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Search: MSC category 17B65 ( Infinite-dimensional Lie (super)algebras [See also 22E65] )

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1. CJM Online first

Allison, Bruce; Faulkner, John; Smirnov, Oleg
Weyl images of Kantor pairs
Kantor pairs arise naturally in the study of $5$-graded Lie algebras. In this article, we introduce and study Kantor pairs with short Peirce gradings and relate them to Lie algebras graded by the root system of type $\mathrm{BC}_2$. This relationship allows us to define so called Weyl images of short Peirce graded Kantor pairs. We use Weyl images to construct new examples of Kantor pairs, including a class of infinite dimensional central simple Kantor pairs over a field of characteristic $\ne 2$ or $3$, as well as a family of forms of a split Kantor pair of type $\mathrm{E}_6$.

Keywords:Kantor pair, graded Lie algebra, Jordan pair
Categories:17B60, 17B70, 17C99, 17B65

2. CJM Online first

Marquis, Timothée; Neeb, Karl-Hermann
Isomorphisms of twisted Hilbert loop algebras
The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a ``minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitly as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$.

Keywords:locally affine Lie algebra, Hilbert-Lie algebra, positive energy representation
Categories:17B65, 17B70, 17B22, 17B10

3. CJM 2015 (vol 68 pp. 258)

Calixto, Lucas; Moura, Adriano; Savage, Alistair
Equivariant Map Queer Lie Superalgebras
An equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) $X$ to a queer Lie superalgebra $\mathfrak{q}$ that are equivariant with respect to the action of a finite group $\Gamma$ acting on $X$ and $\mathfrak{q}$. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that $\Gamma$ is abelian and acts freely on $X$. We show that such representations are parameterized by a certain set of $\Gamma$-equivariant finitely supported maps from $X$ to the set of isomorphism classes of irreducible finite-dimensional representations of $\mathfrak{q}$. In the special case where $X$ is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

Keywords:Lie superalgebra, queer Lie superalgebra, loop superalgebra, equivariant map superalgebra, finite-dimensional representation, finite-dimensional module
Categories:17B65, 17B10

4. CJM 2009 (vol 62 pp. 382)

Lü, Rencai; Zhao, Kaiming
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

5. CJM 2008 (vol 60 pp. 892)

Neeb, Karl-Hermann; Wagemann, Friedrich
The Second Cohomology of Current Algebras of General Lie Algebras
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.

Keywords:current algebra, Lie algebra cohomology, Lie algebra homology, invariant bilinear form, central extension
Categories:17B56, 17B65

6. CJM 2006 (vol 58 pp. 225)

Azam, Saeid
Generalized Reductive Lie Algebras: Connections With Extended Affine Lie Algebras and Lie Tori
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.

Categories:17B65, 17B67, 17B40

7. CJM 2003 (vol 55 pp. 856)

Su, Yucai
Poisson Brackets and Structure of Nongraded Hamiltonian Lie Algebras Related to Locally-Finite Derivations
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.

Categories:17B40, 17B65

8. CJM 2001 (vol 53 pp. 195)

Mokler, Claus
On the Steinberg Map and Steinberg Cross-Section for a Symmetrizable Indefinite Kac-Moody Group
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~$C$.

Categories:22E65, 17B65

9. CJM 1999 (vol 51 pp. 523)

Fabbri, Marc A.; Okoh, Frank
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

10. CJM 1998 (vol 50 pp. 210)

Zhao, Kaiming
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

11. CJM 1997 (vol 49 pp. 820)

Robart, Thierry
Sur l'intégrabilité des sous-algèbres de Lie en dimension infinie
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.

Categories:22E65, 58h05, 17B65

12. CJM 1997 (vol 49 pp. 119)

Osborn, J. Marshall
Automorphisms of the Lie algebras $W^*$ in characteristic $0$
No abstract.

Categories:17B40, 17B65, 17B66, 17B68, 17B70

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