Normal Enveloping Algebras

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

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.

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.

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.

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.

ON THE CONSISTENCY OF TWISTED GENERALIZED WEYL ALGEBRAS

A twisted generalized Weyl algebra A of degree n depends on a. base algebra R, n commuting automorphisms sigma(i) of R, n central elements t(i) of R and on some additional scalar parameters. In a paper by Mazorchuk and Turowska, it is claimed that certain consistency conditions for sigma(i) and t(i) are sufficient for the algebra to be nontrivial. However, in this paper we give all example which shows that this is false. We also correct the statement by finding a new set of consistency conditions and prove that the old and new conditions together are necessary and sufficient for the base algebra R to map injectively into A. In particular they are sufficient for the algebra A to be nontrivial. We speculate that these consistency relations may play a role in other areas of mathematics, analogous to the role played by the Yang-Baxter equation in the theory of integrable systems.

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.

Multiparameter twisted Weyl algebras

We introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordan's simple localization of the multiparameter quantized Weyl algebra and Hayashi's q-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus, we also describe all Whittaker pairs up to isomorphism over a class of twisted generalized Weyl algebras which includes the multiparameter twisted Weyl algebras. (C) 2011 Elsevier Inc. All rights reserved.

Polynomial and Poisson dependence in free Poisson algebras and free Poisson fields

We prove that any two Poisson dependent elements in a free Poisson algebra and a free Poisson field of characteristic zero are algebraically dependent, thus answering positively a question from Makar-Limanov and Umirbaev (2007) [8]. We apply this result to give a new proof of the tameness of automorphisms for free Poisson algebras of rank two (see Makar-Limanov and Umirbaev (2011) [9], Makar-Limanov et al. (2009) [10]). (C) 2011 Elsevier Inc. All rights reserved.

Algebras determined by their supports

In this paper, we introduce and study a class of algebras which we call ada algebras. An artin algebra is ada if every indecomposable projective and every indecomposable injective module lies in the union of the left and the right parts of the module category. We describe the Auslander-Reiten components of an ada algebra which is not quasi-tilted, showing in particular that its representation theory is entirely contained in that of its left and right supports, which are both tilted algebras. Also, we prove that an ada algebra over an algebraically closed field is simply connected if and only if its first Hochschild cohomology group vanishes. (C) 2011 Elsevier B.V. All rights reserved.

Ontogeny of manipulative behavior and nut-cracking in young tufted capuchin monkeys (Cebus apella): a Perception-action perspective

How do capuchin monkeys learn to use stones to crack open nuts? Perception-action theory posits that individuals explore producing varying spatial and force relations among objects and surfaces, thereby learning about affordances of such relations and how to produce them. Such learning supports the discovery of tool use. We present longitudinal developmental data from semifree-ranging tufted capuchin monkeys (Cebus apella) to evaluate predictions arising from Perception-action theory linking manipulative development and the onset of tool-using. Percussive actions bringing an object into contact with a surface appeared within the first year of life. Most infants readily struck nuts and other objects against stones or other surfaces from 6 months of age, but percussive actions alone were not sufficient to produce nut-cracking sequences. Placing the nut on the anvil surface and then releasing it, so that it could be struck with a stone, was the last element necessary for nut-cracking to appear in capuchins. Young chimpanzees may face a different challenge in learning to crack nuts: they readily place objects on surfaces and release them, but rarely vigorously strike objects against surfaces or other objects. Thus the challenges facing the two species in developing the same behavior (nut-cracking using a stone hammer and an anvil) may be quite different. Capuchins must inhibit a strong bias to hold nuts so that they can release them; chimpanzees must generate a percussive action rather than a gentle placing action. Generating the right actions may be as challenging as achieving the right sequence of actions in both species. Our analysis suggests a new direction for studies of social influence on young primates learning sequences of actions involving manipulation of objects in relation to surfaces.

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.

Facies associations and controls on the evolution from a coastal bay to a lagoon system, Santa Catarina Coast, Brazil

The diverse Holocene morphological features along the south coast of the state of Santa Catarina include lagoons and residual lakes, a barrier, a delta (constructed by the Tubarao River), and pre-existing incised valleys that have flooded and filled. This scenario contains the sedimentary record of the transition from a bay to a lagoon system, which occurred during the rise and subsequent semi-stabilisation of the relative sea-level during the Holocene. The geomorphological evolution of this area was investigated using a combination of morphology, stratigraphic analysis of rotary push cores, vibracores and trenches with radiocarbon dating, taxonomic determination and taphonomic characterisation of Holocene fossil molluscs. Palaeogeographic maps were constructed to illustrate how the bay evolved over the last 8000 years. The relative sea-level rise and local sedimentary processes were the prime forcing factors determining the depositional history and palaeogeographic changes. The Holocene sedimentary succession began between 8000 and 5700 cal BP with the deposits of transgressive sandsheets. These deposits correspond to the initial marine flooding surface that was formed while the relative sea-level rose at a higher rate than the input of sediments, prior to the formation of the coastal barrier. The change from a bay to a lagoon system occurred around 5700 and 2500 cal BP during the mid-Holocene highstand with the formation of the barrier and with the achievement of a balance between sea-level rise and sedimentary supply. Until 2500 cal BP, the presence of this barrier, the following gentle decline in sea level and the initial emergence of back-barrier features restricted the hydro-dynamic circulation inside the bay and favoured an increase in the Tubarao River delta progradation rate. The final stage, during the last 2500 years, was marked by the increasing back-barrier width, with the establishment of salt marshes, the arrival of the delta in the back-barrier, and the advance of aeolian dunes along the outer lagoon margins. This study shed light on the mechanisms of coastal bay evolution in a setting existed prior to the beginning of barrier lagoon sedimentation. (C) 2012 Elsevier B.V. All rights reserved.