Borcherds Lie algebra

Borcherds algebra

While a Kac–Moody algebra is generated in a fairly simple way from copies of , a Borcherds or generalized Kac–Moody algebra [a1], [a7], [a9], [a11] can also involve copies of the -dimensional Heisenberg algebra. Nevertheless, it inherits many of the Kac–Moody properties. Borcherds algebras played a key role in the proof of the Monstrous Moonshine conjectures [a4], and also led to the development of a theory of automorphic products [a5].

First recall the definition of a Kac–Moody algebra. By a (symmetrizable) Cartan matrix one means an integral -matrix obeying

C1) and for all ; and

C2) there is a diagonal matrix with each such that is symmetric. A (symmetrizable) Kac–Moody algebra [a10], [a12] is the Lie algebra on generators , obeying the relations:

R1) , , , and , for all ; and

R2) for all .

A Borcherds algebra is defined similarly. By a generalized Cartan matrix one means a (possibly infinite) matrix , , obeying

GC1) either or ;

GC2) for , and when ; and

GC3) there is a diagonal matrix with each such that is symmetric. By the (symmetrizable) universal Borcherds algebra one means the Lie algebra (over say) with generators , subject to the relations [a3]:

GR1) , and , for all ;

GR2) , whenever both and ; and

GR3) whenever .

Note that for each , is isomorphic to when , and to the -dimensional Heisenberg algebra when . Immediate consequences of the definition are that:

i) ;

ii) unless the th and th column of are identical;

iii) the for lie in the centre of . Setting all for gives the definition of the (symmetrizable) Borcherds algebra [a1]. This central extension of is introduced for its role in the characterization of Borcherds algebras below. If has no zero columns, then equals its own universal central extension [a3]. An important technical point is that both and have trivial radical.

The basic structure theorem [a1] is that of Kac–Moody algebras. Let be a symmetrizable Borcherds algebra. Then:

a) has triangular decomposition , where is the subalgebra generated by the , is generated by the , and is the Cartan subalgebra. Also, and .

b) has a root space decomposition: formally calling and , and defining to be the subspace of degree , one gets and , where and ;

c) there is an involution on for which , , and ;

d) and ;

e) there is an invariant symmetric bilinear form on such that for each root , the restriction of to is non-degenerate, and whenever ;

f) there is a linear assignment such that for all , , one has .

The condition that be symmetrizable (i.e. condition GC3)) is necessary for the existence of the bilinear form in e). For representation theory it is common to add derivations, so that the roots will lie in a dual space . In particular, define for any ; then each linear mapping is a derivation, and adjoining these to defines an Abelian algebra . The simple root can be interpreted as the element of obeying and . Construct the induced bilinear form on , obeying .

The properties a)–f) characterize Borcherds algebras. Let be a Lie algebra (over ) satisfying the following conditions:

1) has a -grading (cf. also Lie algebra, graded), and for all ;

2) has an involution sending to and acting as on ;

3) has an invariant bilinear form invariant under such that if , and such that if for . Then there is a homomorphism from some to whose kernel is contained in the centre of , and is the semi-direct product of the image of with a subalgebra of the Abelian subalgebra . That is, is obtained from by modding out some of the centre and adding some commuting derivations. See e.g. [a4] for details.

Define to be the set of all real simple roots, i.e. all with ; the remaining simple roots are the imaginary simple roots . The Weyl group (cf. also Weyl group) of is the group generated by the reflections for each : . It will be a (crystallographic) Coxeter group. The real roots of are defined to be those in ; all other roots are called imaginary. For all real roots, and .

is called an integrable module if

where the weight space , with , and for each with both and are locally nilpotent: i.e. for all and all sufficiently large , . By the character one means the formal sum

Let be the set of all weights obeying whenever , and for all . Define the highest-weight -module in the usual way as the quotient of the Verma module (cf. also Representation of a Lie algebra) by the unique proper graded submodule. Then one obtains the Weyl–Kac–Borcherds character formula: Choose to satisfy

for all , and define , where runs over all sums of and if is the sum of distinct mutually orthogonal imaginary simple roots, each of which is orthogonal to , otherwise . Then

where . is the correction factor due to imaginary simple roots, much as the "extra" terms in the Macdonald identities are due to the imaginary affine roots. Putting gives the denominator identity, as usual.

Thus, Borcherds algebras strongly resemble Kac–Moody algebras and constitute a natural and non-trivial generalization. The main differences are that they can be generated by copies of the Heisenberg algebra as well as , and that there can be imaginary simple roots.

Interesting examples of Borcherds algebras are the Monster Lie algebra [a4], whose (twisted) denominator identity supplied the relations needed to complete the proof of the Monstrous Moonshine conjectures, and the fake Monster [a2]. A Borcherds algebra can be associated to any even Lorentzian lattice. The denominator identities of Borcherds algebras are often automorphic forms on the automorphism group of the even self-dual lattice [a5]. They can serve as "automorphic corrections" to Lorentzian Kac–Moody algebras (see, for instance, [a6]). The space of BPS states in string theory carries a natural structure of a Borcherds-like algebra [a8].

References

[a1]	R.E. Borcherds, "Generalized Kac–Moody algebras" J. Algebra , 115 (1988) pp. 501–512
[a2]	R.E. Borcherds, "The monster Lie algebra" Adv. Math. , 83 (1990) pp. 30–47
[a3]	R.E. Borcherds, "Central extensions of generalized Kac–Moody algebras" J. Algebra , 140 (1991) pp. 330–335
[a4]	R.E. Borcherds, "Monstrous moonshine and monstrous Lie superalgebras" Invent. Math. , 109 (1992) pp. 405–444
[a5]	R.E. Borcherds, "Automorphic forms on and infinite products" Invent. Math. , 120 (1995) pp. 161–213
[a6]	V.A. Gritsenko, V.V. Nikulin, "Siegel automorphic form corrections of some Lorentzian Kac–Moody Lie algebras" Amer. J. Math. , 119 (1997) pp. 181–224
[a7]	K. Harada, M. Miyamoto, H. Yamada, "A generalization of Kac–Moody algebras" , Groups, Difference Sets, and the Monster , de Gruyter (1996)
[a8]	J.A. Harvey, G. Moore, "On the algebras of BPS states" Commun. Math. Phys. , 197 (1998) pp. 489–519
[a9]	E. Jurisich, "An exposition of generalized Kac–Moody algebras" Contemp. Math. , 194 (1996) pp. 121–159
[a10]	V.G. Kac, "Simple irreducible graded Lie algebras of finite growth" Math. USSR Izv. , 2 (1968) pp. 1271–1311
[a11]	V.G. Kac, "Infinite dimensional Lie algebras" , Cambridge Univ. Press (1990) (Edition: Third)
[a12]	R.V. Moody, "A new class of Lie algebras" J. Algebra , 10 (1968) pp. 211–230