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.

Let $A$ be a unital commutative associative algebra over a field of characteristic zero, $\k$ a Lie algebra, and $\zf$ a vector space, considered as a trivial module of the Lie algebra $\gf := A \otimes \kf$. In this paper, we give a description of the cohomology space $H^2(\gf,\zf)$ in terms of easily accessible data associated with $A$ and $\kf$. We also discuss the topological situation, where $A$ and $\kf$ are locally convex algebras.

We investigate a class of Lie algebras which we call {\it generalized reductive Lie algebras}. These are generalizations of semi-simple, reductive, and affine Kac--Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be {\it irreducible\/} and we note that this class of algebras has been under intensive investigation in recent years. They have also been called {\it extended affine Lie algebras}. The larger class of generalized reductive Lie algebras has not been so intensively investigated. We study them in this paper and note that one way they arise is as fixed point subalgebras of finite order automorphisms. We show that the core modulo the center of a generalized reductive Lie algebra is a direct sum of centerless Lie tori. Therefore one can use the results known about the classification of centerless Lie tori to classify the cores modulo centers of generalized reductive Lie algebras.

Xu introduced a class of nongraded Hamiltonian Lie algebras. These Lie algebras have a Poisson bracket structure. In this paper, the isomorphism classes of these Lie algebras are determined by employing a ``sandwich'' method and by studying some features of these Lie algebras. It is obtained that two Hamiltonian Lie algebras are isomorphic if and only if their corresponding Poisson algebras are isomorphic. Furthermore, the derivation algebras and the second cohomology groups are determined.

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~$C$.

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.

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.

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.

No abstract.