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1. CMB Online first
Sur les algÃ¨bres de Lie associÃ©es Ã une connexion Let $\Gamma$ be a connection on a smooth manifold
$M$, in this paper we give some properties of $\Gamma$ by studying
the corresponding Lie algebras. In particular, we compute the
first Chevalley-Eilenberg cohomology space of the horizontal
vector fields Lie algebra on the tangent bundle of $M$, whose
the corresponding Lie derivative of $\Gamma$ is null, and of
the horizontal nullity curvature space.
Keywords:algÃ¨bre de Lie, connexion, cohomologie de Chevalley-Eilenberg, champs dont la dÃ©rivÃ©e de Lie correspondante Ã une connexion est nulle, espace de nullitÃ© de la courbure Categories:17B66, 53B15 |
2. CMB 2012 (vol 56 pp. 606)
Characterization of Simple Highest Weight Modules We prove that for simple complex finite dimensional
Lie algebras, affine Kac-Moody Lie algebras, the
Virasoro algebra and the Heisenberg-Virasoro algebra,
simple highest weight modules are characterized
by the property that all positive root elements
act on these modules locally nilpotently. We
also show that this is not the case for higher rank
Virasoro and for Heisenberg algebras.
Keywords:Lie algebra, highest weight module, triangular decomposition, locally nilpotent action Categories:17B20, 17B65, 17B66, 17B68 |
3. CMB 2002 (vol 45 pp. 672)
A New Class of Representations of EALA Coordinated by Quantum Tori in Two Variables We study the representations of extended affine Lie algebras
$s\ell_{\ell+1} (\mathbb{C}_q)$ where $q$ is $N$-th primitive root of
unity ($\mathbb{C}_q$ is the quantum torus in two variables). We
first prove that $\bigoplus s\ell_{\ell+1} (\mathbb{C})$ for a
suitable number of copies is a quotient of $s\ell_{\ell+1}
(\mathbb{C}_q)$. Thus any finite dimensional irreducible module for
$\bigoplus s\ell_{\ell+1} (\mathbb{C})$ lifts to a representation of
$s\ell_{\ell+1} (\mathbb{C}_q)$. Conversely, we prove that any finite
dimensional irreducible module for $s\ell_{\ell+1} (\mathbb{C}_q)$
comes from above. We then construct modules for the extended affine
Lie algebras $s\ell_{\ell+1} (\mathbb{C}_q) \oplus \mathbb{C} d_1
\oplus \mathbb{C} d_2$ which is integrable and has finite dimensional
weight spaces.
Categories:17B65, 17B66, 17B68 |