian decomposition in the polynomial loop group

Let $G^\min$ be a minimal Kac-Moody group. There is an affine ind-variety structure on $G^\min$ such that multiplication induces a regular isomorphism of $U^- \times B^\min$ with an open subset $G^\...

**2**

votes

**1**answer

132 views

### unitary representations of Kac-Moody algebras

Is there an easily accessible text containing a detailed description and derivation of all unitary representations of affine Kac-Moody algebras?

**2**

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124 views

### Is the set of rays through imaginary roots of a Kac-Moody algebra dense in the imaginary cone?

Let $\mathfrak{g}(A)$ be the Kac-Moody Lie algebra associated to an indecomposable generalized Cartan matrix $A$, and let $Z$ be the convex hull in $\mathfrak{h}^*_{\mathbb{R}}$ of the set $\Delta_+^{...

**5**

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201 views

### Feigin-Frenkel centre and opers for reductive Lie algebras

Edward Frenkel (together with Boris Feigin and others) has proven many interesting results connecting the representation theory of an affine Kac-Moody algebra at the critical level with the geometry ...

**1**

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**0**answers

43 views

### How to improve this argument for a restriction of the universal $R$-matrix to $U_{q}^{+}(\mathfrak{g})\otimes U_{q}^{0}(\mathfrak{g})$?

The standard universal $R$-matrix for quantum group algebra $U_{q}\left(\mathfrak{g}\right)$, where $\mathfrak{g}$ is of finite type, is
$$
R_{q}=exp\left(q\sum_{i,j}\left(B^{-1}\right)_{ij}H_{i}\...

**3**

votes

**0**answers

238 views

### Center of affine W-algebras

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $k$ a complex number. Denote by $\hat{\mathfrak{g}}$ the corresponding affine Lie algebra ($\hat{\mathfrak{g}}=\mathfrak{g}((t)...

**0**

votes

**1**answer

108 views

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

This question is related to one asked earlier about inductive presentations of unipotent radicals in Kac-Moody groups.
Let $\Gamma$ be a coxeter diagram --- i.e. an unoriented graph with $r$ vertices ...

**4**

votes

**0**answers

202 views

### Affine Steinberg groups vs Steinberg groups over Laurent polynomials

Let $R$ be a commutative ring and $\Phi$ be a finite (also called spherical) reduced irreducible root system of rank $\geq 2$. I will denote by $\mathrm{St}(\Phi,R)$ the Steinberg group of type $\Phi$ ...

**3**

votes

**1**answer

271 views

### inductive construction of unipotent radicals

Consider a directed coxeter diagram $\vec{\Gamma}$, i.e. a finite graph where each edge is decorated with one of the integer weights $\big\{3,4,6\big\}$ and those edges with weights $4$ or $6$ are ...

**6**

votes

**0**answers

149 views

### Systematic treatment of folding and valued graphs

I'm going to say beforehand that this question has something of a "am I missing something?" flavor. I'm in that odd position mathematicians often find themselves, where a topic has been addressed ...