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

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### 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 ...

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### 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 $\Phi$ ...

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### 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 ...

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### 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 $H$.
...

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### 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 ...

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### 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 ...

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### 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 $ ...

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### 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, $n > 1$ ...

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### 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 ...

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### 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 ...

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### 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 $g_{(i)}$ ...

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### 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 ...

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### Equivalence of Kahler structures of based loop group and its Grassmannian model

In Pressley-Segal's Loop Groups, we have the following spaces equipped with Kahler structures. Let $G$ be a compact, connected, (simply connected) group with Lie algebra $\mathfrak g$.
Let ...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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### Reference about a formula of coroot in an affine root system

Let $\delta$ be the null of an affine root system and let $\alpha + p\delta$ be a real affine root, $p$ is an integer. It is said that
$$
(\alpha + p\delta)^{\vee} = \alpha^{\vee} + ...

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### exceptional cases in Kazhdan-Lusztig

The Kazhdan-Lusztig story doesn't apply to the four exceptional cases $(E_6)_1$, $(E_7)_1$, $(E_8)_1$, $(E_8)_2$ (see this earlier question of mine).
What's special about those cases?

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### minimal energy of affine Lie algebra reps

Let $\mathfrak g$ be a simple Lie algebra.
Let $\widetilde{L\mathfrak g}$ be the universal central extension of $L\mathfrak g:=\mathfrak g[t,t^{-1}]$. Let $V_\lambda$ be a positive energy ...

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### Cohomological Proof of Serre Relations for a Symmetrizable Kac-Moody Algebra

In 'Infinite Dimensional Lie Algebras, 3rd edition', Kac mentions at the top of p. 170 that 'a simple cohomological proof of Theorem 9.11 was found by O. Mathieu (unpublished)'.
Does anyone know how ...

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### What's the state of affairs concerning the identification between quantum group reps at root of unity, and positive energy affine Lie algebra reps?

In his paper [1], Finkelberg used Kazhdan-Lusztig's massive work [4,5,6,7,8] to prove that $Rep^{ss}(U_q\mathfrak g)$ (the semisimplification of the category of finite dimensional reps of ...

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### Are Schubert varieties for Kac-Moody groups cut out by linear equations?

Let $G$ be a reductive group, and let $X$ be a partial flag variety for $G$. Then it is known that for any projective embedding of $X$, that the equations scheme-theoretically cutting out a Schubert ...

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### What are global sections of the determinant bundle on the Beilinson-Drinfeld Grassmannian?

Let $X$ be a smooth proper algebraic curve over $\mathbb{C}$, and let $G$ be a reductive group over $\mathbb{C}$. Let $Gr_{X,n}$ be the Beilinson-Drinfeld Grassmannian (for n points in $X$), which ...

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### Is is possible to lift an equivariant map of Loop lie algebras to an equivariant map of Loop groups?

For brevity, let $LG=\mathbb{T}\ltimes \tilde{L}G$, the affine loop group and let $G$ be a simple simply conneceted Lie group. I have a map $\phi:L\mathfrak{g} \to L\mathfrak{g}$ that is equivariant. ...

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### Kac Moody algebra defintion

Why is the dimension of the cartan subalgebra $2n-\text{rank}(A)$ in the defintion from Kumar's book. From a few examples I can see why the defintion is the way it is, but, I would like a better ...

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### Are Generalized Verma modules natural w.r.t isometries?

Let $H$ be a subgroup of $G$ with Lie algebras $\mathfrak{h}$ and $\mathfrak{g}$ respectively. If I have 2 representations $V, W$ of $\mathfrak{h}$ equipped with a $\mathfrak{h}$ invariant inner ...

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### level 2,3 characters of affine su(2)

Does anyone know where I can find an explicit formula to compute the level 2 or level 3
characters of affine $su(2)$? I have found several sources that give a formula to compute the
level 1 ...

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### Restriction of highest-weight representations to Heisenberg subalgebras

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $\tilde{\mathfrak{g}}=\mathfrak{g}((t))\oplus \mathbf{C}K\oplus \mathbf{C}d$ its Kac-Moody extension ($K$ is the level and $d$ ...

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### lie algebras, Kac Moody, and quantum mechanics book

Hi all, I've just finished a graduated course on Kac-Moody algebras, and I'm really looking for some reading in regard to their applications to Quantum Mechanics. Can you help?

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### What are the simple Lie superalgebras of type E?

Background
Simple finite dimensional Lie superalgebras over $\Bbb C$ have been classified. There are the Cartan type superalgebras (algebras of purely odd vector fields), two strange families P(n) ...

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### A possible mistake in Kac's “Infinite Dimensional Lie Algebras”

I have a paperback 3rd edition and on page 65 you can find Proposition 5.8. My question is about part (c):
If $A$ is of indefinite type, then
$$ \overline{X} = \{ h \in \{ \frak h_{\mathbb{R}} ...

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### Is the centralizer of a torus in a Kac-Moody algebra always a Borcherds algebra?

If one has a finite dimension simple Lie algebra, one can easily calculate that taking the centralizer of a torus (or toral subalgebra), that is, summing the weight spaces that lie in some proper ...

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### Does Iwahori subalgebra correspond to any Cartan decomposition for affine Kac-Moody algebra?

Let $\hat{\mathfrak{g}}$ be an affine Kac-Moody algebra which is the central extension of $\mathfrak{g}[t,t^{-1}]$(polynomial version). Consider Iwahori subalgebra $I$. My question is whether $I$ ...