Marc A. Rieffel's Deformation Quantization for Actions of rd PDF

By Marc A. Rieffel

ISBN-10: 0821825755

ISBN-13: 9780821825754

This paintings describes a basic development of a deformation quantization for any Poisson bracket on a manifold which comes from an motion of $R^d$ on that manifold. those deformation quantizations are strict, within the feel that the deformed fabricated from any features is back a functionality and that there are corresponding involutions and operator norms. the various recommendations concerned are tailored from the idea of pseudo-differential operators. the development is proven to have many favorable houses. a couple of particular examples are defined, starting from easy ones similar to quantum disks, quantum tori, and quantum spheres, to points of quantum teams.

Show description

Read Online or Download Deformation Quantization for Actions of rd PDF

Similar linear books

Jonathan S. Golan's The Linear Algebra a Beginning Graduate Student Ought to PDF

Linear algebra is a dwelling, lively department of arithmetic that is imperative to nearly all different parts of arithmetic, either natural and utilized, in addition to to computing device technological know-how, to the actual, organic, and social sciences, and to engineering. It encompasses an in depth corpus of theoretical effects in addition to a wide and rapidly-growing physique of computational thoughts.

Recent Developments in Quantum Affine Algebras and Related by Naihuan Jing, Kailash C. Misra PDF

This quantity displays the lawsuits of the overseas convention on Representations of Affine and Quantum Affine Algebras and Their functions held at North Carolina country college (Raleigh). lately, the speculation of affine and quantum affine Lie algebras has turn into an enormous sector of mathematical study with a number of functions in different parts of arithmetic and physics.

Additional info for Deformation Quantization for Actions of rd

Sample text

8 LEMMA. Let {G\} be a net of elements ofBA which converges to G e BA uniformly on compact subsets of V and dominated over all of V by a constant for each semi-norm uniformly in A. Then G\ Xjh converges to G xj h in the same way, for each h G SA; and ( / , G \ Xj K)A converges to (/, G Xj K)A for each f G SA. 4. The second assertion then follows immediately. 9 PROPOSITION. For G e BA and f,h e SA we have (fxjG,h)A = (f,Gxjh)A. 8. 7 (/ xjG,h)A as needed. = lim(/ xj(Gipm),h)A = lim(/, {Gipm) Xjh)A = (f,Gxjh)A, • To conclude this chapter, we derive a useful formula for the deformed product of three functions.

13 it will then be an approximate identity for Cj, and so has the desired property with respect to Aj. But, of course, it is contained in Bj. • We continue by extending a to Aj. 5 that a gives an action of V as automorphisms of A°° for the product Xj, and that this action is strongly continuous for the Frechet topology. 7 it gives an automorphism of Aj. 10 it follows that this action is strongly continuous for the operator norm on Aj. Another way to see all this is to use the embedding of A°° into BA by a i—• a, equivariant for a and r, and the fact that r on BA corresponds to conjugation on SA by translation, which is "unitary" and "strongly continuous".

LeJJ)A + (fJ)A, D For later use there is one more detail which we need to clarify here. Let r denote the action of V on CU(V,A). 8 to this algebra and action to get a C*-norm on the space of smooth vectors, that is, on BA. We will need later the fact that this norm is the same as that on BA defined directly by its action on the A- rigged space SA by Xj. Unwinding the definitions, we see that the former norm is defined as follows. Define F by F(t) = rt(F), so that F can be viewed as a function on V x V.

Download PDF sample

Deformation Quantization for Actions of rd by Marc A. Rieffel


by William
4.0

Rated 4.81 of 5 – based on 44 votes