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
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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 ![]() |
[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 |
Borcherds Lie algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borcherds_Lie_algebra&oldid=42263