Thin Lie algebras with diamonds of finite and infinite type (original) (raw)

The structure of thin Lie algebras up to the second diamond

2008

Thin Lie algebras are graded Lie algebras L = L ∞ i=1 Li with dim Li ≤ 2 for all i, and satisfying a more stringent but natural narrowness condition modeled on an analogous condition for prop groups. The two-dimensional homogeneous components of L, which include L1, are named diamonds. Infinite-dimensional thin Lie algebras with various diamond patterns have been produced, over fields of positive characteristic, as loop algebras of suitable finite-dimensional simple Lie algebras, of classical or of Cartan type depending on the location of the second diamond. The goal of this paper is a description of the initial structure of a thin Lie algebra, up to the second diamond. Specifically, if L k is the second diamond of L, then the quotient L/L k is a graded Lie algebras of maximal class. In odd characteristic p, the quotient L/L k is known to be metabelian, and hence uniquely determined up to isomorphism by its dimension k, which ranges in an explicitly known set of possible values: 3, 5, a power of p, or one less than twice a power of p. However, the quotient L/L k need not be metabelian in characteristic two. We describe here all the possibilities for L/L k up to isomorphism. In particular, we prove that k + 1 equals a power of two.

Some thin Lie algebras related to Albert-Frank algebras and algebras of maximal class

Journal of the Australian Mathematical Society, 1999

We investigate a class of infinite-dimensional, modular, graded Lie algebra in which the homogeneous components have dimension at most two. A subclass of these algebras can be obtained via a twisted loop algebra construction from certain finite-dimensional, simple Lie algebras of Albert-Frank type.

Diamonds of finite type in thin Lie algebras

2005

Borrowing some terminology from pro-p groups, thin Lie algebras are N-graded Lie algebras of width two and obliquity zero, generated in degree one. In particular, their homogeneous components have degree one or two, and they are termed diamonds in the latter case. In one of the two main subclasses of thin Lie algebras the earliest diamond after that in degree

Gradings of non-graded Hamiltonian Lie algebras

Journal of the Australian Mathematical Society, 2005

A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2 : n; ω 2 ) (of dimension one less than a power of p) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of H(2 : n; ω 2 ) with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).

NOTTINGHAM LIE ALGEBRAS WITH DIAMONDS OF FINITE TYPE

International Journal of Algebra and Computation, 2004

We study a class of positively graded Lie algebras with a pattern of homogeneous components similar to that of the graded Lie algebra associated to the Nottingham group with respect to its lower central series.

On some classes of {\mathbb {Z}}$$-graded Lie algebras

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 2020

We study finite dimensional almost-and quasi-effective prolongations of nilpotent Z-graded Lie algebras, especially focusing on those having a decomposable reductive structural subalgebra. Our assumptions generalize effectiveness and algebraicity and are appropriate to obtain Levi-Malčev and Levi-Chevalley decompositions and precisions on the heigth and other properties of the prolongations in a very natural way. In a last section we consider the semisimple case and discuss some examples in which the structural algebras are central extensions of orthogonal Lie algebras and their degree (− 1) components arise from spin representations.

1 Vertex Ring-Indexed Lie Algebras

2005

Infinite-dimensional Lie algebras are introduced, which are only partially graded, and are specified by indices lying on cyclotomic rings. They may be thought of as generalizations of the Onsager algebra, but unlike it, or its sl(n) generalizations, they are not subalgebras of the loop algebras associated with sl(n). In a particular interesting case associated with sl(3), their indices lie on the Eisenstein integer triangular lattice, and these algebras are expected to underlie vertex operator combinations in CFT, brane physics, and graphite monolayers.

A Characterization of Certain Loop Lie Algebras

Journal of Algebra, 1999

In this paper, we study Z × Z-graded Lie algebras A = i j∈Z A i j with dim A i j ≤ 1 satisfying (I) dim A ±1 0 = dim A 0 ±1 = 1, and A is generated by A ±1 0 A 0 ±1 ; (II) i∈Z A 0 j sl 2 ; (III) A −1 0 A 1 0 = 0, and adA ±1 0 act faithfully on j∈Z A j 1. We show that A is necessarily isomorphic to one of the two smallest loop algebras F t t −1 ⊗ sl 2 andŝl 2 3 with appropriate grading.