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Search: All articles in the CMB digital archive with keyword Lie algebra

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1. CMB Online first

Buijs, Urtzi; Félix, Yves; Murillo, Aniceto; Tanré, Daniel
 Maurer-Cartan elements in the Lie models of finite simplicial complexes In a previous work, we have associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we have also a realization functor from the category of complete differential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex. Keywords:complete differential graded Lie algebra, Maurer-Cartan element, rational homotopy theoryCategory:16E45

2. CMB Online first

Lee, Tsiu-Kwen
 Ad-nilpotent elements of semiprime rings with involution Let $R$ be an $n!$-torsion free semiprime ring with involution $*$ and with extended centroid $C$, where $n\gt 1$ is a positive integer. We characterize $a\in K$, the Lie algebra of skew elements in $R$, satisfying $(\operatorname{ad}_a)^n=0$ on $K$. This generalizes both Martindale and Miers' theorem and the theorem of Brox et al. To prove it we first prove that if $a, b\in R$ satisfy $(\operatorname{ad}_a)^n=\operatorname{ad}_b$ on $R$, where either $n$ is even or $b=0$, then $\big(a-\lambda\big)^{[\frac{n+1}{2}]}=0$ for some $\lambda\in C$. Keywords:Semiprime ring, Lie algebra, Jordan algebra, faithful $f$-free, involution, skew (symmetric) element, ad-nilpotent element, Jordan elementCategories:16N60, 16W10, 17B60

3. CMB 2014 (vol 57 pp. 735)

Cagliero, Leandro; Szechtman, Fernando
 On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras We describe of all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let $K/F$ be a finite separable field extension and let $x,y\in K$. When is $F[x,y]=F[\alpha x+\beta y]$ for some non-zero elements $\alpha,\beta\in F$? Keywords:uniserial module, Lie algebra, associative algebra, primitive elementCategories:17B10, 13C05, 12F10, 12E20

4. CMB 2012 (vol 56 pp. 606)

Mazorchuk, Volodymyr; Zhao, Kaiming
 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 actionCategories:17B20, 17B65, 17B66, 17B68

5. CMB 2011 (vol 55 pp. 579)

Ndogmo, J. C.
 Casimir Operators and Nilpotent Radicals It is shown that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators. A different proof is given in the well known special case of an abelian radical. A result relating the number of invariants to the dimension of the Cartan subalgebra is also established. Keywords:nilpotent radical, Casimir operators, algebraic Lie algebras, Cartan subalgebras, number of invariantsCategories:16W25, 17B45, 16S30

6. CMB 2011 (vol 54 pp. 442)

García, Esther; Lozano, Miguel Gómez; Neher, Erhard
 Nondegeneracy for Lie Triple Systems and Kantor Pairs We study the transfer of nondegeneracy between Lie triple systems and their standard Lie algebra envelopes as well as between Kantor pairs, their associated Lie triple systems, and their Lie algebra envelopes. We also show that simple Kantor pairs and Lie triple systems in characteristic $0$ are nondegenerate. Keywords:Kantor pairs, Lie triple systems, Lie algebrasCategories:17A40, 17B60, 17B99

7. CMB 2011 (vol 54 pp. 472)

Iacono, Donatella
 A Semiregularity Map Annihilating Obstructions to Deforming Holomorphic Maps We study infinitesimal deformations of holomorphic maps of compact, complex, KÃ¤hler manifolds. In particular, we describe a generalization of Bloch's semiregularity map that annihilates obstructions to deform holomorphic maps with fixed codomain. Keywords:semiregularity map, obstruction theory, functors of Artin rings, differential graded Lie algebrasCategories:13D10, 14D15, 14B10

8. CMB 2008 (vol 51 pp. 298)

Tocón, Maribel
 The Kostrikin Radical and the Invariance of the Core of Reduced Extended Affine Lie Algebras In this paper we prove that the Kostrikin radical of an extended affine Lie algebra of reduced type coincides with the center of its core, and use this characterization to get a type-free description of the core of such algebras. As a consequence we get that the core of an extended affine Lie algebra of reduced type is invariant under the automorphisms of the algebra. Keywords:extended affine Lie algebra, Lie torus, core, Kostrikin radicalCategories:17B05, 17B65

9. CMB 2000 (vol 43 pp. 3)