Quantization maps, algebra representation, and non-commutative Fourier transform for Lie groups

Quantization of Lie groups and Lie algebras

The Algebraic Bethe Ansatz, which is the essence of the quantum inverse scattering method, emerges as a natural development of the following different directions in mathematical physics: the inverse scattering method for solving nonlinear equations of evolution [GGK1967], quantum theory of magnets [Bet1931], the method of commuting transfer-matrices in classical statistical mechanics [Bax1982]]...

Non-commutative combinatorial algebra

Deformation Quantization of non Regular Orbits of Compact Lie Groups

In this paper we construct a deformation quantization of the algebra of polynomials of an arbitrary (regular and non regular) coadjoint orbit of a compact semisimple Lie group. The deformed algebra is given as a quotient of the enveloping algebra by a suitable ideal.

Fractional Fourier Transform and Geometric Quantization

Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of the phase-space: no linear structure is necessary. It is shown that the “fractional Fourier transform” provides a simple example of this construction. As an ap...

Geometric Quantization and the Generalized Segal–Bargmann Transform for Lie Groups of Compact Type

Let K be a connected Lie group of compact type and let T ∗(K) be its cotangent bundle. This paper considers geometric quantization of T ∗(K), first using the vertical polarization and then using a natural Kähler polarization obtained by identifying T ∗(K) with the complexified group KC. The first main result is that the Hilbert space obtained by using the Kähler polarization is naturally identi...