Structure of the Level One Standard Modules for the Affine Lie Algebras Blp

161 Pages·1987·7.1 MB·English

by  Marly Mandia

Memoirs of the American Mathematical Society
Number 362
Marly Mandia
Structure of the
level one standard modules
for the affine Lie algebras
(1) (1) (1)
B , F and G
e 4 2
Published by the
AMERICAN MATHEMATICAL SOCIETY
Providence, Rhode Island, USA
January 1987 • Volume 65 • Number 362 (end of volume)
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TABLE OF CONTENTS
Page
INTRODUCTION vii
CHAPTER 1: PRELIMINARIES
§1.1 The principal picture of the affine Lie
algebras of type X 1
§1.2 The algebra ^i(x) anc* tne vacuum space .. 6
§1.3 The generalized commutation relations
f°r ZL(X) n
§1.4 The principal character for the affine Lie
algebras, and the basic modules 13
CHAPTER 2: STRUCTURE OF THE STANDARD B^-MODULES
OF LEVEL ONE
§2.1 B „ as a subalgebra of D„ 22
n
§2.2 The principal Cartan subalgebras of D ,
and B and the corresponding principal
automorphisms 26
§2.3 The standard B -modules of level 1 39
S.
CHAPTER 3: STRUCTURE OF THE STANDARD (^-MODULES
OF LEVEL ONE
§3.1 G« as a subslgebra of D, 59
§3.2 The principal Cartan subalgebras of D,
and G« and the corresponding principal
automorphisms 63
§3.3 The generalized anticommutation relations
for Z /** when L(X) is a standard
LI A. )
GL-module of level one 73
§3.4 The standard G«-modules of level one 94
CHAPTER 4: STRUCTURE OF THE STANDARD F -MODULES
OF LEVEL ONE
§4.1 F, as a subalgebra of E 102
&
§4.2 The principal Cartan subalgebras of E^
and F, and the corresponding principal
automorphisms 106
IV CONTENTS
Page
4.3 The generalized anticommutation
relations for Z , * when L(X)
is a standard F.-module of level one ....118
4.4 The standard F.-modules of level one ....139
4
REFERENCES 144
ABSTRACT
This work gives a realization of all the level one
standard modules for the affine Kac-Moody algebras
B^ , F^ and G^ in the "principal picture," by
constructing explicit bases of the vacuum spaces for the
corresponding Heisenberg subalgebras. Viewing each such
standard module V as a subspace of a basic module for
an affine Lie algebra of type D or E, we are able to
obtain generalized anticommutation relations for the
Z -algebra. For the level one standard F , and
G« -modules we also use the generalized commutation
relations for the Z -algebras, obtained by J. Lepowsky
and R.L. Wilson, as well as the principal character of
the vacuum spaces and the classical Rogers-Ramanujan
identi ties.
1980 Mathematics Subject Classification. 17B65, 17B25, 17B10,
Library of Congress Cataloging-in-Publication Data
Mandia, Marly, 1947-
Structure of the level one standard modules for the
affine lie algebras B^\ F<x> and G (1>.
4 2
(Memoirs of the American Mathematical Society,
ISSN 0065-9266; no. 362)
On t.p. "£" is subscript.
"January 1987, volume 65, number 362."
Bibliography: p.
1. Lie algebras. 2. Modules (Algebra) I. Title.
II. Series.
QA3.A57 no. 362 510s [512'.55] 86-28797
ISBN 0-8218-2423-6
v
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INTRODUCTION
One of the most important problems in the representa
tion theory of affine Kac-Moody algebras is the construc
tion of their "standard" modules. The first construction
was that of the standard modules of "level one" for
A-, by Lepowsky and Wilson in [15], where "vertex
operators" were first introduced in mathematics. In this
approach, each module was realized as a polynomial
algebra G)[x, ,x«,...] on infinitely many variables and
the Lie algebra as an algebra of differential operators
on C[x, ,x« , . . . 3 . This A-. construction was generalized
by Kac, Kazhdan, Lepowsky and Wilson in [9] to all afine
Lie algebras of types A,D,E. The modules constructed
in [93 are called "basic modules" and this construction
is referred to as the "principal picture" because the
algebras are realized via "principal automorphisms" in the
sense of [113. The same basic modules were constructed
in a new picture, the "homogeneous picture," by Frenkel
and Kac in [53 and also by Segal in [24]. Spin
constructions of the level one standard D and
B -modules were found by Frenkel in [3] and [4], and
also by Kac and Peterson in [103.
Lepowsky and Wilson, working in the principal picture,
constructed all the standard modules of levels two and
three for A-. in [173, interpreting and Li e-atgebrai cal ly
vii
viii M. MANDIA
proving the classical Rogers-Ramanujan identities in
the process. In [16] they introduced "Z-algebras"
and obtained constructions of all the standard
A-, -modules in the principal picture (see [18] and
[19]). By means of Z-algebras, Misra in [20], [21]
and [22] obtained the construction of some standard
modules for A and C in the principal picture,
n n '
and, in particular, gave explicit constructions of all
level one standard modules for C« and C~ . Also
using Z-algebras, Lepowsky and Prime, working in the
homogeneous picture, constructed the level one standard
B -modules in [13], and in [14] they obtained new
constructions of all the standard modules for A-.
A more detailed historical description of the construc
tions of standard modules for affine Lie algebras can be
found in [25].
In this work we give the structure of all the level one
standard modules for B„ , F. and G in the
0
8. ' 4 2
principal picture. We use Z-algebras to obtain an
explicit basis of the "vacuum space" for the "principal
Heisenberg subalgebra" in each case.
We view our Lie algebra g as a subalgebra of a Lie
algebra m of type A,D or E (actually type D or
E), following Mitzman's approach in [23] (see [5] and [24]
for the construction of the finite-dimensional Lie
algebras of types A,D,E, and [4] and [13] for the
LEVEL ONE STANDARD MODULES FOR B^ , F ^ , G£ ix
construction of the Lie algebras of type Bn). In
this way we are able to view a level one standard
g -module V as a subspace of a basic m -module.
Using information about the basic m -modules given
in [9] and [18], we obtain generalized anticommutation
relations for the Z -algebra for g in each case.
Chapter 1 discusses the preliminaries.
Chapter 2 describes the case B . We view B as
a subalgebra of D . and a level one standard
n
Z+l
B. -module V as a subspace of a basic D -.-module.
Z nZ+l
We obtain the anticommutation relations for the (Clifford)
algebra Z (Theorem 2.3.7) which allow us to give an
explicit construction of V (Theorem 2.3.8).
Chapter 3 analyzes the case G«. We view G~ as a
subalgebra of D, and a level one standard G« -module
V as a subspace of a basic D, -module W. In
4
Theorem 3.3.9 we obtain generalized anticommutation
relations for the algebra Z through information
about W. These relations together with the generalized
commutation relations for Z due to Lepowsky and
Wilson [18], enable us to find a spanning set of the
vacuum space of V. We finally give an explicit
construction of V (Theorem 3.4.7) by using the principal
character of the vacuum space of V and the classical
Rogers-Ramanujan identities.
Chapter 4 treats the case F,. The same technique