Loop algebras of Zassenhaus algebras in characteristic three

The simple, modular Lie algebras of Zassenhaus have peculiar features in characteristic three. Their second cohomology groups are larger than in characteristicp>3, and they possess a non-degenerate associative form. These properties are reflected in the presentations of certain loop algebras of these algebras, that arise naturally in analogy with the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Partially supported by MURST, Italy. The author is a member of CNR-GNSAGA, Italy. The author is grateful to the Mathematisches Forschungsinstitut Oberwolfach for the kind hospitality while part of this work was being written.     

Cotangent bundles of 4-dimensional hypercomplex Lie groups

 We study geometrical structures on the cotangent bundle T ^* G of a Lie group G which are left-invariant with respect to the Lie group structure on T ^* G determined by a left-invariant affine structure ∇ on G. In particular, we investigate the existence of conformally hyper-K?hler metrics and hyper-K?hler with torsion (HKT) structures on the cotangent bundle of hypercomplex 4-dimensional Lie groups. By applying In?nü-Wigner contractions to compact semisimple Lie algebras we obtain non semisimple Lie algebras endowed with invariant HKT structures. Received: 4 February 2002 / Revised version: 20 August 2002 Research partially supported by MURST and GNSAGA (Indam) of Italy Mathematics Subject Classification (2000): 53C26, 22E25     

The orders of nonsingular derivations of modular Lie algebras

We extend the results of Shalev [Sh] on the orders of nonsingular derivations of finite-dimensional non-nilpotent modular Lie algebras. The author is grateful to Ministero dell’Università e della Ricerca Scientifica, Italy, for financial support to the project “Graded Lie algebras and pro-p-groups of finite width”.     

Generators of simple Lie algebras in arbitrary characteristics

In this paper we study the minimal number of generators for simple Lie algebras in characteristic 0 or p > 3. We show that any such algebra can be generated by 2 elements. We also examine the ‘one and a half generation’ property, i.e. when every non-zero element can be completed to a generating pair. We show that classical simple algebras have this property, and that the only simple Cartan type algebras of type W which have this property are the Zassenhaus algebras. The author was partially supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00287 (RTN Network “K-Theory, Algebraic Groups and Related Structures”) and a long-term research grant from the D.A.A.D.     

The orders of nonsingular derivations of Lie algebras of characteristic two

Nonsingular derivations of modular Lie algebras which have finite multiplicative order play a role in the coclass theory for pro-p groups and Lie algebras. A study of the set of positive integers which occur as orders of nonsingular derivations of finite-dimensional nonnilpotent Lie algebras of characteristic p > 0 was initiated by Shalev and continued by the present author. In this paper we continue this study in the case of characteristic two. Among other results, we prove that any divisor n of 2^k − 1 with n ⁴ > (2^k − n)³ belongs to . Our methods consist of elementary arguments with polynomials over finite fields and a little character theory of finite groups. This work was partially supported by Ministero dell’Istruzione e dell’Università, Italy, through PRIN “Graded Lie algebras and pro-p-groups of finite width”.     

Strongly Flat and PO-Flat S-Posets

Abstract

The main purpose of this paper is to study Lie algebras L such that if a subalgebra U of L has a maximal subalgebra of dimension one then every maximal subalgebra of U has dimension one. Such an L is called lm(0)-algebra. This class of Lie algebras emerges when it is imposed on the lattice of subalgebras of a Lie algebra the condition that every atom is lower modular. We see that the effect of that condition is highly sensitive to the ground field F. If F is algebraically closed, then every Lie algebra is lm(0). By contrast, for every algebraically non-closed field there exist simple Lie algebras which are not lm(0). For the real field, the semisimple lm(0)-algebras are just the Lie algebras whose Killing form is negative-definite. Also, we study when the simple Lie algebras having a maximal subalgebra of codimension one are lm(0), provided that char(F) ≠ 2. Moreover, lm(0)-algebras lead us to consider certain other classes of Lie algebras and the largest ideal of an arbitrary Lie algebra L on which the action of every element of L is split, which might have some interest by themselves.     

Nonisothermal filtration and seismic acoustics in porous soil: Thermoviscoelastic equations and Lamé equations

We study applications of a new class of infinite-dimensional Lie algebras, called Lax operator algebras, which goes back to the works by I. Krichever and S. Novikov on finite-zone integration related to holomorphic vector bundles and on Lie algebras on Riemann surfaces. Lax operator algebras are almost graded Lie algebras of current type. They were introduced by I. Krichever and the author as a development of the theory of Lax operators on Riemann surfaces due to I. Krichever, and further investigated in a joint paper by M. Schlichenmaier and the author. In this article we construct integrable hierarchies of Lax equations of that type. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 216–226. Dedicated to S.P. Novikov on the occasion of his 70th birthday     

Cohomology of graded lie algebras of maximal class with coefficients in the adjoint representation

We compute explicitly the adjoint cohomology of two ℕ-graded Lie algebras of maximal class (infinite-dimensional filiform Lie algebras) m₀ and m₂. It is known that up to an isomorphism there are only three ℕ-graded Lie algebras of maximal class. The third algebra from this list is the “positive” part L ₁ of the Witt (or Virasoro) algebra, and its adjoint cohomology was computed earlier by Feigin and Fuchs. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 106–119.     

On Whittaker modules over a class of algebras similar to U(sl 2)

Motivated by the study of invariant rings of finite groups on the first Weyl algebras A ₁ and finding interesting families of new noetherian rings, a class of algebras similar to U(sl ₂) was introduced and studied by Smith. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained. But it seems that not much has been done on the part of nonweight modules. In this paper, we generalize Kostant’s results on the Whittaker model for the universal enveloping algebras U(g) of finite dimensional semisimple Lie algebras g to Smith’s algebras. As a result, a complete classification of irreducible Whittaker modules (which are definitely infinite dimensional) for Smith’s algebras is obtained, and the submodule structure of any Whittaker module is also explicitly described.      

On one-dimensional stochastic differential equations driven by stable processes

We consider the one-dimensional stochastic differential equation dX _t=b(t, X_t−) dZ _t, whereZ is a symmetric α-stable Lévy process with α ε (1, 2] andb is a Borel function. We give sufficient conditions under which the equation has a weak nonexploding solution. Partially supported by Programma Professori Visitatori of G. N. A. F. A. (Italy). Partially supported by MURST (Italy). The present research was completed while the second author was visiting the Institute of Mathematics and Informatics (Vilnius, Lithuania) in spring of 1999. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 361–385, July–September, 2000. Translated by H. Pragarauskas     

Poisson color algebras of arbitrary degree

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g₀, where g₀ is some element of the grading group G such that g₀ = 0 or 4g₀≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.     

Simple and Nearly Simple Deep Matrix Algebras

Deep matrix algebras were originally created by Cuntz (Comm. Math. Phys. 57:173–185, 1977) and McCrimmon (2006). Further study of the associative case was done by the author in Kennedy (2004) and Kennedy (Algebr. Represent. Theory 9:525–537, 2006). In this paper, the associative algebra DM(X,\mathbbK){\mathcal{DM}}(X,{\mathbb{K}}) based on a set X over a field \mathbbK{\mathbb{K}} and various of its subalgebras are studied for the purpose of determining the structure of the associated Lie algebra \mathfrakgld(X,\mathbbK){\mathfrak{gld}}(X,{\mathbb{K}}) and its subalgebras. Several key examples of deep matrix Lie algebras are constructed. These are shown to be either simple or nearly simple depending on the cardinality of the set X. Cartan subalgebras are constructed and two of the key Lie algebras are then decomposed with respect to the adjoint action of these subalgebras. In the process, an infinite dimensional analogue to \mathfraksl₂(\mathbbK)\mathfrak{sl}_2({\mathbb{K}}) is naturally realized as a key subalgebra in deep matrix Lie algebras.     

Representations of four-derivation Lie algebras of Block type

Block introduced certain analogues of the Zassenhaus algebras over a field of characteristic 0. The nongraded infinite-dimensional simple Lie algebras of Block type constructed by Xu can be viewed as generalizations of the Block algebras. In this paper, we construct a family of irreducible modules in terms of multiplication and differentiation operators on ＂polynomials＂ for four-devivation nongraded Lie algebras of Block type based on the finite-dimensional irreducible weight modules with multiplicity one of general linear Lie algebras. We also find a new series of submodules from which some irreducible quotient modules are obtained.     

A note on the tensor product of Lie soluble algebras

D. M. Riley proved in [3] that, if A and B are either Lie nilpotent or Lie metabelian algebras, then their tensor product A ⊗ B is Lie soluble and obtained bounds on the Lie derived length of A ⊗ B. The aim of the present note is to improve Riley’s bounds; moreover we consider also the cases in which A and B are either strongly Lie soluble or strongly Lie nilpotent algebras. Received: 5 April 2006 The first two authors partially supported by MIUR-Italy via PRIN “Group theory and applications”.     

Five-dimensional K-contact Lie algebras

We introduce a general approach to the study of left-invariant K-contact structures on Lie groups and we obtain a full classification in dimension five. We show that Sasakian structures on five-dimensional Lie algebras with non-trivial center are a relatively rare phenomenon with respect to K-contact structures. We also prove that a five-dimensional solvmanifold with a left-invariant K-contact (not Sasakian) structure is a \mathbb S¹{\mathbb S^1} -bundle over a symplectic solvmanifold. Rigidity results are then obtained for five-dimensional K-contact Lie algebras with trivial center and for K-contact η-Einstein structures. Moreover, five-dimensional Sasakian φ-symmetric Lie algebras are completely classified, and some explicit examples of five-dimensional Sasakian pseudo-metric Lie algebras are provided.     

Generalized Poisson brackets and lie algebras of type H in characteristic 0

The Lie algebra of Cartan type H which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials . We show in this paper that these generalizations of Cartan type H algebras are isomorphic to certain generalizations of the classical algebra of Poisson brackets, and that it can be generalized further. In turn, these algebras can be recast in a form that is an adaption of a class of Lie algebras of characteristic p that was defined in 1958 be R. Block. A further generalization of these algebras is the main topic of this paper. We show when these algebras are simple, find their derivations, and determine all possible isomorphisms between two of these algebras. Received December 20, 1996; in final form September 15, 1997     

Finite vs affine W-algebras

In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the λ-bracket. In Section 2 we construct, in the most general framework, the Zhu algebra Zhu_ΓV, an associative algebra which “controls” Γ-twisted representations of the vertex algebra V with a given Hamiltonian operator H. An important special case of this construction is the H-twisted Zhu algebra Zhu_H V. In Section 3 we review the theory of non-linear Lie conformal algebras (respectively non-linear Lie algebras). Their universal enveloping vertex algebras (resp. universal enveloping algebras) form an important class of freely generated vertex algebras (resp. PBW generated associative algebras). We also introduce the H-twisted Zhu non-linear Lie algebra Zhu_H R of a non-linear Lie conformal algebra R and we show that its universal enveloping algebra is isomorphic to the H-twisted Zhu algebra of the universal enveloping vertex algebra of R. After a discussion of the necessary cohomological material in Section 4, we review in Section 5 the construction and basic properties of affine and finite W-algebras, obtained by the method of quantum Hamiltonian reduction. Those are some of the most intensively studied examples of freely generated vertex algebras and PBW generated associative algebras. Applying the machinery developed in Sections 3 and 4, we then show that the H-twisted Zhu algebra of an affine W-algebra is isomorphic to the finite W-algebra, attached to the same data. In Section 6 we define the Zhu algebra of a Poisson vertex algebra, and we discuss quasiclassical limits. In the Appendix, the equivalence of three definitions of a finite W-algebra is established. “I am an old man, and I know that a definition cannot be so complicated.” I.M. Gelfand (after a talk on vertex algebras in his Rutgers seminar)     

Intermediate Growth of Solvable Lie Superalgebras

Finitely generated solvable Lie algebras have an intermediate growth between polynomial and exponential. Recently the second author suggested the scale to measure such an intermediate growth of Lie algebras. The growth was specified for solvable Lie algebras F(A ^q , k) with a finite number of generators k, and which are free with respect to a fixed solubility length q. Later, an application of generating functions allowed us to obtain more precise asymptotic. These results were obtained in the generality of polynilpotent Lie algebras. Now we consider the case of Lie superalgebras; we announce that main results and describe the methods. Our goal is to compute the growth for F(A ^q , m, k), the free solvable Lie superalgebra of length q with m even and k odd generators. The proof is based upon a precise formula of the generating function for this algebra obtained earlier. The result is obtained in the generality of free polynilpotent Lie superalgebras. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 14, Algebra, 2004.     

Simple algebras of Weyl type

Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceA[D] =A⊗F[D] from any pair of a commutative associative algebra,A with an identity element and the polynomial algebraF[D] of a commutative derivation subalgebraD ofA We prove thatA[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only ifA isD-simple andA[D] acts faithfully onA. Thus we obtain a lot of simple algebras. Su, Y., Zhao, K., Second cohornology group of generalized Witt type Lie algebras and certain representations, submitted to publication