Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy

Oblatum 21-I-1991 & 16-IX-1991     

Central extensions of Lie algebras graded by finite root systems

Lie algebras graded by finite irreducible reduced root systems have been classified up to central extensions by Berman and Moody, Benkart and Zelmanov, and Neher. In this paper we determine the central extensions of these Lie algebras and hence describe them completely up to isomorphism. Received: 22 May 1997 / in final form: 13 January 1999     

Leibniz algebras graded by finite root systems of type C l

We study Leibniz algebras graded by finite root systems of type C _l .     

Locally finite Lie algebras with root decomposition

Let K be an algebraically closed field of characteristic zero, $\frak {g}$ be a countably dimensional locally finite Lie algebra over K, and $\frak {h} \subset \frak {g}$ be a (a priori non-abelian) locally nilpotent subalgebra of $\frak {g}$ which coincides with its zero Fitting component. We classify all such pairs $(\frak {g}, \frak {h})$ under the assumptions that the locally solvable radical of $\frak {g}$ equals zero and that $\frak {g}$ admits a root decomposition with respect to $\frak {h}$. More precisely, we prove that $\frak {g}$ is the union of reductive subalgebras $\frak {g}_n$ such that the intersections $\frak {g}_n \cap \frak {h}$ are nested Cartan subalgebras of $\frak {g}_n$ with compatible root decompositions. This implies that $\frak {g}$ is root-reductive and that $\frak {h}$ is abelian. Root-reductive locally finite Lie algebras are classified in [6]. The result of the present note is a more general version of the main classification theorem in [9] and is at the same time a new criterion for a locally finite Lie algebra to be root-reductive. Finally we give an explicit example of an abelian selfnormalizing subalgebra $\frak {h}$ of $\frak {g} = \frak {sl}(\infty)$ with respect to which $\frak {g}$ does not admit a root decomposition.Work Supported in Part by the University of Hamburg and the Max Planck Institute for Mathematics, Bonn     

Classification of simple graded Lie algebras of finite growth

Oblatum 27-VII-1990 & 5-VIII-1991     

The centroid of extended affine and root graded Lie algebras

We develop general results on centroids of Lie algebras and apply them to determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody Lie algebras, and Lie algebras graded by finite root systems.     

Integrable Hamiltonian systems connected with graded Lie algebras

In this paper there is given a geometric scheme for constructing integrable Hamiltonian systems based on Lie groups, generalizing the construction of M. Adler. The operation of this scheme is considered for parabolic decompositions of semisimple Lie groups. Fundamental examples of integrable systems are connected with graded Lie algebras. Among them are the generalized periodic chains of Toda, multidimensional tops, and the motion of a point on various homogeneous spaces in a quadratic potential.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 95, pp. 3–54, 1980.     

On split Lie algebras with symmetric root systems

We develop techniques of connections of roots for split Lie algebras with symmetric root systems. We show that any of such algebras L is of the form L = + Σ _j I _j with a subspace of the abelian Lie algebra H and any I _j a well described ideal of L, satisfying [I _j , I _k ] = 0 if j ≠ k. Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its minimal ideals, each one being a simple split Lie algebra with a symmetric root system and having all its nonzero roots connected.     

Nonlinear equations and graded Lie algebras

The survey is devoted to a new method of constructing a broad class of exactly and completely integrable one- and two-dimensional nonlinear dynamical systems and finding explicit solutions of them.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 22, pp. 101–136, 1984.     

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)).     

A radical for graded Lie algebras

For an arbitrary group?G and a G-graded Lie algebra L over a field of characteristic zero we show that the Kostrikin radical of?L is graded and coincides with the graded Kostrikin radical of?L. As an important tool for our proof we show that the graded Kostrikin radical is the intersection of all graded-strongly prime ideals of?L. In particular, graded-nondegenerate Lie algebras are subdirect products of graded-strongly prime Lie algebras.     

On varieties parametrizing graded complex Lie algebras

We study spaces parametrizing graded complex Lie algebras from geometric as well as algebraic point of view. If R is a finite-dimensional complex Lie algebra, which is graded by a finite abelian group of order n, then a graded contraction of R, denoted by , is defined by a complex n × n-matrix , i, j = 1, . . . , n. In order for to be a Lie algebra, should satisfy certain homogeneous equations. In turn, these equations determine a projective variety X _R . We compute the first homology group of an irreducible component M of X _R , under some assumptions on M. We look into algebraic properties of graded Lie algebras where .      

Algebras of quotients of graded Lie algebras

In this paper we explore graded algebras of quotients of Lie algebras with special emphasis on the 3-graded case and answer some natural questions concerning its relation to maximal Jordan systems of quotients.     

Lie algebras determined by finite valued Auslander-Reiten quivers

Let г denote a connected valued Auslander-Reiten quiver, let ℒ(γ) denote the free abelian group generated by the vertex setγ ₀ and let ℒ(Γ) be the universal cover ofг with fundamental groupG. It is proved that whenγ is a finite connected valued Auslander-Reiten quiver,ℒ(γ) is a Lie subalgebra ofℋ(г), and is just the “orbit” Lie algebra ℒ( )/G, where ℋ (г)₁ is the degenerate Hall algebra ofг and ℒ( )/G is the “orbit” Lie algebra induced by .