Conformal algebras, recently introduced by Kac, encode an axiomatic description of the singular part of the operator product expansion in conformal field theory. The objective of this paper is to develop the theory of ``multi-dimensional'' analogues of conformal algebras. They are defined as Lie algebras in a certain ``pseudotensor'' category instead of the category of vector spaces. A pseudotensor category (as introduced by Lambek, and by Beilinson and Drinfeld) is a category equipped with ``polylinear maps'' and a way to compose them. This allows for the definition of Lie algebras, representations, cohomology, etc. An instance of such a category can be constructed starting from any cocommutative (or more generally, quasitriangular) Hopf algebra $H$. The Lie algebras in this category are called Lie $H$-pseudoalgebras. The main result of this paper is the classification of all simple and all semisimple Lie $H$-pseudoalgebras which are finitely generated as $H$-modules. We also start developing the representation theory of Lie pseudoalgebras; in particular, we prove analogues of the Lie, Engel, and Cartan-Jacobson Theorems. We show that the cohomology theory of Lie pseudoalgebras describes extensions and deformations and is closely related to Gelfand-Fuchs cohomology. Lie pseudoalgebras are closely related to solutions of the classical Yang-Baxter equation, to differential Lie algebras (introduced by Ritt), and to Hamiltonian formalism in the theory of nonlinear evolution equations. As an application of our results, we derive a classification of simple and semisimple linear Poisson brackets in any finite number of indeterminates.Comment: 102 pages, 7 figures, AMS late

In this thesis I gave a classification of simple and semi-simple conformal algebras of finite rank, and studied their representation theory, trying to prove or disprove the analogue of the classical Lie algebra representation theory results. I re-expressed the operator product expansion (OPE) of two formal distributions by means of a generating series which I call "A-bracket" and studied the properties of the resulting algebraic structure. The above classification describes finite systems of pairwise local fields closed under the OPE.Thesis Supervisor: Victor G. Kac Title: Professor of Mathematics AcknowledgmentsThe few people I would like to thank are those who delayed my thesis the most. Even though I might have been able to finish my thesis earlier, my time at MIT would also have been much less pleasant without these people, with whom I shared joy and sadness, interest and idleness, relaxation and stress.And although putting up with me as a roommate, playing a terrible bridge hand without blaming the awful contract on me, lending me books and CDs without expecting me to give them back, sending me birthday cards, hiding a huge aversion towards the most intolerable aspects of my character with a smile, sharing a trip in a van on a road to nowhere, sipping tea with me while listening to Schubert, helping me explore Boston at night, or giving me a call in the moment I need -it the most, may all appear as the most ordinary abilities (and they are not), I know that I could never have survived the past four years without them.

not the corresponding Lie-Cartan algebra, as in other cases, but an irreducible central extension. When the action of the center of the annihilation algebra is trivial, this complex is related to work by M. Eastwood [6] on conformally symplectic geometry, and we call it conformally symplectic pseudo de Rham complex.

Macroscopic aspects are variable, and final diagnosis is made from the typical histological and immunohistochemical pattern. The therapy of choice is gross total surgical resection since subtotal removal can lead to local regrowth, but the time to recurrence is unpredictable and highly variable. The role of adjuvant therapy is currently undefined.

ABSTRACT. We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of approximating the identity component as well as the maximal normal (in the sense of Wang) connected subgroup by introducing canonical, but possibly transfinite, sequences of subgroups. These sequences have a trivial behaviour in the classical case. We give examples, arising as free products, where the identity component is not normal and the associated sequence has length 1.We give necessary and sufficient conditions for normality of the identity component and finiteness or profiniteness of the quantum component group. Among them, we introduce an ascending chain condition on the representation ring, called Lie property, which characterizes Lie groups in the commutative case and reduces to group Noetherianity of the dual in the cocommutative case. It is weaker than ring Noetherianity but ensures existence of a generating representation. The Lie property and ring Noetherianity are inherited by quotient quantum groups. We show that A u (F ) is not of Lie type. We discuss an example arising from the compact real form of U q (sl 2 ) for q < 0.