# Twining characters and orbit Lie algebras Page: 3 of 7

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1 Generalized Kac-Moody algebras

Generalized Kac-Moody algebras constitute a class of Lie algebras which comprise many Lie algebras

that describe symmetries in physical systems. In particular, they include the finite-dimensional simple

Lie algebras (i.e. the four series of classical Lie algebras and the five exceptional simple Lie algebras),

twisted and untwisted affine Lie algebras (i.e. centrally extended loop algebras), hyperbolic Lie algebras

(such as E10), as well as the Monster Lie algebra and its various relatives. Moreover, any chiral algebra

of a conformal field theory, i.e. any vertex operator algebra, gives rise to a generalized Kac-Moody

algebra.

Any finite-dimensional simple Lie algebra is generated (as a Lie algebra) by several copies of sl(2),

one copy for each simple root. This is also true for ordinary Kac-Moody algebras; a generalized

Kac-Moody algebra, however, is generated by copies of sl(2) and of the Heisenberg algebra which

has a basis {e, f, K} with K a central element and non-trivial Lie bracket [e, f] = K. Nonetheless,

generalized Kac-Moody algebras can still be characterized by a square matrix, the Cartan matrix

A = (auj)ijEI. The index set I can either be finite, I = {1, 2, ... , n}, or countably infinite, I = Z+.

In case of finite-dimensional simple Lie algebras the entries of A are integers; here we allow for real

entries, but still we keep the following properties of A:

(i) ai j< 0 if i j; (ii) aE Z if ai i> 0;

(iii) if a3 = 0, then aji = 0; (1.1)

(iv) there exists a diagonal matrix D = diag(ei, ..., en), with ei a positive

real number for all i, such that DA is symmetric.

From these data the generalized Kac-Moody algebra is constructed by the same procedure that is

used to construct a finite-dimensional simple Lie algebra from its Cartan matrix. One starts with an

abelian Lie algebra go of dimension greater or equal to n and fixes n linearly independent elements

hl, ..., hn of go and n linear forms aj on go, the simple roots, such that aj(hi) = aid. The generalized

Kac-Moody algebra g = g(A) with Cartan matrix A and Cartan subalgebra go is then the Lie algebra

generated by ei, fi , i E I, and go, modulo the relations

[ei, fj] = 81j hi , [h, ei] = ai(h)ei , [h, fi] = -ai(h)fi

(ad ei)1-2aU/Qiiej = 0 = (ad fi)1-2aii/aiifj if aii > 0, (1.2)

[ei,ej] = 0 = [fi, fj] if ai3 = 0.

2 Automorphisms of generalized Kac-Moody algebras

We now turn to the main object of our interest: a class of outer automorphisms of generalized

Kac-Moody algebras and some new structures associated to them. Such automorphisms occur natu-

rally in many physical applications; for some examples see the last section and [5]. We start with a

permutation W of finite order of the index set I which leaves the Cartan matrix A invariant:

abi = ai . (2.3)

If the generalized Kac-Moody algebra has a Dynkin diagram, W corresponds to a symmetry of the

Dynkin diagram. Such a permutation W induces an automorphism w : g -> g of the Lie algebra g,

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Fuchs, Jurgen; Ray, Urmie; Schellekens, Bert & Schweigert, Christoph. Twining characters and orbit Lie algebras, article, December 5, 1996; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc932245/m1/3/: accessed November 14, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.