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Mathematics Stack Exchange

Questions tagged [kac-moody-algebras]

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

Kac–Moody algebras are , usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix. These algebras form a generalization of finite-dimensional semi-simple Lie algebras, and many properties related to the structure of a Lie algebra such as its root system, irreducible representations, and connection to flag manifolds have natural analogues in the Kac–Moody setting.

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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 datum, can we construct the corresponding Lie…
Joppy
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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's book Infinite dimensional Lie algebras,…
KJA
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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 be sure to have linearly independence between the…
TJ123
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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 highest-weight vector of the module $L(\lambda)$ (the…
njlieta
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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 by the monster group. The monster Lie algebra is…
Aidan Backus
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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 $\Phi_+$ are the positive root system and $e(\lambda)$ is…
KJA
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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 $a_{ij} = 0 \iff a_{ji} = 0$. Then there is an abstract Weyl group $W_A$…
Joppy
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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 vector space and $\mathfrak{h}^\ast$ be its dual space,…
zemora
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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) and $(h,\Pi,\Pi^{\lor})$ a realization of $A$…
Goulag
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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 that this algebra is the Kac-Moody algebra…
NDewolf
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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 is taken over the Weyl group. Here…
njlieta
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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 $V=\bigoplus_{\lambda\in \mathfrak{h}^*}V_\lambda$ with…
KJA
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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 understood from different books, and ask you to help me…
soap
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Questions about the quotient of extended Weyl group and the isomorphism of extended Weyl group

When I am reading a paper An Algebraic Characterization of the Affine Canonical Basis and I have some questions about some notations. In the paper, we assume that $\mathfrak{g}$ be a simple Lie algebra with rank $n$ and $I=\{1,\cdots,n\}$. Let…
fusheng
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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 $\mathbb{C}\textbf{1}$ of $\mathfrak{g}((t)) =…
toyr99
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