Twisted cocycles of lie algebras and corresponding invariant functions

We consider finite-dimensional complex Lie algebras. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie algebras. A special case is generalization of 1-cocycles with respect to the adjoint representation - so called (α,β,γ)-derivations. Parametric sets of spaces of cocycles allow us to define complex functions which are invariant under Lie isomorphisms. Such complex functions thus represent useful invariants - we show how they classify three and four-dimensional Lie algebras as well as how they apply to some eight-dimensional one-parametric nilpotent continua of Lie algebras. These functions also provide necessary criteria for existence of 1-parametric continuous contraction.     

Integration of Geodesic Flows on Homogeneous Spaces: The Case of a Wild Lie Group

We obtain necessary and sufficient conditions for the integrability in quadratures of geodesic flows on homogeneous spaces M with invariant and central metrics. The proposed integration algorithm consists in using a special canonical transformation in the space T ^* M based on constructing the canonical coordinates on the orbits of the coadjoint representation and on the simplectic sheets of the Poisson algebra of invariant functions. This algorithm is applicable to integrating geodesic flows on homogeneous spaces of a wild Lie group.     

Special Quasigraded Lie Algebras and Integrable Hamiltonian Systems

We construct a family of special quasigraded Lie algebras of functions of one complex variables with values in finite-dimensional Lie algebra , labeled by the special 2-cocycles F on . The main property of the constructed Lie algebras is that they admit Kostant-Adler-Symes scheme. Using them we obtain new integrable finite-dimensional Hamiltonian systems and new hierarchies of soliton equations.     

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.     

c-Graded filiform Lie algebras

In this paper we generalize naturally graded filiform Lie algebras as well as filiform Lie algebras admitting a connected gradation of maximal length, by introducing the concept of c-graded complex filiform Lie algebras. We deal with the particular case of 3-graded filiform Lie algebras and we obtain their classification in arbitrary dimension. We finally show a link among derived algebras, graded filiform and rigid solvable Lie algebras.     

Complete involutive algebras of functions on cotangent bundles of homogeneous spaces

Homogeneous spaces of all compact Lie groups admit Riemannian metrics with completely integrable geodesic flows by means of C –smooth integrals [9, 10]. The purpose of this paper is to give some constructions of complete involutive algebras of analytic functions, polynomial in velocities, on the (co)tangent bundles of homogeneous spaces of compact Lie groups. This allows us to obtain new integrable Riemannian and sub-Riemannian geodesic flows on various homogeneous spaces, such as Stiefel manifolds, flag manifolds and orbits of the adjoint actions of compact Lie groups. Mathematics Subject Classification (2000): 70H06, 37J35, 53D17, 53D25     

Leibniz Central Extensions on Some Infinite-Dimensional Lie Algebras

Abstract

In this paper, all one-dimensional Leibniz central extensions on the algebras of differential operators over C[t, t ^?1] and C((t)), as well as on the quantum 2-torus, the Virasoro-like algebra and its q-analog are studied. We determine all nontrivial Leibniz 2-cocycles on these infinite dimensional Lie algebras.     

Exponentiation of infinite dimensional Z-graded lie algebras

In this article we give a new technique for exponentiating infinite dimensional graded representations of graded Lie algebras that allows for the exponentiation of some non-locally nilpotent elements. Our technique is to naturally extend the representation of the Lie algebra g on the space V naturally to a representation on a subspace £ of the dual space V ^*. After introducing the technique, we prove that it enables the exponentiation of all elements of free Lie Algebras and afhne Kac-Moody Lie algebras.     

Direct Unions of Lie Tori (Realization of Locally Extended Affine Lie Algebras)

In this work, we consider realizations of locally extended affine Lie algebras, in the level of core modulo center. We provide a framework similar to the one for extended affine Lie algebras by “direct unions.” Our approach suggests that the direct union of existing realizations of extended affine Lie algebras, in a rigorous mathematical sense, would lead to a complete realization of locally extended affine Lie algebras, in the level of core modulo center. As an application of our results, we realize centerless cores of locally extended affine Lie algebras with specific root systems of types A₁, B, C, and BC.     

Cohomology of Oriented Tree Diagram Lie Algebras

Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural analogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this article, we use Hodge Laplacian to study the cohomology of these Lie algebras. The “total rank conjecture” and “b ₂-conjecture” for the algebras are proved. Moreover, we find the generating functions of the Betti numbers by means of Young tableaux for the Lie algebras associated with certain tree diagrams of single branch point. By these functions and Euler–Poincaré principle, we obtain analogues of the denominator identity for finite-dimensional simple Lie algebras. The result is a natural generalization of the Bott's classical result in the case of special linear Lie algebras.     

On Exact Courant Algebras

In this note, we will show that exact Courant algebras over a Lie algebra 𝔤 can be characterized via Leibniz 2-cocycles, and the automorphism group of a given exact Courant algebra is in a one-to-one correspondence with first Leibniz cohomology space of 𝔤.     

Group algebras of Lie nilpotency index 12 and 13

Let G be a group and let K be a field of characteristic p>0. Lie nilpotent group algebras of strong Lie nilpotency index up to 11 have already been classified. In this paper, our aim is to classify the group algebras KG which are strongly Lie nilpotent of index 12 or 13.     

SECOND COHOMOLOGY GROUP OF GENERALIZED WITT TYPE LIE ALGEBRAS AND CERTAIN REPRESENTATIONS

ABSTRACT

We determine the second cohomology groups of Lie algebras of generalized Witt type which are some Lie algebras defined by Passman and Jordan, more general than those defined by Dokovic and Zhao, and slightly more general than those defined by Xu. Among all the 2-cocycles, there is a special one we think interesting. Using this 2-cocycle, we define the so-called Virasoro-like algebras. Then we give a class of their representations.     

On Lie nilpotent modular group algebras

Let K be a field of characteristic p>0 and let KG be the group algebra of an arbitrary group G over K. It is known that if KG is Lie nilpotent, then its lower as well as upper Lie nilpotency index is at least p+1. The group algebras KG for which these indices are p+1 or 2p or 3p?1 or 4p?2 have already been determined. In this paper, we classify the group algebras KG for which the upper Lie nilpotency index is 5p?3, 6p?4 or 7p?5.     

Depth Generated Simple Lie Algebras

A Lie module algebra for a Lie algebra L is an algebra and L-module A such that L acts on A by derivations. The depth Lie algebra of a Lie algebra L with Lie module algebra A acts on a corresponding depth Lie module algebra . This determines a depth functor from the category of Lie module algebra pairs to itself. Remarkably, this functor preserves central simplicity. It follows that the Lie algebras corresponding to faithful central simple Lie module algebra pairs (A,L) with A commutative are simple. Upon iteration at such (A,L), the Lie algebras are simple for all i ∈ ω. In particular, the (i ∈ ω) corresponding to central simple Jordan Lie algops (A,L) are simple Lie algebras. Presented by Don Passman.     

On c-nilpotent multiplier and c-covers of a pair of Lie algebras

In this paper, we characterize the structure of the c-nilpotent multiplier of a pair of Lie algebras concerning the c-covering pairs and discuss some results on the existence of c-covers of a pair of Lie algebras. Moreover, we show that every nilpotent pair of Lie algebras of class at most k with non-trivial c-nilpotent multiplier does not admit any c-covering pair, if c>k. Also, we obtain the structure of c-covers of a c-capable pair of Lie algebras.     

Contact and Frobenius solvable Lie algebras with abelian nilradical

The aim of this work is to characterize the families of Frobenius (respectively, contact) solvable Lie algebras that satisfies the following condition: 𝔤 = 𝔥?V, where 𝔥?𝔤𝔩(V), |dim V?dim 𝔤|≤1 and NilRad(𝔤) = V, V being a finite dimensional vector space. In particular, it is proved that every complex Frobenius solvable Lie algebra is decomposable, whereas that in the real case there are only two indecomposable Frobenius solvable Lie algebras.     

Solvable Indecomposable Extensions of Two Nilpotent Lie Algebras

A pair of sequences of nilpotent Lie algebras denoted by N_{n, 7} and N_{n, 16} are introduced. Here, n denotes the dimension of the algebras that are defined for n ≥ 6; the first terms in the sequences are denoted by 6.7 and 6.16, respectively, in the standard list of six-dimensional Lie algebras. For each of them, all possible solvable extensions are constructed so that N_{n, 7} and N_{n, 16} serve as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program of investigating solvable Lie algebras using special properties rather than trying to extend one dimension at a time.     

The hypoelliptic Laplacian on a compact Lie group

Let G be a compact Lie group, and let g be its Lie algebra. In this paper, we produce a hypoelliptic Laplacian on G×g, which interpolates between the classical Laplacian of G and the geodesic flow. This deformation is obtained by producing a suitable deformation of the Dirac operator of Kostant. We show that various Poisson formulas for the heat kernel can be proved using this interpolation by methods of local index theory. The paper was motivated by papers by Atiyah and Frenkel, in connection with localization formulas in equivariant cohomology and with Kac's character formulas for affine Lie algebras. In a companion paper, we will use similar methods in the context of Selberg's trace formula.     

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.