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1. 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 |
2. CMB 2003 (vol 46 pp. 529)
Representations of the Twisted Heisenberg--Virasoro Algebra at Level Zero We describe the structure of the irreducible highest weight modules
for the twisted Heisenberg--Virasoro Lie algebra at level zero. We
prove that either a Verma module is irreducible or its maximal
submodule is cyclic.
Categories:17B68, 17B65 |
3. CMB 2002 (vol 45 pp. 567)
Subalgebras of $\gc_N$ and Jacobi Polynomials We classify the subalgebras of the general Lie conformal algebra
$\gc_N$ that act irreducibly on $\mathbb{C} [\partial]^N$ and that
are normalized by the sl$_2$-part of a Virasoro element. The
problem turns out to be closely related to classical Jacobi
polynomials $P_n^{(-\sigma,\sigma)}$, $\sigma \in \mathbb{C}$. The
connection goes both ways---we use in our classification some
classical properties of Jacobi polynomials, and we derive from the
theory of conformal algebras some apparently new properties of
Jacobi polynomials.
Categories:17B65, 17B68, 17B69, 33C45 |
4. 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 |