Expand all Collapse all | Results 26 - 43 of 43 |
26. CJM 2001 (vol 53 pp. 195)
On the Steinberg Map and Steinberg Cross-Section for a Symmetrizable Indefinite Kac-Moody Group Let $G$ be a symmetrizable indefinite Kac-Moody group over $\C$. Let
$\Tr_{\La_1},\dots,\Tr_{\La_{2n-l}}$ be the characters of the
fundamental irreducible representations of $G$, defined as convergent
series on a certain part $G^{\tralg} \subseteq G$. Following
Steinberg in the classical case and Br\"uchert in the affine case, we
define the Steinberg map $\chi := (\Tr_{\La_1},\dots,
\Tr_{\La_{2n-l}})$ as well as the Steinberg cross section $C$,
together with a natural parametrisation $\omega \colon \C^{n} \times
(\C^\times)^{\,n-l} \to C$. We investigate the local behaviour of
$\chi$ on $C$ near $\omega \bigl( (0,\dots,0) \times (1,\dots,1)
\bigr)$, and we show that there exists a neighborhood of $(0,\dots,0)
\times (1,\dots,1)$, on which $\chi \circ \omega$ is a regular
analytical map, satisfying a certain functional identity. This
identity has its origin in an action of the center of $G$ on~$C$.
Categories:22E65, 17B65 |
27. 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 |
28. CJM 2000 (vol 52 pp. 141)
Numerical Ranges Arising from Simple Lie Algebras A unified formulation is given to various generalizations of the
classical numerical range including the $c$-numerical range,
congruence numerical range, $q$-numerical range and von Neumann
range. Attention is given to those cases having connections with
classical simple real Lie algebras. Convexity and inclusion
relation involving those generalized numerical ranges are
investigated. The underlying geometry is emphasized.
Keywords:numerical range, convexity, inclusion relation Categories:15A60, 17B20 |
29. CJM 1999 (vol 51 pp. 506)
On Polynomial Invariants of Exceptional Simple Algebraic Groups We study polynomial invariants of systems of vectors with respect
to exceptional simple algebraic groups in their minimal linear
representations. For each type we prove that the algebra of
invariants is integral over the subalgebra of trace polynomials
for a suitable algebraic system (\cf\ \cite{Schw1},
\cite{Schw2}, \cite{Ilt}).
Categories:15A72, 17C20 |
30. CJM 1999 (vol 51 pp. 658)
Nilpotency of Some Lie Algebras Associated with $p$-Groups Let $ L=L_0+L_1$ be a $\mathbb{Z}_2$-graded Lie algebra over a
commutative ring with unity in which $2$ is invertible. Suppose
that $L_0$ is abelian and $L$ is generated by finitely many
homogeneous elements $a_1,\dots,a_k$ such that every commutator in
$a_1,\dots,a_k$ is ad-nilpotent. We prove that $L$ is nilpotent.
This implies that any periodic residually finite $2'$-group $G$
admitting an involutory automorphism $\phi$ with $C_G(\phi)$
abelian is locally finite.
Categories:17B70, 20F50 |
31. CJM 1999 (vol 51 pp. 523)
Representations of Virasoro-Heisenberg Algebras and Virasoro-Toroidal Algebras Virasoro-toroidal algebras, $\tilde\mathcal{T}_{[n]}$, are
semi-direct products of toroidal algebras $\mathcal{T}_{[n]}$ and
the Virasoro algebra. The toroidal algebras are, in turn,
multi-loop versions of affine Kac-Moody algebras. Let $\Gamma$ be
an extension of a simply laced lattice $\dot{Q}$ by a hyperbolic
lattice of rank two. There is a Fock space $V(\Gamma)$
corresponding to $\Gamma$ with a decomposition as a complex vector
space: $V(\Gamma) = \coprod_{m \in \mathbf{Z}}K(m)$. Fabbri and
Moody have shown that when $m \neq 0$, $K(m)$ is an irreducible
representation of $\tilde\mathcal{T}_{[2]}$. In this paper we
produce a filtration of $\tilde\mathcal{T}_{[2]}$-submodules of
$K(0)$. When $L$ is an arbitrary geometric lattice and $n$ is a
positive integer, we construct a Virasoro-Heisenberg algebra
$\tilde\mathcal{H}(L,n)$. Let $Q$ be an extension of $\dot{Q}$ by
a degenerate rank one lattice. We determine the components of
$V(\Gamma)$ that are irreducible $\tilde\mathcal{H}(Q,1)$-modules
and we show that the reducible components have a filtration of
$\tilde\mathcal{H}(Q,1)$-submodules with completely reducible
quotients. Analogous results are obtained for $\tilde\mathcal{H}
(\dot{Q},2)$. These results complement and extend results of
Fabbri and Moody.
Categories:17B65, 17B68 |
32. CJM 1998 (vol 50 pp. 1323)
L'invariant de Hasse-Witt de la forme de Killing Nous montrons que l'invariant de Hasse-Witt de la forme de Killing
d'une alg{\`e}bre de Lie semi-simple $L$ s'exprime {\`a} l'aide de
l'invariant de Tits de la repr{\'e}sentation irr{\'e}ductible de
$L$ de poids dominant $\rho=\frac{1}{2}$ (somme des racines
positives), et des invariants associ{\'e}s au groupe des
sym{\'e}tries du diagramme de Dynkin de $L$.
Categories:11E04, 11E72, 17B10, 17B20, 11E88, 15A66 |
33. CJM 1998 (vol 50 pp. 929)
Decomposition varieties in semisimple Lie algebras The notion of decompositon class in a semisimple Lie algebra is a
common generalization of nilpotent orbits and the set of
regular semisimple elements. We prove that the closure of a
decomposition class has many properties in common with nilpotent
varieties, \eg, its normalization has rational singularities.
The famous Grothendieck simultaneous resolution is related to the
decomposition class of regular semisimple elements. We study the
properties of the analogous commutative diagrams associated to
an arbitrary decomposition class.
Categories:14L30, 14M17, 15A30, 17B45 |
34. 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 |
35. CJM 1998 (vol 50 pp. 816)
Tableaux realization of generalized Verma modules We construct the tableaux realization of generalized Verma modules
over the Lie algebra $\sl(3,{\bbd C})$. By the same procedure we
construct and investigate the structure of a new family of
generalized Verma modules over $\sl(n,{\bbd C})$.
Categories:17B35, 17B10 |
36. CJM 1998 (vol 50 pp. 266)
The torsion free Pieri formula Central to the study of simple infinite dimensional
$g\ell(n, \Bbb C)$-modules having finite dimensional weight spaces are the
torsion free modules. All degree $1$ torsion free modules are known.
Torsion free modules of arbitrary degree can be constructed by tensoring
torsion free modules of degree $1$ with finite dimensional simple modules.
In this paper, the central characters of such a tensor product module are
shown to be given by a Pieri-like formula, complete reducibility is
established when these central characters are distinct and an example
is presented illustrating the existence of a nonsimple indecomposable
submodule when these characters are not distinct.
Category:17B10 |
37. CJM 1998 (vol 50 pp. 356)
Some norms on universal enveloping algebras The universal enveloping algebra, $U(\frak g)$, of a Lie algebra $\frak g$
supports some norms and seminorms that have arisen naturally in the
context of heat kernel analysis on Lie groups. These norms and seminorms
are investigated here from an algebraic viewpoint. It is shown
that the norms corresponding to heat kernels on the associated Lie
groups decompose as product norms under the natural isomorphism
$U(\frak g_1 \oplus \frak g_2) \cong U(\frak g_1) \otimes U(\frak
g_2)$. The seminorms corresponding to Green's functions are
examined at a purely Lie algebra level for $\rmsl(2,\Bbb C)$. It
is also shown that the algebraic dual space $U'$ is spanned by its
finite rank elements if and only if $\frak g$ is nilpotent.
Categories:17B35, 16S30, 22E30 |
38. CJM 1998 (vol 50 pp. 225)
Derivations and invariant forms of Lie algebras graded by finite root systems Lie algebras graded by finite reduced root systems have been
classified up to isomorphism. In this paper we describe the derivation
algebras of these Lie algebras and determine when they possess invariant
bilinear forms. The results which we develop to do this are much more
general and apply to Lie algebras that are completely reducible with
respect to the adjoint action of a finite-dimensional subalgebra.
Categories:17B20, 17B70, 17B25 |
39. CJM 1998 (vol 50 pp. 210)
Isomorphisms between generalized Cartan type $W$ Lie algebras in characteristic $0$ In this paper, we determine when two simple generalized Cartan
type $W$ Lie algebras $W_d (A, T, \varphi)$ are isomorphic, and discuss
the relationship between the Jacobian conjecture and the generalized
Cartan type $W$ Lie algebras.
Keywords:Simple Lie algebras, the general Lie algebra, generalized Cartan type $W$ Lie algebras, isomorphism, Jacobian conjecture Categories:17B40, 17B65, 17B56, 17B68 |
40. CJM 1997 (vol 49 pp. 1206)
Subalgebras which appear in quantum Iwasawa decompositions Let $g$ be a semisimple Lie algebra. Quantum analogs of the
enveloping algebra of the fixed Lie subalgebra are introduced for
involutions corresponding to the negative of a diagram automorphism.
These subalgebras of the quantized enveloping algebra specialize to
their classical counterparts. They are used to form an Iwasawa type
decompostition and begin a study of quantum Harish-Chandra modules.
Category:17B37 |
41. CJM 1997 (vol 49 pp. 772)
Finite dimensional representations of $U_t\bigl(\rmsl (2)\bigr)$ at roots of unity All finite dimensional indecomposable representations of
$U_t (\rmsl (2))$ at roots of $1$ are determined.
Categories:16G10, 16G70, 17B37 |
42. CJM 1997 (vol 49 pp. 820)
Sur l'intÃ©grabilitÃ© des sous-algÃ¨bres de Lie en dimension infinie Une des questions fondamentales de la th\'eorie des groupes de
Lie de dimension infinie concerne l'int\'egrabilit\'e des
sous-alg\`ebres de Lie topologiques $\cal H$ de l'alg\`ebre
de Lie $\cal G$ d'un groupe de Lie $G$ de dimension infinie
au sens de Milnor. Par contraste avec ce qui se passe en
th\'eorie classique il peut exister des sous-alg\`ebres de Lie
ferm\'ees $\cal H$ de $\cal G$ non-int\'egrables en un
sous-groupe de Lie. C'est le cas des alg\`ebres de Lie de champs
de vecteurs $C^{\infty}$ d'une vari\'et\'e compacte qui ne
d\'efinissent pas un feuilletage de Stefan. Heureusement cette
``imperfection" de la th\'eorie n'est pas partag\'ee par tous les
groupes de Lie int\'eressants. C'est ce que montre cet article
en exhibant une tr\`es large classe de groupes de Lie de
dimension infinie exempte de cette imperfection. Cela permet de
traiter compl\`etement le second probl\`eme fondamental de
Sophus Lie pour les groupes de jauge de la
physique-math\'ematique et les groupes formels de
diff\'eomorphismes lisses de $\R^n$ qui fixent l'origine.
Categories:22E65, 58h05, 17B65 |
43. CJM 1997 (vol 49 pp. 119)
Automorphisms of the Lie algebras $W^*$ in characteristic $0$ No abstract.
Categories:17B40, 17B65, 17B66, 17B68, 17B70 |