Expand all Collapse all | Results 1 - 4 of 4 |
1. CJM 2009 (vol 62 pp. 382)
Verma Modules over Quantum Torus Lie Algebras Representations of various one-dimensional central
extensions of quantum tori (called quantum torus Lie algebras) were
studied by several authors. Now we define a central extension of
quantum tori so that all known representations can be regarded as
representations of the new quantum torus Lie algebras $\mathfrak{L}_q$. The
center of $\mathfrak{L}_q$ now is generally infinite dimensional.
In this paper, $\mathbb{Z}$-graded Verma modules $\widetilde{V}(\varphi)$ over $\mathfrak{L}_q$
and their corresponding irreducible highest weight modules
$V(\varphi)$ are defined for some linear functions $\varphi$.
Necessary and sufficient conditions for $V(\varphi)$ to have all
finite dimensional weight spaces are given. Also necessary and
sufficient conditions for Verma modules $\widetilde{V}(\varphi)$ to
be irreducible are obtained.
Categories:17B10, 17B65, 17B68 |
2. 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 |
3. 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 |
4. CJM 1997 (vol 49 pp. 119)
Automorphisms of the Lie algebras $W^*$ in characteristic $0$ No abstract.
Categories:17B40, 17B65, 17B66, 17B68, 17B70 |