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1. CJM 2013 (vol 65 pp. 1287)
$K$-theory of Furstenberg Transformation Group $C^*$-algebras The paper studies the $K$-theoretic invariants of the crossed product
$C^{*}$-algebras associated with an important family of homeomorphisms
of the tori $\mathbb{T}^{n}$ called Furstenberg transformations.
Using the Pimsner-Voiculescu theorem, we prove that given $n$, the
$K$-groups of those crossed products, whose corresponding $n\times n$
integer matrices are unipotent of maximal degree, always have the same
rank $a_{n}$. We show using the theory developed here that a claim
made in the literature about the torsion subgroups of these $K$-groups
is false. Using the representation theory of the simple Lie algebra
$\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a
combinatorial significance. For example, every $a_{2n+1}$ is just the
number of ways that $0$ can be represented as a sum of integers
between $-n$ and $n$ (with no repetitions). By adapting an argument
of van Lint (in which he answered a question of ErdÅs), a simple,
explicit formula for the asymptotic behavior of the sequence
$\{a_{n}\}$ is given. Finally, we describe the order structure of the
$K_{0}$-groups of an important class of Furstenberg crossed products,
obtaining their complete Elliott invariant using classification
results of H. Lin and N. C. Phillips.
Keywords:$K$-theory, transformation group $C^*$-algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism Categories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20 |
2. CJM 2003 (vol 55 pp. 1155)
The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair $(\SO_{p+q},\SO_p\times\SO_q)$ |
The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair $(\SO_{p+q},\SO_p\times\SO_q)$ The main problem that is solved in this paper has the following simple
formulation (which is not used in its solution). The group $K =
\mathrm{O}_p ({\bf C}) \times \mathrm{O}_q ({\bf C})$ acts on the
space $M_{p,q}$ of $p\times q$ complex matrices by $(a,b) \cdot x =
axb^{-1}$, and so does its identity component $K^0 = \SO_p ({\bf C})
\times \SO_q ({\bf C})$. A $K$-orbit (or $K^0$-orbit) in $M_{p,q}$ is said
to be nilpotent if its closure contains the zero matrix. The closure,
$\overline{\mathcal{O}}$, of a nilpotent $K$-orbit (resp.\ $K^0$-orbit)
${\mathcal{O}}$ in $M_{p,q}$ is a union of ${\mathcal{O}}$ and some
nilpotent $K$-orbits (resp.\ $K^0$-orbits) of smaller dimensions. The
description of the closure of nilpotent $K$-orbits has been known for
some time, but not so for the nilpotent $K^0$-orbits. A conjecture
describing the closure of nilpotent $K^0$-orbits was proposed in
\cite{DLS} and verified when $\min(p,q) \le 7$. In this paper we
prove the conjecture. The proof is based on a study of two
prehomogeneous vector spaces attached to $\mathcal{O}$ and
determination of the basic relative invariants of these spaces.
The above problem is equivalent to the problem of describing the
closure of nilpotent orbits in the real Lie algebra $\mathfrak{so}
(p,q)$ under the adjoint action of the identity component of the real
orthogonal group $\mathrm{O}(p,q)$.
Keywords:orthogonal $ab$-diagrams, prehomogeneous vector spaces, relative invariants Categories:17B20, 17B45, 22E47 |
3. 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 |
4. 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 |
5. 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 |