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     

Zur konstruktion von Lie-Algebren aus Jordan-Tripelsystemen

In this paper we study certain Lie algebras which are constructed from the (-1)-eigenspaees of an involution of a Jordan algebra. The construction is a generalisation of the Koecher-Tits-construction. We give necessary conditions in terms of the Jordan algebras for the Lie algebras being simple. If the (-1)-spaces are Peirce-1/2-components then we obtain a close relation between the Lie algebras under consideration and the structure algebras of Jordan algebras. We finally give a list of those types of simple Lie algebras which can be formed by this construction; among them are Lie algebras of type E₆ and E₇.Of fundamental importance for our considerations is a close connection between the constructed Lie algebras and the standard imbeddings of Lie triple systems.     

ON LIE ALGEBRAS WHOSE NILRADICAL IS (n — p)-FILIFORM

We prove first that every (n — p)-filiform Lie algebra, p ≤ 3, is the nilradical of a solvable, nonnilpotent rigid Lie algebra. We also analize how this result extends to (n — 4)-filiform Lie algebras. For this purpose, we give a classificaction of these algebras and then determine which of the obtained classes appear as the nilradical of a rigid algebra.     

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)     

Restricted Lie 2-algebras

In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-algebras from A(m)-algebras, restricted Leibniz algebras, restricted right-symmetric algebras. Finally, we prove that there is a one-to-one correspondence between strict restricted Lie 2-algebras and crossed modules of restricted Lie algebras.     

A unified view of some vertex operator constructions

We present a general vertex operator construction based on the Fock space for affine Lie algebras of typeA. This construction allows us to give a unified treatment for both the homogeneous and principle realizations of the affine Lie algebras as well as for some extended affine Lie algebras coordinatized by certain quantum tori.     

Narrow Positively Graded Lie Algebras

The present paper is devoted to the classification of infinite-dimensional naturally graded Lie algebras that are narrow in the sense of Zelmanov and Shalev [9]. Such Lie algebras are Lie algebras of slow linear growth. In the theory of nonlinear hyperbolic partial differential equations the notion of the characteristic Lie algebra of equation is introduced [3]. Two graded Lie algebras n₁ and n₂ from our list, that are positive parts of the affine Kac–Moody algebras A₁⁽¹⁾ and A₂⁽²⁾, respectively, are isomophic to the characteristic Lie algebras of the sinh-Gordon and Tzitzeika equations [6]. We also note that questions relating to narrow and slowly growing Lie algebras have been extensively studied in the case of a field of positive characteristic [2].

     

Group Algebras of Finite Groups as Lie Algebras

We consider the natural Lie algebra structure on the (associative) group algebra of a finite group G, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a decomposition in simple factors of these Lie algebras, in terms of the ordinary representations of G.     

Schunck Classes of Soluble Leibniz Algebras

I set out the theory of Schunck classes and projectors for soluble Leibniz algebras, parallel to that for Lie algebras. Primitive Leibniz algebras come in pairs, one (Lie) symmetric, the other antisymmetric. A Schunck formation containing one member of a pair also contains the other. If ? is a Schunck formation and H is an ?-projector of the Leibniz algebra L, then H is intravariant in L. An example is given to show that the assumption that the Schunck class ? is a formation cannot be omitted.     

Some notes on the second homology of Leibniz algebras

Some properties of the second homology and cover of Leibniz algebras are established. By constructing a stem cover, the second Leibniz homology and cover of abelian, Heisenberg Lie algebras and cyclic Leibniz algebras are described. Also, for the dimension of a non-cyclic nilpotent Leibniz algebra L, we obtain dim(HL₂(L))≥2.     

Lie algebras with few centralizers

We will characterize all finite dimensional Lie algebras with at most |F|²+|F|+2 centralizers, where F is the underlying field of Lie algebras under consideration.     

Co-Frobenius Hopf algebras and the coradical filtration

We prove that a Hopf algebra with a finite coradical filtration is co-Frobenius. We also characterize co-Frobenius Hopf algebras with coradical a Hopf subalgebra. Let H be a Hopf algebra whose coradical is a Hopf algebra. Let gr H be the associated graded coalgebra and let R be the diagram of H, c. f. [2]. Then the following are equivalent: (1) H is co-Frobenius; (2) gr H is co-Frobenius; (3) R is finite dimensional; (4) the coradical filtration of H is finite. This Theorem allows us to give systematically examples of co-Frobenius Hopf algebras, and opens the way to the classification of ample classes of such Hopf algebras. Received: 8 March 2001 / Published online: 8 November 2002 RID="*" ID="*" Parts of this work were done during visits of the second author to the University of Córdoba, Argentina, in May 2000; and of the first author to the University of Bucharest in November 2000. We thank the FOMEC and CNCSIS (Grant C12) for support to these visits. The first author also thanks ANPCyT, CONICET, Agencia Córdoba Ciencia and Secyt (UNC) for partial support. The second author was partiall y supported by Grant SM10/01 of the Research Administration of Kuwait University.     

Triangulated categories and Kac-Moody algebras

By using the Ringel-Hall algebra approach, we find a Lie algebra arising in each triangulated category with T ²=1, where T is the translation functor. In particular, the generic form of the Lie algebras determined by the root categories, the 2-period orbit categories of the derived categories of finite dimensional hereditary associative algebras, gives a realization of all symmetrizable Kac-Moody Lie algebras. Oblatum 4-XII-1998 & 11-XI-1999?Published online: 21 February 2000     

Metric Lie algebras and quadratic extensions

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a nondegenerate invariant symmetric bilinear form. We show that any metric Lie algebra g without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary Lie algebra l by an orthogonal l-module a in a canonical way. Identifying equivalence classes of quadratic extensions of l by a with a certain cohomology set H²_Q(l,a), we obtain a classification scheme for general metric Lie algebras and a complete classification of metric Lie algebras of index 3.     

Heisenberg Lie Color Algebras

In this article, we give a sufficient condition for a Lie color algebra to be complete. The color derivation algebra Der(?) and the holomorph L of finite dimensional Heisenberg Lie color algebra ? graded by a torsion-free abelian group over an algebraically closed field of characteristic zero are determined. We prove that Der(?) and Der(L) are simple complete Lie color algebras, but L is not a complete Lie color algebra.     

Sub-semi-Riemannian geometry of general H-type groups

We study a special class of nilpotent Lie groups of step 2, that generalizes the class of the so-called H(eisenberg)-type groups, defined by A. Kaplan in 1980. We change the presence of an inner product to an arbitrary scalar product and relate the construction to the composition of quadratic forms. We present the geodesic equation for sub-semi-Riemannian metric on nilpotent Lie groups of step 2 and solve them for the case of general H-type groups. We also present some results on sectional curvature and the Ricci tensor of semi-Riemannian general H-type groups.     

Lie ideals of graded associative algebras

We study the Lie structure of graded associative algebras. Essentially, we analyze the relation between Lie and associative graded ideals, and between Lie and associative graded derivations. Gathering together results on both directions, we compute maximal graded algebras of quotients of graded Lie algebras that arise from associative algebras. We also show that the Lie algebra Der _gr (A) of graded derivations of a graded semiprime associative algebra is strongly non-degenerate (modulo a certain ideal containing the center of Der _gr (A)).