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Kac-Moody algebra in nLab

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Context

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

∞\infty-Lie groupoids

∞\infty-Lie groups

∞\infty-Lie algebroids

∞\infty-Lie algebras

Contents

Idea

The notion of Kac-Moody Lie algebra is a generalization of that of semisimple Lie algebra to infinite dimension of the underlying vector space.

Definition

Kac-Moody algebras

Recall that the Lie algebra of 𝔰𝔩 2 \mathfrak{sl}_2 admits a linear basis (h,e,f)(h,e,f) on which the non-vanishing Lie brackets are

[e,f]=hand[h,e] = −2e [h,f] = −2f. [e,f] \;=\; h \;\;\;\text{and}\;\;\; \begin{array}{rcl} [h,e] &=& \phantom{-}2 e \\ [h,f] &=& -2 f \mathrlap{\,.} \end{array}

A Kac-Moody Lie algebra is generated from a set of r∈ℕr \in \mathbb{N} such 𝔰𝔩 2\mathfrak{sl}_2 generators, (h i,e i,f i) i=1 r(h_i, e_i, f_i)_{i = 1}^r, which in addition interact among each other as specified by a prescribed generalized Cartan matrix A=(A ij) i,j=1 rA = (A_{i j})_{i,j=1}^r:

(1)∀i,j A ij∈ℤ ∀i A ii=2 ∀i≠j A ij≤0 ∀i,j A ij=0⇔A ji=0 ∃D∈DiagMat r×r(ℤ)S∈SymMat r×r(ℤ) A=DS \begin{array}{cl} \underset{i,j}{\forall} & A_{i j} \,\in\, \mathbb{Z} \\ \underset{i}{\forall} & A_{i i} = 2 \\ \underset{i \neq j}{\forall} & A_{i j} \leq 0 \\ \underset{i, j}{\forall} & A_{i j} = 0 \;\Leftrightarrow\; A_{j i} = 0 \\ \underset{ { D \in DiagMat_{r \times r}(\mathbb{Z}) } \atop { S \in SymMat_{r \times r}(\mathbb{Z}) } }{\exists} & A = D S \end{array}

(where in the last line DD is a diagonal matrix and SS a symmetric matrix)

via the following Chevalley relations:

(2)[h i,h j]=0 [h i,e j]=−A ije j [h i,f j]=−A ijf j [e i,f i]=δ ijh j \begin{array}{l} [h_i, h_j] \,=\, 0 \\ [h_i, e_j] \,=\, \phantom{-}A_{i j} e_j \\ [h_i, f_j] \,=\, -A_{i j} f_j \\ [e_i, f_i] \,=\, \delta_{i j} h_j \end{array}

and subject to the following Serre relations for the adjoint action ad(e i)(−)∶−[e i,−]ad(e_i)(-) \,\coloneq\,[e_i,-]:

(3)ad(e i) 1−A ij(e j)=0 ad(f i) 1−A ij(e j)=0. \begin{array}{l} ad(e_i)^{1 - A_{ij}}(e_j) \,=\, 0 \\ ad(f_i)^{1 - A_{ij}}(e_j) \,=\, 0 \,. \end{array}

The generalized Cartan matrix AA, and hence 𝔤 A\mathfrak{g}_A, is called:

  • simply-laced if ∀i≠jA ij∈{0,−1}\underset{i \neq j}{\forall}\; A_{i j} \,\in\, \{0,-1\}

  • symmetric if ∀i,jA ij=A ji\underset{i, j}{\forall}\; A_{i j} = A_{j i}.

A symmetric generalized Cartan matrix is equivalently encoded in the Dynkin diagram that it is the coincidence matrix of: The undirected graph with rr vertices and A ijA_{i j} edges between the iith and the jjth vertex.

Maximal-compact (involutory) subalgebras

Definition

Given a Kac-Moody algebra 𝔤 A\mathfrak{g}_A (according to Def. ), its Cartan-Chevalley involution is the Lie algebra endomorphism given on generators by

(4)𝔤 A →θ 𝔤 A e i ↦ −f i f i ↦ −e i h i ↦ −h i \begin{array}{ccc} \mathfrak{g}_A &\xrightarrow{\;\; \theta \;\;}& \mathfrak{g}_A \\ e_i &\mapsto& - f_i \\ f_i &\mapsto& - e_i \\ h_i &\mapsto& - h_i \end{array}

It is clear that (4) is an involution: θ∘θ=id\theta \circ \theta \,=\, \mathrm{id}.
Definition

The maximal compact (or involutory) subalgebra 𝔨 A⊂𝔤 A\mathfrak{k}_A \,\subset\, \mathfrak{g}_A is the fixed locus of the Cartan-Chevalley involution (Def. ):

𝔨 A≔𝔨(𝔤 A)≔{x∈𝔤 A|θ(x)=x}. \mathfrak{k}_A \,\coloneqq\, \mathfrak{k}(\mathfrak{g}_A) \;\coloneqq\; \big\{ x \,\in\, \mathfrak{g}_A \,\big\vert\, \theta(x) = x \big\} \,.

For Kac-Moody algebras which are Lie algebras of classical Lie groups, the maximal compact subalgebra (Def. ) is indeed the Lie algebra of the maximal compact subgroup:

Generalizing from this, it may often be useful to think of the involutory subalgebra as consisting of the anti-Hermitian elements.

Examples

The sequence of exceptional semisimple Lie algebras 𝔢 6 \mathfrak{e}_6 , 𝔢 7 \mathfrak{e}_7 𝔢 8 \mathfrak{e}_8 may be continued to the Kac-Moody algebras:

References

General

On the Wess-Zumino-Witten model 2d CFT via Kac-Moody algebra and Virasoro algebra:

Lecture notes:

With an eye towards U-duality in 11D supergravity:

A standard textbook account:

Collections of articles:

  • N. Sthanumoorty, K. Misra (eds.): Kac-Moody Lie algebras and related topics, Contemporary Mathematics 343 AMS (2002) [ISBN:978-0-8218-7933-7, ams:conm-343]

See also:

The EE-series

Surveys:

  • Wikipedia, En

The fact that every simply laced hyperbolic Kac-Moody algebra appears as a subalgebra of E 10 E_{10} :

Affine Lie algebras

For more see also at affine Lie algebra.

In supergravity

The following references discuss aspects of the Kac-Moody exceptional geometry of supergravity theories.

(for much more see the references at U-duality and exceptional field theory)

  • Hermann Nicolai, Infinite dimensional symmetries (2009) (pdf)

  • Paul Cook, Connections between Kac-Moody algebras and M-theory PhD thesis (arXiv:0711.3498)

  • Daniel Persson, Nassiba Tabti, Lectures on Kac-Moody algebras with applications in (Super-)Gravity (pdf)

Maximal compact subalgebras

On non-trivial finite-dimensional representations of involutary (“maximal compact”) subalgebras 𝔨\mathfrak{k}:

Last revised on December 9, 2024 at 08:17:58. See the history of this page for a list of all contributions to it.