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Questions tagged [kac-moody-algebras]

For questions regarding the definition, properties and types of the Kac-Moody algebras.

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The action of Minimal Kac-Moody group on its Lie algebra

In the textbook "Kac-Moody Groups, their Flag Varieties and Representation Theory" by Shrawan Kumar, he defines the minimal Kac-Moody groups, $\mathcal{G}_{min}$, as the group generated by a ...
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Producing a nontrivial proper ideal in the (Kac-Moody) Lie algebra $\mathfrak{g}(A)$

Proposition 1.7 in Kac's book Infinite dimensional Lie algebras states that if $A=(a_{ij})$ is any complex matrix (that is, not necessarily a generalized Cartan matrix) then the Lie algebra $\mathfrak{...
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Central Charge Calculation of $SL_k(2,\mathbb{R})$ WZW Model

According to P. Francesco et al. conformal field theory book the conformal charge of the enveloping Virasoro algebra of the affine Lie algebra $\hat{g}_k$ corresponding with Lie algebra $g$ which ...
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Jordan and Dynkin quivers

I have a simple question. If I am not wrong, we can define $\tilde{A}_l$ affine Dynkin quiver of type A, for $l\geq1$. It has $l+1$ vertices that we can order from $0$ to $l$ such that $i$ is ...
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A certain enveloping algebra of an affine Kac-Moody algebra

This is my context: I have a simple Lie algebra $\mathfrak{g}$ defined over $\mathbb{C}$ , and the corresponding affine Kac-Moody algebra $\mathfrak{g}_{k}$ , defined as the central extension by $\...
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The closure of the Tits' cone of a Kac-Moody Lie algebra

The question is based on Proposition $5.8$ in 'Infinite dimensional Lie algebras' by Victor G. Kac. The Proposition describes the closure of the tits cone in the $\textit{metric topology}$ of $\...
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When is a sum of two real roots of a symmetrizable Kac-Moody algebra is again a real root?

In general I would like to know when the sum $\alpha+\beta$ of two real roots $\alpha$ and $\beta$ of a symmetrizable Kac-Moody algebra $\mathfrak{g}(A)$ is again a real root (e.g. if $A$ is of finite ...
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Kac denominator formula and convergence

The following is the Kac denominator formula: $\prod_{\alpha \in \Phi^+} (1 - e_{-\alpha})^{m_\alpha} = \sum_{w \in W} \varepsilon (w) e_{w(\rho) - \rho}$ (Where the equality is in the ring $\mathfrak{...
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About imaginary root of a symmetrizable Kac Moody algebra

The question is from Infinite dimensional Lie algebras by Victor G Kac (ex 5.17, Page 74) Let $\mathfrak{g}=\mathfrak{g}(A)$ be a symmetrizable Kac Moody algebra and let $\Delta$ be its set of roots. ...
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Determinie the multiplicity of a root in $\mathfrak{g}(A)$ where $2\times 2$-matrix $A$ given (or arbitrary)

I'm trying to solve the exercises on Infinite-dimensional Lie algebras by Victor G.Kac. Exercise 1.6.: Let $A = \begin{pmatrix}2&-3\\-3&2\end{pmatrix}$. Show that $\operatorname{mult} (2\...
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Definition affine kac moody algebra

Take $\mathfrak{g}$ a simple Lie algebra and $k$ the Killing form of $\mathfrak{g}$. Now I found two definitions of affine Kac Moody algebra associated to $\mathfrak{g}$. The first: a one-dimensional ...
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What is the classification of the Witt algebra?

I'm learning about Kac-Moody algebras. The simplest example of an affine KM algebra that comes up is the Virasoro algebra, which is a central extension of the Witt algebra: $$ [d_l, d_m] = (l-m) d_{l+...
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What makes an affine Kac-Moody algebra 'untwisted' or 'twisted'

I'm learning about affine Lie algebras partly from J. Fuchs' Affine Lie Algebras and Quantum Groups. Actually I'm using Kerf and Bauerle but for this question I find Fuchs' labelling of the Dynkin ...
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Kac-Moody algebras: why $Z \subset$ span$\Pi^V$

I am working through Volume 1 of Kerf and Bäuerle's book (which I generally find excellent for a humble physicist like myself to learn from), and I'm unfortunately stuck. In Lemma 11.2.1c, they state ...
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Why do we introduce a realisation of a generalized Cartan matrix?

When introducing the generalized Cartan matrix, one also introduces the the corresponding realisation. Why is this necessary? I've read that if we didn't introduce the realisation, then we might not ...
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Highest-weight module over $\mathfrak{sl}_2$

I'm reading the following example, which I have trouble understanding: Let $\mathfrak{g}=\mathfrak{sl}_2$ and $\lambda=m\Lambda_1$. Consider the irreducible module $V$. Then the vectors in the ...
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How does one find subalgebras using the Dynkin diagram of a Lie algebra

The following is taken from these lecture notes on page 8.16: [...] $\mathfrak{so}(p+q)$ has as subalgebras $\mathfrak{so}(p)$ and $\mathfrak{so}(q)$ as well as their direct sum $\mathfrak{so}(p)\...
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$\dim V_\lambda=\dim V_{w(\lambda)}$ when $V$ is an integrable module over a Kac-Moody algebra

Let $V$ be an integrable module over a Kac-Moody algebra. Then $\dim V_\lambda=\dim V_{w(\lambda)}$ for each $\lambda\in\mathfrak{h}^*$ and $w\in W$ (the Weyl group). It's a proposition stated in Kac'...
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Deriving Jacobi triple from the Weyl-Kac denominator formula of affine $\mathfrak{sl}_2$

The Weyl-Kac denominator formula states $$\prod_{\alpha\in\Phi^+}(1-e(-\alpha))^{mult(\alpha)}=\sum_{w\in W}\epsilon(w)e(w(\rho)-\rho))$$ Where the product is taken over the positive roots and the sum ...
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$f_i^N(uv_\lambda)=\sum_{k=0}^N {{N}\choose{k}}((ad\ f_i)^ku)(f_i^{N-k}v_\lambda)$

Carter states that the following holds $$f_i^N(uv_\lambda)=\sum_{k=0}^N {{N}\choose{k}}((ad\ f_i)^ku)(f_i^{N-k}v_\lambda)$$ where $u\in\mathfrak{g}$ the Kac-Moody algebra and $v_\lambda$ is the ...
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Character formula of Verma module over a Kac-Moody algebra

The character formula of a Verma module over a Kac-Moody algebra is given by $$\textrm{ch}\ M(\Lambda)=\frac{e(\Lambda)}{\prod_{\alpha\in\Phi+}(1-e(-\alpha))^{\textrm{mult}(\alpha)}}$$ Here $\Phi_+$ ...
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Why is the highest weight module in $\mathcal{O}$ category?

The $\mathcal{O}$ category is defined as follows: Let $\mathfrak{g}$ be a Kac-Moody algebra and $V$ a $\mathfrak{g}$-module. $V$ is an object in $\mathcal{O}$ if $V$ has decomposition $V=\bigoplus_{\...
KJA's user avatar
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Category $\mathcal{O}$ is closed under tensor product

I'm reading the proof of lemma 19.1 in Carter's Lie algebras of finite and affine type, which states that the category $\mathcal{O}$ is closed under tensor products. I'm having a bit of trouble ...
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$[h_i,{(ad\ f_i)^t\ f_j}]=-(t\alpha_i+\alpha_j)(h_i){(ad\ f_i)^t\ f_j}$ in a Kac-Moody algebra

In Carter's Lie algebras of finite and affine type, when constructing the Weyl group in the setting of a Kac-Moody algebras, Carter uses an identity $[h_i,{(ad\ f_i)^t(f_j)}]=-(t\alpha_i+\alpha_j)(h_i)...
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Is $SL(n,F)$ a Kac-Moody group?

Is $SL(n,F)$ a Kac-Moody group? Do Kac-Moody groups generalize the classical Lie groups or are they a different family of groups? I can't find an exposition of the material that I can understand, and ...
FireFenix777's user avatar
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Is $\mathfrak{gl}_2$ a Kac-Moody Lie Algebra?

Is $\mathfrak{gl}_2$ a Kac-Moody Lie Algebra? For the definition of Kac-Moody algebra, I was using "Introduction to Quantum Groups and Crystal Bases" by Hong and Kang. Edit: On page 150 in ...
Andy Nguyen's user avatar
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Explicit description of closure of Tits cone of hyperbolic case

This question arises from a formula in Kac’s book Infinite dimensional Lie algebras, 3rd edition. Let $A=(a_{ij})_{n\times n}$ be a generalized Cartan matrix of hyperbolic type(which is symmetrizable) ...
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Basic representation of a simple Lie algebra and its highest weight

I have come across the terms "basic representation" of a semisimple Lie algebra, but I am finding it hard to find a clear definition of this representation. Can someone provide me a ...
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Galois Cohomology and Loop Groups

I am trying to understand problem 8.5 in Kac's Infinite dimensional Lie algebras. It goes as follows. Let $G$ be a semisimple algebraic group, let $\alpha$ be an automorphism of $G$ of order $m$, and ...
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The set of weights of a module in the $\mathcal O$ category has a maximal element

First, some definitions: Given a Kac-Moody algebra $\mathfrak g$, the category $\mathcal O$ of $\mathfrak g$ is the category whose objects are $\mathfrak g$-modules $V$ which are weight modules ($...
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Faithfullness of Weyl group action

Let $I$ be a finite indexing set, and $A \in \operatorname{Mat}_I(\mathbb{Z})$ be a generalised Cartan matrix, i.e. $a_{ii} = 2$, $a_{ij} \leq 0$ for $i \neq j$, and $a_{ij} = 0 \iff a_{ji} = 0$. ...
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Write imaginary roots as linear combination of fundamental weights on a Kac-Moody Lie algebra

Let $\mathfrak{g}$ be a Kac-Moody algebra of type $\widehat{A_2}$. I want to write $\delta$, the basic imaginary root, as linear combination of the fundamental weights. How can I do it? Any help ...
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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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Different cartan subalgebras for loop algebras

In [1] they say that the Cartan subalgebra of a loop algebra $\mathring{g} = \mathbb C[t,t^{-1}]\otimes g$ is $\mathring h = \mathbb C[t,t^{-1}]\otimes h$, where $h$ is the Cartan subalgebra of the ...
soap's user avatar
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Kac-Moody algebras/groups for reductive groups

The question in brief is: can $\mathfrak{gl}_n$ be constructed as a Kac-Moody algebra, and can $\operatorname{GL}_n$ be constructed as a Kac-Moody group in a compatible way? In general, given any root ...
Joppy's user avatar
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How to make a category of irreducible representations a monoidal category?

Consider the category $\mathcal O$ for a Lie algebra $\mathfrak g$. As I understand it, we can define such a category whenever $\mathfrak g$ has a triangular decomposition. So $\mathfrak g$ can be for ...
soap's user avatar
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Structure of affine Lie algebras

It is well-known that to every simple Lie algebra $\mathfrak{g}$ one can associate an affine Kac-Moody algebra by a double extension (once by a 2-cocycle and once by a derivation). One can then show ...
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Definition of the $O$ category for Kac Moody algebras

In Carter's book "Lie algebras of finite and affine type", he defines the $O$ category for Kac Moody algebras as follows: I do not understand in what sense $\lambda<\lambda_i$ on the last part of ...
soap's user avatar
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Clarification on Kac Moody algebras and the different meanings in mathematics and physics

I am confused by the way that mathematicians and physicists use the words "Kac Moody algebra", and "loop algebra", and how exactly these concepts relate to one another. I will write down what I ...
soap's user avatar
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6 votes
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What is the matrix of the monster Lie algebra?

In Richard Borcherds' proof of monstrous moonshine, he constructs a "monster Lie algebra", which is a $\mathbb Z^2$-graded, infinite-dimensional Lie algebra with a contravariant bilinear form acted on ...
Aidan Backus's user avatar
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The elements $f_1,\cdots, f_n$ generate $\mathfrak{\tilde n_{-} }$ freely

I've started studying Kac-Moody algebras and free lie algebras is a really new thing for me. I am trying to understand the statement (b) of theorem 1.2 in the following book Theorem 1.2, statement (b)...
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Construction of Lie algebra $g(A)$ in Victor Kac's book

In Kac's book "Infinite Dimensional Lie Algebras" Chapter I, he constructed an infinite Lie algebra $g(A)$ starts from any $n\times n$ complex matrix $A$ as follows: Let $\mathfrak{h}$ be a ...
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