Questions tagged [kac-moody-algebras]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3
votes
0answers
59 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 ...
5
votes
0answers
74 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
1answer
93 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 complete ...
3
votes
0answers
49 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
0answers
161 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
0answers
46 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
1answer
72 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
0answers
139 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
0answers
93 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
0answers
111 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
0answers
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
votes
0answers
60 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
0answers
690 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 ...
5
votes
1answer
503 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
1answer
566 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
1answer
149 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
1answer
84 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
0answers
94 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
0answers
72 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
1answer
412 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
vote
0answers
75 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 $-...
8
votes
1answer
345 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
3answers
325 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
votes
0answers
74 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
1answer
275 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
1answer
292 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
1answer
100 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)\...
3
votes
0answers
87 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
1answer
120 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
votes
0answers
106 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
votes
0answers
177 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
vote
0answers
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
0answers
219 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
1answer
100 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
0answers
201 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
1answer
264 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
0answers
143 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 ...
5
votes
0answers
180 views

Subquotients of Jantzen Filtration for Kac-Moody algebras

Let $\mathfrak{g}$ be a complex symmetrizable Kac-Moody algebra, with triangular decomposition $\mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+$. Let $\lambda \in \mathfrak{h}^*$, and $M(\...
1
vote
0answers
88 views

Simple Diophantine equations for Cartan matrices of Kac-Moody algebras

Let us consider the matrix $A$ defined as follows: $A_{i i} =2$, $A_{i j} = - \frac{2 d_i}{d - 1 - d_{j}}, \qquad i \neq j$; $i, j = 1, \dots, n$. Here $d_1,...,d_n$ are natural numbers, $n > 1$ ...
1
vote
0answers
68 views

Extra-Lorentzian Kac-Moody algebras

My question is about Kac-Moody (KM) algebras of finite rank with symmetrized Cartan matrices $B = C A$ ($A$ is Cartan matrix) of signatures $(-,-,+,...,+)$, $(-,-,-,+,...,+)$, etc. i.e. with $2$,...
2
votes
0answers
146 views

product of root multiplicities in Kac Moody Algebras

Let $\mathfrak{g}$ be a Kac-Moody Algebra with GCM $A$. Let $\alpha$ and $\beta$ be two roots not necessarily real and $g_\alpha$ and $g_\beta$ be the corresponding weight spaces of dimension $ \text{...
-3
votes
1answer
67 views

If a g-module is sum of irreps then is direct sum of irreps

In Infinite dimensional Lie algebras book by Victor G Kac, In prop.3.6 He proves that, any integrable $g(A)$ - module $V$ is direct sum of finite dimensional, irreducible, $h$ - invariant $g_{(i)}$ ...
4
votes
0answers
201 views

Milnor's model of $EG$ and Kac-Moody groups

I am working with non-compact Kac-Moody groups $\mathcal{K}$. We can use Milnor's join model for $E\mathcal{K}=\varinjlim \mathcal{K}^{*n}$, where $\mathcal{K}^{*n}$ is the iterated join (see page 20 ...
2
votes
0answers
99 views

Conjugacy classes of involutions in Kac-Moody groups

Let $A=(a_{s,s'})_{s,s'\in S}$ be a generalized Cartan matrix. Let $G=G(A)$ be the corresponding simply connected complex Kac-Moody group with Cartan subgroup $H$ and Weyl group $W$ acting on $H$. ...
1
vote
1answer
147 views

Normalized invariant form on a Kac-Moody Algebra

For a symmetrizable Kac-Moody Algebra, we can define a normalized invariant form that performs the same role as the Killing form in the finite dimensional case. My question is, do these forms coincide,...
0
votes
0answers
97 views

Kac moody algebras and Weyl groups

1.Let $\pi$ be an integrable representation of a Kac Moody algebra $g(A)$ on a vector space $V$. For $i = 1,2 ,...,n$ set $r_i^{\pi} = (exp fi)(exp (-ei))(exp fi)$. Then how to prove that $r_i^{\pi}(...
4
votes
1answer
453 views

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

Let $\mathfrak{g}$ be a Kac-Moody algebra with Cartan subalgebra $\mathfrak{h}$, Weyl group $W$, and simple roots and coroots $\alpha_i, \check{\alpha_i}, i \in I$, respectively. Let $L$ be an ...
2
votes
1answer
811 views

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

The CFT of the SU(N) level k Kac-Moody current algebra has many Kac-Moody primary fields. I wonder if any one has calculated the quantum dimensions of those Kac-Moody primary fields. I know that, ...
1
vote
0answers
56 views

Reference for using an algebra of meromorphic functions to extend a Lie algebra

For example, let $\mathfrak{g}=\mathfrak{sl}_{2}\left(\mathbb{C}\right)$, let $s_{0}=1$, $s_{1}=-1$, $s_{2}$=0, $s_{3}=\infty$ in $\mathbb{P}_{1}\left(\mathbb{C}\right)$ and $\mathcal{R}$ is the ...
2
votes
1answer
169 views

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?

Context Let $V$ be a 2-dimensional evaluation representation of the quantum loop algebra $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)$ with $a=q$. Also, for $m\in\mathbb{Z}$, the ...