By K. D. Elworthy (auth.), Yoshiaki Maeda, Takushiro Ochiai, Peter Michor, Akira Yoshioka (eds.)

This quantity consists of invited expository articles via famous mathematicians in differential geometry and mathematical physics which were prepared in social gathering of Hideki Omori's fresh retirement from Tokyo college of technological know-how and in honor of his primary contributions to those areas.

The papers specialize in fresh tendencies and destiny instructions in symplectic and Poisson geometry, international research, infinite-dimensional Lie workforce conception, quantizations and noncommutative geometry, in addition to functions of partial differential equations and variational the way to geometry. those articles will entice graduate scholars in arithmetic and quantum mechanics, in addition to researchers, differential geometers, and mathematical physicists.

Contributors comprise: M. Cahen, D. Elworthy, A. Fujioka, M. Goto, J. Grabowski, S. Gutt, J. Inoguchi, M. Karasev, O. Kobayashi, Y. Maeda, ok. Mikami, N. Miyazaki, T. Mizutani, H. Moriyoshi, H. Omori, T. Sasai, D. Sternheimer, A. Weinstein, ok. Yamaguchi, T. Yatsui, and A. Yoshioka.

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**Extra resources for From Geometry to Quantum Mechanics: In Honor of Hideki Omori**

**Example text**

34 N. 5 1) Suppose that : CU → CU is a contact Weyl diffeomorphism which induces the identity map on the base space. Then, there exists uniquely a Weyl function f # (ν 2 ) of the form f # = f 0 + ν 2 f +# (ν 2 ) ( f 0 ∈ R, f + (ν 2 ) ∈ C ∞ (U )[[ν 2 ]]), 1 (15) such that = ead ν { f0 +ν f+ (ν )} . 2) If induces the identity map on WU , then there exists a unique element c(ν 2 ) ∈ 1 2 R[[ν 2 ]] with c(ν 2 ) = c(ν 2 ), such that 5 = ead ν c(ν ) . 5, we have the following. 6 For any modiﬁed contact Weyl diffeomorphism : CU → CU which induces the identity map on the base space, there exists a Weyl function f # (ν 2 ) of the form f # (ν 2 ) = f 0 + ν 2 f +# (ν 2 ) ( f 0 ∈ R, f + (ν 2 ) ∈ C ∞ (U )[[ν 2 ]]), and smooth function g(ν 2 ) ∈ C ∞ (U )[[ν 2 ]] such that 1 = ead( ν {g(ν 2 )+ f # (ν 2 )}) (16) .

42]). However, it was already known in [31] that a Banach–Lie group acting effectively on a ﬁnite dimensional smooth manifold is necessarily ﬁnite dimensional. So there is no way to model the diffeomorphism group on a Banach space as a manifold. Under the situation above, at the end of the 1960s, Omori initiated the theory of inﬁnite-dimensional Lie groups, called “ILB-Lie groups”, beyond Banach–Lie groups, taking ILB-chains as model spaces in order to treat the diffeomorphism group on a manifold (see [32] for the precise deﬁnition).

For example, Etingof and Kazhdan proved every Poisson–Lie group can be quantized in the sense above, and investigated quantum groups as deformation quantization of Poisson–Lie groups. After their works, for a while, there were no speciﬁc developments for existence problems of deformation quantization on any Poisson manifold. The situation drastically changed when M. Kontsevich [10] proved his celebrated formality theorem. As a collorary, he showed that deformation quantization exists on any Poisson manifold.