Diamonds of finite type in thin Lie algebras

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 one occurs in degree 2q-1, where q is a power of the characteristic. This paper is a contribution to a classification project of this subclass of thin Lie algebras. Specifically, we prove that, under certain technical assumptions, the degree of the earliest diamond of finite type in such a Lie algebra can only have a certain form, which does occur in explicit examples constructed elsewhere.Comment: 19 page

Gradings of non-graded Hamiltonian Lie algebras

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\colon\n;\omega_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\colon\n;\omega_2) with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).Comment: 36 pages, to be published in J. Austral. Math. Soc. Ser.

Nottingham Lie algebras with diamonds of finite and infinite type

We consider a class of infinite-dimensional, modular, graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. We identify two subclasses of Nottingham Lie algebras as loop algebras of finite-dimensional simple Lie algebras of Hamiltonian Cartan type. A property of Laguerre polynomials of derivations, which is related to toral switching, plays a crucial role in our constructions.Comment: 17 pages; minor changes from the previous versio

Free Lie algebroids and the space of paths

We construct algebraic and algebro-geometric models for the spaces of unparametrized paths. This is done by considering a path as a holonomy functional on indeterminate connections. For a manifold X, we construct a Lie algebroid P which serves as the tangent space to X (punctual paths) inside the space of all unparametrized paths. It serves as a natural receptacle of all "covariant derivatives of the curvature" for all bundles with connections on X. If X is an algebraic variety, we integrate P to a formal groupoid G which can be seen as the formal neighborhood of X inside the space of paths. We establish a relation of G with the stable map spaces of Kontsevich.Comment: 42 pages, revised version, to appear in Selecta Mat

Parabolic subalgebras, parabolic buildings and parabolic projection

Reductive (or semisimple) algebraic groups, Lie groups and Lie algebras have a rich geometry determined by their parabolic subgroups and subalgebras, which carry the structure of a building in the sense of J. Tits. We present herein an elementary approach to the geometry of parabolic subalgebras, over an arbitrary field of characteristic zero, which does not rely upon the structure theory of semisimple Lie algebras. Indeed we derive such structure theory, from root systems to the Bruhat decomposition, from the properties of parabolic subalgebras. As well as constructing the Tits building of a reductive Lie algebra, we establish a "parabolic projection" process which sends parabolic subalgebras of a reductive Lie algebra to parabolic subalgebras of a Levi subquotient. We indicate how these ideas may be used to study geometric configurations and their moduli.Comment: 26 pages, v2 minor clarification