A preparation theorem for the prime spectrum of a semisimple Lie algebra

Publisher Summary The orbit method pioneered by Kirillov and Kostant seeks to construct irreducible unitary representations by analogy with quantization procedures in mechanics. A classical physical system may sometimes be modeled by a configuration space M, a manifold the points of which represent the possible positions of the bodies in the system. The corresponding phase space is the cotangent bundle T'M, the points of which represent the possible states (positions and momenta) of the bodies in the system. In this setting, a classical observable is a function on T*M. The quantum mechanical analog of this system is based on the Hilbert space L2(M) of square-integrable half-densities on M. A quantum mechanical observable is an operator T on L2(M); the scalar product (Tv, v) is the expected value of the observable on the state v. Some connection between classical and quantum observables is provided by a symbol calculus; an interesting quantum observable is often a differential operator on M, and the corresponding classical observable is related to the symbol of the differential operator.

The Orbit Method and Unitary Representations for Reductive Lie Groups

We develop the theory of rational ideals for arbitrary associative algebras R without assuming the standard finiteness conditions, noetherianness or the Goldie property. The Amitsur-Martindale ring of quotients replaces the classical ring of quotients which underlies the previous definition of rational ideals but is not available in a general setting. Our main result concerns rational actions of an affine algebraic group G on R. Working over an algebraically closed base field, we prove an existence and uniqueness result for generic rational ideals in the sense of Dixmier: for every G-rational ideal I of R, the closed subset of the rational spectrum RatR that is defined by I is the closure of a unique G-orbit in RatR. Under additional Goldie hypotheses, this was established earlier by Mœglin and Rentschler (in characteristic 0) and by Vonessen (in arbitrary characteristic), answering a question of Dixmier.

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Group actions and rational ideals

This is a brief survey of some recent developments in the study of infinite dimen- sional Hopf algebras which are either noetherian or have finite Gelfand-Kirillov dimension. A number of open questions are listed.

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Hopf algebras under finiteness conditions

Noetherian algebras and the Nullstellensatz

Idéaux primitifs dans les algébres enveloppantes

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Idéaux primitifs dans les algébres enveloppantes