Advances in Geometry by Alexander Astashkevich (auth.), Jean-Luc Brylinski, Ranee

By Alexander Astashkevich (auth.), Jean-Luc Brylinski, Ranee Brylinski, Victor Nistor, Boris Tsygan, Ping Xu (eds.)

This ebook is an outgrowth of the actions of the heart for Geometry and Mathematical Physics (CGMP) at Penn country from 1996 to 1998. the guts was once created within the arithmetic division at Penn kingdom within the fall of 1996 for the aim of marketing and helping the actions of researchers and scholars in and round geometry and physics on the collage. The CGMP brings many viewers to Penn nation and has ties with different examine teams; it organizes weekly seminars in addition to annual workshops The booklet comprises 17 contributed articles on present learn themes in various fields: symplectic geometry, quantization, quantum teams, algebraic geometry, algebraic teams and invariant conception, and personality­ istic sessions. lots of the 20 authors have talked at Penn nation approximately their learn. Their articles current new effects or talk about fascinating perspec­ tives on contemporary paintings. the entire articles were refereed within the normal type of fine clinical journals. Symplectic geometry, quantization and quantum teams is one major topic of the booklet. a number of authors learn deformation quantization. As­ tashkevich generalizes Karabegov's deformation quantization of Kahler manifolds to symplectic manifolds admitting transverse polarizations, and reports the instant map in relation to semisimple coadjoint orbits. Bieliavsky constructs an specific star-product on holonomy reducible sym­ metric coadjoint orbits of an easy Lie crew, and he indicates the way to con­ struct a star-representation which has attention-grabbing holomorphic properties.

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We say that ¢ E Rd(T* X) quantizes into D if D E Vd(X) has order d and the principal symbol of D is ¢. Unless X is smooth and affine, there is no guarantee that a given symbol will quantize. In this paper we quantize the symbols r x into order 4 differential operators Dx on 0 in a manner equivariant with respect to both the G-action and the Euler C* -action. We show that this equivariant quantization is unique. In our next paper [A-B3], we use these same operators Dx to quantize 0 by quantizing the map (6).

Let u . v be a symmetric non-degenerate complex bilinear form on CN. We have a complex linear Lie bracket on /\2C N given by [a /\ b, c /\ d] = (a· c)b /\ d + (b. d)a /\ c - (a· d)b /\ c - (b· c)a /\ d. (98) Each vector u /\ v defines a skew-symmetric linear transformation L uAv on C n by LUAV(a) = (u. a)v - (v. a)u. (99) Extending linearly, we obtain a natural complex Lie algebra isomorphism /\2C N -+ so(N, C), z 1--+ L z • We use this to identify 9 with /\2C N . Now the minimal nilpotent orbit 0 is 0= {u /\ v E 9 = /\2C N Iu· u = u· v = V· v = 0, u /\ v'" O}.

C@P. The algebra R(V) identifies with the symmetric algebra S(V*). Properties (ii) and (iii) say that the natural graded algebra homomorphism ( : S(V*) -t R(X), (16) defined by restriction of polynomial functions from V to X, is surjective and the pth graded map (p : SP(V*) - t Rp(X) induces a G-linear isomorphism (p : (V*)~P - t Rp(X). Consequently, Rp(X) = 0 for p < 0 and Ro(X) = c. (17) Also R(X) is generated by Rl(X). In degree 1, ( gives the G-linear isomorphism (18) where fa(w) = a(w). Also property (ii) implies that D(X) = D(CI(X)).

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