nLab Kac-Moody algebra

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

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

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

(1)$$\begin{array}{cl} \underset{i,j}{\forall} & A_{i j} \,\in\, \mathbb{Z} \\ \underset{i}{\forall} & A_{i i} = 2 \\ \underset{i \neq j}{\forall} & A_{i j} \leq 0 \\ \underset{i, j}{\forall} & A_{i j} = 0 \;\Leftrightarrow\; A_{j i} = 0 \\ \underset{ { D \in DiagMat_{r \times r}(\mathbb{Z}) } \atop { S \in SymMat_{r \times r}(\mathbb{Z}) } }{\exists} & A = D S \end{array}$$

(where in the last line $D$ is a diagonal matrix and $S$ a symmetric matrix)

via the following Chevalley relations:

(2)$$\begin{array}{l} [h_i, h_j] \,=\, 0 \\ [h_i, e_j] \,=\, \phantom{-}A_{i j} e_j \\ [h_i, f_j] \,=\, -A_{i j} f_j \\ [e_i, f_i] \,=\, \delta_{i j} h_j \end{array}$$

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

(3)$$\begin{array}{l} ad(e_i)^{1 - A_{ij}}(e_j) \,=\, 0 \\ ad(f_i)^{1 - A_{ij}}(e_j) \,=\, 0 \,. \end{array}$$

Definition

A Kac-Moody algebra of rank $r \,\in\,\mathbb{N}$ is a Lie algebra isomorphic to a quotient of the free Lie algebra on generators $(h_i, e_i, f_i)_{i=1}^r$ by the Lie ideal generated by the Chevalley relations (2) and the Serre relations (3) for some generalized Cartan matrix (1).

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

• simply-laced if $\underset{i \neq j}{\forall}\; A_{i j} \,\in\, \{0,-1\}$

• symmetric if $\underset{i, j}{\forall}\; A_{i j} = A_{j i}$.

A symmetric generalized Cartan matrix is equivalently encoded in the Dynkin diagram that it is the coincidence matrix of: The undirected graph with $r$ vertices and $A_{i j}$ edges between the $i$th and the $j$th vertex.

Maximal-compact (involutory) subalgebras

Definition

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

(4)$$\begin{array}{ccc} \mathfrak{g}_A &\xrightarrow{\;\; \theta \;\;}& \mathfrak{g}_A \\ e_i &\mapsto& - f_i \\ f_i &\mapsto& - e_i \\ h_i &\mapsto& - h_i \end{array}$$

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

Definition

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

$$\mathfrak{k}_A \,\coloneqq\, \mathfrak{k}(\mathfrak{g}_A) \;\coloneqq\; \big\{ x \,\in\, \mathfrak{g}_A \,\big\vert\, \theta(x) = x \big\} \,.$$

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:

Example

The Cartan-Chevalley involution on $\mathfrak{sl}_n$ is, in its canonical matrix Lie algebra-incarnation, given by passing to negative transpose matrices:

$$\theta(x) \;=\; -x^T \,,$$

whence the fixed locus is $\mathfrak{so}_n$:

$$\mathfrak{k}\big( \mathfrak{sl}_n(\mathbb{R}) \big) \;\simeq\; \mathfrak{so}_n(\mathbb{R}) \,.$$

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 $\mathfrak{e}_6$, $\mathfrak{e}_7$$\mathfrak{e}_8$ may be continued to the Kac-Moody algebras:

• affine: E9

• hyperbolic: E10,

• Lorentzian: E11, …

Related concepts

• Kac-Moody group

• higher Kac-Moody algebra

• Borcherds algebra

References

General

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

• Peter Goddard, David Olive, Kac-Moody and Virasoro algebras in relation to quantum physics, International Journal of Modern Physics A 01 02 (1986) 303-414 [doi:10.1142/S0217751X86000149, spire:18583]

Lecture notes:

• Antony Wassermann: Kac-Moody and Virasoro algebras, course notes (2011) [arXiv:1004.1287]

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

• Hermann Nicolai (notes by Oliver Schlotterer): Infinite dimensional symmetries, lecture at Saalburg summer school in Wolfersdorf, Thuringia (2009) [pdf]

A standard textbook account:

• Victor G. Kac, Infinite Dimensional Lie Algebras, Progress in Mathematics 44 Springer 1983 [doi:10.1007/978-1-4757-1382-4], Cambridge University Press (1990) [doi:10.1017/CBO9780511626234]

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:

• Wikipedia: Kac-Moody algebra

The $E$-series

• E7, E8, E9, E10, E11, …

Surveys:

• Wikipedia, En

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

• Sankaran Viswanath: Embeddings of hyperbolic Kac-Moody algebras into $E_{10}$ [arXiv:0801.2586, doi:10.1007/s11005-007-0214-7, doi:10.1007/s11005-007-0214-7]

Affine Lie algebras

For more see also at affine Lie algebra.

• David Hernandez: An introduction to affine Kac-Moody algebras, lecture notes (2006) [pdf, pdf]

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 $\mathfrak{k}$:

• Axel Kleinschmidt, Hermann Nicolai, Adriano Viganò: On spinorial representations of involutory subalgebras of Kac-Moody algebras, In: Partition Functions and Automorphic Forms, Moscow Lectures 5, Springer (2020) [arXiv:1811.11659, doi:10.1007/978-3-030-42400-8_4]

• Axel Kleinschmidt, Ralf Köhl, Robin Lautenbacher, Hermann Nicolai: Representations of involutory subalgebras of affine Kac-Moody algebras, Commun. Math. Phys. 392 (2022) 89–123 [arXiv:2102.00870, doi:10.1007/s00220-022-04342-9]

• Axel Kleinschmidt, Hermann Nicolai, Jakob Palmkvist: $K(E_9)$ from $K(E_{10})$, Journal of High Energy Physics 2007 JHEP06 (2007) [arXiv:hep-th/0611314, doi:10.1088/1126-6708/2007/06/051]

• Robin Lautenbacher, Ralf Köhl: Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras [arXiv:2409.07247]