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Search: MSC category 17B70 ( Graded Lie (super)algebras )

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1. CJM Online first

Iacono, Donatella; Manetti, Marco
On deformations of pairs (manifold, coherent sheaf)
We analyse infinitesimal deformations of pairs $(X,\mathcal F)$ with $\mathcal F$ a coherent sheaf on a smooth projective variety $X$ over an algebraically closed field of characteristic $0$. We describe a differential graded Lie algebra controlling the deformation problem, and we prove an analog of a Mukai-Artamkin Theorem about the trace map.

Keywords:deformation of manifold and coherent sheaf, differential graded Lie algebra
Categories:14D15, 13D10, 17B70, 18G50

2. CJM 2017 (vol 69 pp. 721)

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

3. CJM 2016 (vol 69 pp. 453)

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

4. CJM 2014 (vol 67 pp. 55)

Barron, Tatyana; Kerner, Dmitry; Tvalavadze, Marina
On Varieties of Lie Algebras of Maximal Class
We study complex projective varieties that parametrize (finite-dimensional) filiform Lie algebras over ${\mathbb C}$, using equations derived by Millionshchikov. In the infinite-dimensional case we concentrate our attention on ${\mathbb N}$-graded Lie algebras of maximal class. As shown by A. Fialowski there are only three isomorphism types of $\mathbb{N}$-graded Lie algebras $L=\oplus^{\infty}_{i=1} L_i$ of maximal class generated by $L_1$ and $L_2$, $L=\langle L_1, L_2 \rangle$. Vergne described the structure of these algebras with the property $L=\langle L_1 \rangle$. In this paper we study those generated by the first and $q$-th components where $q\gt 2$, $L=\langle L_1, L_q \rangle$. Under some technical condition, there can only be one isomorphism type of such algebras. For $q=3$ we fully classify them. This gives a partial answer to a question posed by Millionshchikov.

Keywords:filiform Lie algebras, graded Lie algebras, projective varieties, topology, classification
Categories:17B70, 14F45

5. CJM 2012 (vol 65 pp. 82)

Félix, Yves; Halperin, Steve; Thomas, Jean-Claude
The Ranks of the Homotopy Groups of a Finite Dimensional Complex
Let $X$ be an $n$-dimensional, finite, simply connected CW complex and set $\alpha_X =\limsup_i \frac{\log\mbox{ rank}\, \pi_i(X)}{i}$. When $0\lt \alpha_X\lt \infty$, we give upper and lower bound for $ \sum_{i=k+2}^{k+n} \textrm{rank}\, \pi_i(X) $ for $k$ sufficiently large. We show also for any $r$ that $\alpha_X$ can be estimated from the integers rk$\,\pi_i(X)$, $i\leq nr$ with an error bound depending explicitly on $r$.

Keywords:homotopy groups, graded Lie algebra, exponential growth, LS category
Categories:55P35, 55P62, , , , 17B70

6. CJM 2007 (vol 59 pp. 696)

Bangoura, Momo
Algèbres de Lie d'homotopie associées à une proto-bigèbre de Lie
On associe \`a toute structure de proto-big\`ebre de Lie sur un espace vectoriel $F$ de dimension finie des structures d'alg\`ebre de Lie d'homotopie d\'efinies respectivement sur la suspension de l'alg\`ebre ext\'erieure de $F$ et celle de son dual $F^*$. Dans ces alg\`ebres, tous les crochets $n$-aires sont nuls pour $n \geq 4$ du fait qu'ils proviennent d'une structure de proto-big\`ebre de Lie. Plus g\'en\'eralement, on associe \`a un \'el\'ement de degr\'e impair de l'alg\`ebre ext\'erieure de la somme directe de $F$ et $F^*$, une collection d'applications multilin\'eaires antisym\'etriques sur l'alg\`ebre ext\'erieure de $F$ (resp.\ $F^*$), qui v\'erifient les identit\'es de Jacobi g\'en\'eralis\'ees, d\'efinissant les alg\`ebres de Lie d'homotopie, si l'\'el\'ement donn\'e est de carr\'e nul pour le grand crochet de l'alg\`ebre ext\'erieure de la somme directe de $F$ et de~$F^*$. To any proto-Lie algebra structure on a finite-dimensional vector space~$F$, we associate homotopy Lie algebra structures defined on the suspension of the exterior algebra of $F$ and that of its dual $F^*$, respectively. In these algebras, all $n$-ary brackets for $n \geq 4$ vanish because the brackets are defined by the proto-Lie algebra structure. More generally, to any element of odd degree in the exterior algebra of the direct sum of $F$ and $F^*$, we associate a set of multilinear skew-symmetric mappings on the suspension of the exterior algebra of $F$ (resp.\ $F^*$), which satisfy the generalized Jacobi identities, defining the homotopy Lie algebras, if the given element is of square zero with respect to the big bracket of the exterior algebra of the direct sum of $F$ and~$F^*$.

Keywords:algèbre de Lie d'homotopie, bigèbre de Lie, quasi-bigèbre de Lie, proto-bigèbre de Lie, crochet dérivé, jacobiateur
Categories:17B70, 17A30

7. CJM 2006 (vol 58 pp. 1291)

Weimar-Woods, Evelyn
The General Structure of $G$-Graded Contractions of Lie Algebras I. The Classification
We give the general structure of complex (resp., real) $G$-graded contractions of Lie algebras where $G$ is an arbitrary finite Abelian group. For this purpose, we introduce a number of concepts, such as pseudobasis, higher-order identities, and sign invariants. We characterize the equivalence classes of $G$-graded contractions by showing that our set of invariants (support, higher-order identities, and sign invariants) is complete, which yields a classification.

Keywords:Lie algebras, graded contractions
Categories:17B05, 17B70

8. CJM 1999 (vol 51 pp. 658)

Shumyatsky, Pavel
Nilpotency of Some Lie Algebras Associated with $p$-Groups
Let $ L=L_0+L_1$ be a $\mathbb{Z}_2$-graded Lie algebra over a commutative ring with unity in which $2$ is invertible. Suppose that $L_0$ is abelian and $L$ is generated by finitely many homogeneous elements $a_1,\dots,a_k$ such that every commutator in $a_1,\dots,a_k$ is ad-nilpotent. We prove that $L$ is nilpotent. This implies that any periodic residually finite $2'$-group $G$ admitting an involutory automorphism $\phi$ with $C_G(\phi)$ abelian is locally finite.

Categories:17B70, 20F50

9. CJM 1998 (vol 50 pp. 225)

Benkart, Georgia
Derivations and invariant forms of Lie algebras graded by finite root systems
Lie algebras graded by finite reduced root systems have been classified up to isomorphism. In this paper we describe the derivation algebras of these Lie algebras and determine when they possess invariant bilinear forms. The results which we develop to do this are much more general and apply to Lie algebras that are completely reducible with respect to the adjoint action of a finite-dimensional subalgebra.

Categories:17B20, 17B70, 17B25

10. 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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