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Mathematik für Wirtschaftswissenschaftler 2: Lineare Algebra, Funktionen mehrerer Variablen

Mathematik gehört zu den Grundfächern für jeden Studierenden der Wirtschafts- und Sozialwissenschaften. Er benötigt Kenntnisse der research, der Linearen Algebra sowie der Funktionen einer und mehrerer Variablen. Das zweibändige Taschenbuch, hervorgegangen aus Vorlesungen des Autors an der Universität Regensburg, stellt den Studienstoff sehr anschaulich dar, unterstützt durch eine Vielzahl von Beispielen und Abbildungen.

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1 Quantum mechanics Before explaining the basic setting of quantum mechanics, let us present the basic setting of classical mechanics in a form convenient for quantization. t /. Similarly, the basic setting of Hamiltonian quantum mechanics is as follows. We have a (noncommutative) algebra A of quantum observables, which acts (faithfully) on a space of states H (a complex Hilbert space). e. an operator on H , self-adjoint, and usually unbounded). t/, 2 H , is governed by the Schrödinger equation P D iH ; h where h > 0 is the Planck constant.

3 The Levasseur–Stafford theorem 31 the Harish-Chandra homomorpism (as it was first constructed by Harish-Chandra). Recall that we have the classical Harish-Chandra isomorphism W CŒgg ! CŒhW , defined simply by restricting g-invariant functions on g to the Cartan subalgebra h. g/g on CŒhW , which is clearly given by W -invariant differential operators. However, these operators will, in general, have poles on the reflection hyperplanes. g/g ! hreg /W . 2/ it computes the radial parts of rotationally invariant differential operators on R3 in spherical coordinates.

Both parts of this theorem are quite nontrivial. The first part was proved by Harish-Chandra using analytic methods, and the second part by Levasseur and Stafford using the theory of D-modules. 17. 17. We start the proof with the following proposition, valid for any reductive Lie algebra. 5. Sg/g ! Sh/W . Proof. Without loss of generality, we may assume that g is simple. 6. Let D be the Laplacian g of g corresponding to an invariant form. D/ is the Laplacian h . 32 4 Moment maps, Hamiltonian reduction and the Levasseur–Stafford theorem Proof.