Generalizations of Lie Algebras

The generalizations of Lie algebras appeared in the modern mathematics and mathematical physics. In this paper we consider recent developments and remaining open problems on the subject. Some of that developments have been influenced by lectures given by Professor Jaime Keller in his research seminar. The survey includes Lie superalgebras, color Lie algebras, Lie algebras in symmetric categories, free Lie tau-algebras, and some generalizations with non-associative enveloping algebras: tangent algebras to analytic loops, bialgebras and primitive elements, non-associative Hopf algebras.

COMMUTATIVE ALGEBRAS IN DRINFELD CATEGORIES OF ABELIAN LIE ALGEBRAS

We describe (braided-) commutative algebras with non-degenerate multiplicative form in certain braided monoidal categories, corresponding to abelian metric Lie algebras (so-called Drinfeld categories). We also describe local modules over these algebras and classify commutative algebras with a finite number of simple local modules.

On simple Lie algebras over a field of characteristic 2

We prove that the simple Lie algebras constructed by G. Jurman (2004) in 121 are isomorphic to Hamiltonian algebras. As a corollary we answer all questions formulated in G. Jurman (2004) [2] about isomorphisms of these algebras. (C) 2012 Elsevier Inc. All rights reserved.

Normal Enveloping Algebras

A full characterization is given of ordinary and restricted enveloping algebras which are normal with respect to the principal involution.

Maximal Subgroups of Compact Lie Groups

This report aims at giving a general overview on the classification of the maximal subgroups of compact Lie groups (not necessarily connected). In the first part, it is shown that these fall naturally into three types: (1) those of trivial type, which are simply defined as inverse images of maximal subgroups of the corresponding component group under the canonical projection and whose classification constitutes a problem in finite group theory, (2) those of normal type, whose connected one-component is a normal subgroup, and (3) those of normalizer type, which are the normalizers of their own connected one-component. It is also shown how to reduce the classification of maximal subgroups of the last two types to: (2) the classification of the finite maximal Sigma-invariant subgroups of centerfree connected compact simple Lie groups and (3) the classification of the Sigma-primitive subalgebras of compact simple Lie algebras, where Sigma is a subgroup of the corresponding outer automorphism group. In the second part, we explicitly compute the normalizers of the primitive subalgebras of the compact classical Lie algebras (in the corresponding classical groups), thus arriving at the complete classification of all (non-discrete) maximal subgroups of the compact classical Lie groups.

On delta-derivations of n-ary algebras

We give a description of delta-derivations of (n + 1)-dimensional n-ary Filippov algebras and, as a consequence, of simple finite-dimensional Filippov algebras over an algebraically closed field of characteristic zero. We also give new examples of non-trivial delta-derivations of Filippov algebras and show that there are no non-trivial delta-derivations of the simple ternary Mal'tsev algebra M-8.

Virasoro Action on Imaginary Verma Modules and the Operator Form of the KZ-Equation

We define the Virasoro algebra action on imaginary Verma modules for affine and construct an analogue of the Knizhnik-Zamolodchikov equation in the operator form. Both these results are based on a realization of imaginary Verma modules in terms of sums of partial differential operators.

Higher identities for the ternary commutator

We use computer algebra to study polynomial identities for the trilinear operation [a, b, c] = abc - acb - bac + bca + cab - cba in the free associative algebra. It is known that [a, b, c] satisfies the alternating property in degree 3, no new identities in degree 5, a multilinear identity in degree 7 which alternates in 6 arguments, and no new identities in degree 9. We use the representation theory of the symmetric group to demonstrate the existence of new identities in degree 11. The only irreducible representations of dimension <400 with new identities correspond to partitions 2(5), 1 and 2(4), 1(3) and have dimensions 132 and 165. We construct an explicit new multilinear identity for partition 2(5), 1 and we demonstrate the existence of a new non-multilinear identity in which the underlying variables are permutations of a(2)b(2)c(2)d(2)e(2) f.

delta-Superderivations of Semisimple Finite-Dimensional Jordan Superalgebras

In the paper, a complete description of the delta-derivations and the delta-superderivations of semisimple finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic p not equal 2 is given. In particular, new examples of nontrivial (1/2)-derivations and odd (1/2)-superderivations are given that are not operators of right multiplication by an element of the superalgebra.

OBSERVING BEHAVIOR AND ATYPICALLY RESTRICTED STIMULUS CONTROL

Restricted stimulus control refers to discrimination learning with atypical limitations in the range of controlling stimuli or stimulus features In the study reported here 4 normally capable individuals and 10 individuals with Intellectual disabilities (ID) performed two-sample delayed matching to sample Sample stimulus observing was recorded with an eye tracking apparatus High accuracy scores indicated stimulus control by both sample stimuli for the 4 nondisabled participants and 4 participants with ID and eye tracking data showed reliable observing of all stimuli Intermediate accuracy scores indicated restricted stimulus control for the remaining 6 participants Their eye tracking data showed that errors were related to failures to observe sample stimuli and relatively brief observing durations Five of these participants were then given interventions designed to improve observing behavior For 4 participants the interventions resulted initially in elimination of observing failures increased observing durations and Increased accuracy For 2 of these participants contingencies sufficient to maintain adequate observing were not always sufficient to maintain high accuracy subsequent procedure modifications restored It however For the 5th participant initial improvements in observing were not accompanied by improved accuracy in apparent Instance of observing without attending accuracy improved only after an additional intervention that imposed contingencies on observing behavior Thus interventions that control observing behavior seem necessary but may not always be sufficient for the remediation of restricted stimulus control

Special identities for Bol algebras

Bol algebras appear as the tangent algebra of Bol loops. A (left) Bol algebra is a vector space equipped with a binary operation [a, b] and a ternary operation {a, b, c} that satisfy five defining identities. If A is a left or right alternative algebra then A(b) is a Bol algebra, where [a, b] := ab - ba is the commutator and {a, b, c} := < b, c, a > is the Jordan associator. A special identity is an identity satisfied by Ab for all right alternative algebras A, but not satisfied by the free Bol algebra. We show that there are no special identities of degree <= 7, but there are special identities of degree 8. We obtain all the special identities of degree 8 in partition six-two. (C) 2011 Elsevier Inc. All rights reserved.

Population divergence of Melipona scutellaris (Hymenoptera: Meliponina) in two restricted areas in Bahia, Brazil

Melipona scutellaris Latreille has great economic and ecological importance, especially because it is a pollinator of native plant species. Despite the importance of this species, there is little information about the conservation status of their populations. The objective of this study was to assess the diversity in populations of M. scutellaris coming from a Semideciduous Forest Fragment and an Atlantic Forest Fragment in the Northeast Brazil, through geometric morphometric analysis of wings in worker bees. In each area, worker bees were collected from 10 colonies, 10 workers per colony. To assess the diversity on the right wings of worker bees, 15 landmarks were plotted and the measures were used in analysis of variance and multivariate analysis, principal component analysis, discriminant analysis and clustering analysis. There were significant differences in the shape of the wing venation patterns between colonies of two sites (Wilk's lambda = 0.000006; p < 0.000001), which is probably due to the geographical distance between places of origin which impedes the gene flow between them. It indicates that inter and intrapopulation morphometric variability exists (p < 0.000001) in M. scutellaris coming from two different biomes, revealing the existence of diversity in these populations, which is necessary for the conservation of this bee species.

FREE GROUP ALGEBRAS IN DIVISION RINGS

Let k be an algebraically closed field of characteristic zero and let L be an algebraic function field over k. Let sigma : L -> L be a k-automorphism of infinite order, and let D be the skew field of fractions of the skew polynomial ring L[t; sigma]. We show that D contains the group algebra kF of the free group F of rank 2.

Symmetry insights for design of supercomputer network topologies: roots and weights lattices

We present a family of networks whose local interconnection topologies are generated by the root vectors of a semi-simple complex Lie algebra. Cartan classification theorem of those algebras ensures those families of interconnection topologies to be exhaustive. The global arrangement of the network is defined in terms of integer or half-integer weight lattices. The mesh or torus topologies that network millions of processing cores, such as those in the IBM BlueGene series, are the simplest member of that category. The symmetries of the root systems of an algebra, manifested by their Weyl group, lends great convenience for the design and analysis of hardware architecture, algorithms and programs.

Serum from Calorie-Restricted Rats Activates Vascular Cell eNOS through Enhanced Insulin Signaling Mediated by Adiponectin

eNOS activation resulting in mitochondrial biogenesis is believed to play a central role in life span extension promoted by calorie restriction (CR). We investigated the mechanism of this activation by treating vascular cells with serum from CR rats and found increased Akt and eNOS phosphorylation, in addition to enhanced nitrite release. Inhibiting Akt phosphorylation or immunoprecipitating adiponectin (found in high quantities in CR serum) completely prevented the increment in nitrite release and eNOS activation. Overall, we demonstrate that adiponectin in the serum from CR animals increases NO center dot signaling by activating the insulin pathway. These results suggest this hormone may be a determinant regulator of the beneficial effects of CR.