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1. CJM 2006 (vol 58 pp. 225)
Generalized Reductive Lie Algebras: Connections With Extended Affine Lie Algebras and Lie Tori We investigate a class of Lie algebras which we call {\it generalized reductive
Lie algebras}. These are generalizations of semi-simple, reductive, and affine
Kac--Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible
root system is said to be {\it irreducible\/} and we note that this class of algebras
has been under intensive investigation in recent years. They have also been called
{\it extended affine Lie algebras}. The larger class of generalized reductive Lie
algebras has not been so intensively investigated. We study them in this paper and note
that one way they arise is as fixed point subalgebras of finite order automorphisms. We
show that the core modulo the center of a generalized reductive Lie algebra is a direct
sum of centerless Lie tori. Therefore one can use the results known about the
classification of centerless Lie tori to classify the cores modulo centers of
generalized reductive Lie algebras.
Categories:17B65, 17B67, 17B40 |
2. CJM 2000 (vol 52 pp. 503)
The Level 2 and 3 Modular Invariants for the Orthogonal Algebras The `1-loop partition function' of a rational conformal field theory
is a sesquilinear combination of characters, invariant under a natural
action of $\SL_2(\bbZ)$, and obeying an integrality condition.
Classifying these is a clearly defined mathematical problem, and at
least for the affine Kac-Moody algebras tends to have interesting
solutions. This paper finds for each affine algebra $B_r^{(1)}$ and
$D_r^{(1)}$ all of these at level $k\le 3$. Previously, only those at
level 1 were classified. An extraordinary number of exceptionals
appear at level 2---the $B_r^{(1)}$, $D_r^{(1)}$ level 2
classification is easily the most anomalous one known and this
uniqueness is the primary motivation for this paper. The only level 3
exceptionals occur for $B_2^{(1)} \cong C_2^{(1)}$ and $D_7^{(1)}$.
The $B_{2,3}$ and $D_{7,3}$ exceptionals are cousins of the ${\cal
E}_6$-exceptional and $\E_8$-exceptional, respectively, in the
A-D-E classification for $A_1^{(1)}$, while the level 2 exceptionals
are related to the lattice invariants of affine~$u(1)$.
Keywords:Kac-Moody algebra, conformal field theory, modular invariants Categories:17B67, 81T40 |
3. CJM 1998 (vol 50 pp. 972)
Trace class elements and cross-sections in Kac-Moody groups Let $G$ be an affine Kac-Moody group, $\pi_0,\dots,\pi_r,\pi_{\delta}$
its fundamental irreducible representations and $\chi_0, \dots,
\chi_r, \chi_{\delta}$ their characters. We determine the set of all
group elements $x$ such that all $\pi_i(x)$ act as trace class
operators, \ie, such that $\chi_i(x)$ exists, then prove that the
$\chi_i$ are class functions. Thus, $\chi:=(\chi_0, \dots, \chi_r,
\chi_{\delta})$ factors to an adjoint quotient $\bar{\chi}$ for $G$.
In a second part, following Steinberg, we define a cross-section $C$
for the potential regular classes in $G$. We prove that the
restriction $\chi|_C$ behaves well algebraically. Moreover, we obtain
an action of $\hbox{\Bbbvii C}^{\times}$ on $C$, which leads to a
functional identity for $\chi|_C$ which shows that $\chi|_C$ is
quasi-homogeneous.
Categories:22E65, 17B67 |