Deformation quantization for actions of Kählerian Lie groups. (English) Zbl 1323.22005

Mem. Am. Math. Soc. 1115, v, 154 p. (2015).
M. Rieffel considered the abelian Lie group \(\mathbb B=\mathbb R^n\) with a standard translation invariant symplectic form \(\omega^0(x,y)\) and proved that for any continuous and isometric action \(\alpha\) of \(\mathbb B\) on any Fréchet algebra or a C*-algebra \(A\), the dense subspace \(A^\infty\), consisting of smooth vectors, is deformed with parameter \(\theta\) as the corresponding Moyal product, which can be written in the form of an oscillatory integral \[ a\star^\alpha_\theta b = \int_{\mathbb B \times \mathbb B} K_\theta(x,y) \alpha_x(a)\alpha_y(b)dxdy, \] where \[ K_\theta(x,y) := \theta^{-2n}\exp\{\frac{i}{\theta}\omega^0(x,y)\}. \] He proved also many properties, namely the continuity of the field of deformed C*-algebras, invariance with respect to the \(K\)-theory, etc….
In the Memoir under review, the authors propose some non-abelian generalization of the deformation theory of C*-algebras endowed with an isometric action of a negatively curved Kählerian Lie group \(\mathbb B\). The first candidate of such a non-abelian \(\mathbb B\) is the so-called elementary nornmal \(\mathbf j\)-group \(\tilde{\mathbb S} = (\mathbb R \rtimes \mathbb R^{2d}) \rtimes \mathbb R\). In Chapters 3 and 4, a generalization is constructed for these groups. The groups of this kind have some global parametization coordinates and are diffeomorphic with \(\mathbb R^{2d+2}\). In particular, the authors prove in Theorem 4.5 that there exists a corresponding family of kernel \[ \{K_\theta(x,y)\}=(\pi\theta)^{-2(d+1)}A_{\theta,\tau}(x_1,x_2)\exp\{\frac{2i}{\theta} S^{\mathbb S}_{can}(x_1,x_2) \} \] with the exact amplitude (Definition 3.15) for the point \((x_1,x_2)\), \(x_i = (a_i,v,t_i), i=1,2\), \(\omega^\mathbb S = 2da\wedge dt + \omega^0\), \[ A_{\theta,\tau}(x_1,x_2) = A_{can}^{\mathbb S}{\theta,\tau}(x_1,x_2) \exp\{\tau(\frac{2}{\theta}\sinh 2a_1)+ \tau(-\frac{2}{\theta}\sinh 2a_2) -\tau(\frac{2}{\theta}\sinh (2a_1-2a_2))\}, \]
\[ A_{can}^{\mathbb S}{\theta,\tau}(x_1,x_2)= (\cosh a_1\cosh a_2\cos(a_1-a_2))^d(\cosh2a_1 \cosh 2a_2\cosh(2a_1-2a_2)^{1/2} \] and phase (Definition 3.15) \[ S^{\mathbb S}_{can}(x-1,x_2) = t_2\sinh 2a_1 -t_1\sinh 2a_2 + \omega^0(v_1,v_2)\cosh a_1\cosh a_2. \] Next, the authors considere more general cases of negatively curved Kählerian Lie normal \(\mathbf j\)-algebras and \(\mathbf j\)-groups, which are obtained from this by iterating the semi-direct product (Proposition 3.5): every normal \(\mathbf j\)-algebra \(\mathfrak b\) in the sense of Pyatetskii-Shapiro can be decomposed as a split extension of elementary normal \(\mathbf j\)-algebras \(\mathfrak b = (\dots(\mathfrak s_N \ltimes \mathfrak s_{N-1}) \ltimes \dots \ltimes \mathfrak s_2) \ltimes \mathfrak s_1\) and the same for a nornal \(\mathbf j\)-group, which can be obtained as repeated split extensions of elementary normal \(\mathbf j\)-groups \(\mathbb B = (\dots(\mathbb S_N \ltimes \mathbb S_{N-1}) \ltimes\dots \ltimes \mathbb S_2) \ltimes \mathbb S_1.\) The general theory is therefore constructed in Chapters 5, 6, 7. The technical preparation was done in Chapter 2. Chapter 8 is reserved for an application to the problem of quantization, continuity of deformation and \(K\)-theory invariance. The book is interesting and readable, and the readers can access the subject easily.


22E30 Analysis on real and complex Lie groups
46L87 Noncommutative differential geometry
81R60 Noncommutative geometry in quantum theory
58B34 Noncommutative geometry (à la Connes)
81R30 Coherent states
53C35 Differential geometry of symmetric spaces
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
53D55 Deformation quantization, star products
Full Text: DOI arXiv


[1] P. Bieliavsky, Espaces symétriques symplectiques. Ph.D. thesis, Université Libre de Bruxelles (1995); math.DG/0703358.
[2] Pierre Bieliavsky, Strict quantization of solvable symmetric spaces, J. Symplectic Geom. 1 (2002), no. 2, 269-320. · Zbl 1032.53080
[3] P. Bieliavsky, “Non-formal deformation quantizations of solvable Ricci-type symplectic symmetric spaces”, J. Phys. Conf. Ser. 103 (2008) 012001.
[4] P. Bieliavsky, Ph. Bonneau, F. D’Andrea, V. Gayral, Y. Maeda and Y. Voglaire, “Multiplicative unitaries and locally compact quantum Kählerian groups”, in preparation.
[5] P. Bieliavsky, M. Cahen, and S. Gutt, A class of homogeneous symplectic manifolds, Geometry and nature (Madeira, 1995) Contemp. Math., vol. 203, Amer. Math. Soc., Providence, RI, 1997, pp. 241-255. · Zbl 0879.53026
[6] Pierre Bieliavsky, Laurent Claessens, Daniel Sternheimer, and Yannick Voglaire, Quantized anti de Sitter spaces and non-formal deformation quantizations of symplectic symmetric spaces, Poisson geometry in mathematics and physics, Contemp. Math., vol. 450, Amer. Math. Soc., Providence, RI, 2008, pp. 1-24. · Zbl 1150.53029
[7] P. Bieliavsky, V. Gayral and B. Iochum, “Non-unital spectral triples on quantum symplectic symmetric spaces”, in preparation.
[8] Pierre Bieliavsky and Marc Massar, Oscillatory integral formulae for left-invariant star products on a class of Lie groups, Lett. Math. Phys. 58 (2001), no. 2, 115-128. · Zbl 0998.53059
[9] Ernst Binz, Reinhard Honegger, and Alfred Rieckers, Infinite dimensional Heisenberg group algebra and field-theoretic strict deformation quantization, Int. J. Pure Appl. Math. 38 (2007), no. 1, 43-78. · Zbl 1151.81025
[10] Ernst Binz, Reinhard Honegger, and Alfred Rieckers, Field-theoretic Weyl quantization as a strict and continuous deformation quantization, Ann. Henri Poincaré 5 (2004), no. 2, 327-346. · Zbl 1088.81066
[11] Bruce Blackadar, \(K\)-theory for operator algebras, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. · Zbl 0913.46054
[12] José F. Cariñena, José M. Gracia-Bondía, and Joseph C. Várilly, Relativistic quantum kinematics in the Moyal representation, J. Phys. A 23 (1990), no. 6, 901-933. · Zbl 0706.60108
[13] Alain Connes and Michel Dubois-Violette, Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples, Comm. Math. Phys. 230 (2002), no. 3, 539-579. · Zbl 1026.58005
[14] V. G. Drinfel\(^{\prime}\)d, Quasi-Hopf algebras, Algebra i Analiz 1 (1989), no. 6, 114-148 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 6, 1419-1457.
[15] V. Gayral, J. M. Gracia-Bondía, B. Iochum, T. Schücker, and J. C. Várilly, Moyal planes are spectral triples, Comm. Math. Phys. 246 (2004), no. 3, 569-623. · Zbl 1084.58008
[16] Victor Gayral, José M. Gracia-Bondía, and Joseph C. Várilly, Fourier analysis on the affine group, quantization and noncompact Connes geometries, J. Noncommut. Geom. 2 (2008), no. 2, 215-261. · Zbl 1148.43005
[17] Murray Gerstenhaber, Anthony Giaquinto, and Samuel D. Schack, Quantum symmetry, Quantum groups (Leningrad, 1990) Lecture Notes in Math., vol. 1510, Springer, Berlin, 1992, pp. 9-46. · Zbl 0762.17013
[18] José M. Gracia-Bondía, Joseph C. Várilly, and Héctor Figueroa, Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Inc., Boston, MA, 2001. · Zbl 0958.46039
[19] A. Grothendieck, Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires, Ann. Inst. Fourier Grenoble 4 (1952), 73-112 (1954) (French). · Zbl 0055.09705
[20] G. Lechner and S. Waldmann, “Strict deformation quantization of locally convex algebras and modules”, arXiv:1109.5950. · Zbl 1330.53117
[21] Ottmar Loos, Symmetric spaces. I: General theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. · Zbl 0228.32012
[22] I. I. Pyateskii-Shapiro, Automorphic functions and the geometry of classical domains, Translated from the Russian. Mathematics and Its Applications, Vol. 8, Gordon and Breach Science Publishers, New York-London-Paris, 1969.
[23] Z. Qian, Groupoids, midpoints and quantization, Ph.D. thesis, UC. Berkeley (1997).
[24] Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace \(C^*\)-algebras, Mathematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence, RI, 1998. · Zbl 0922.46050
[25] Marc A. Rieffel, Proper actions of groups on \(C^*\)-algebras, Mappings of operator algebras (Philadelphia, PA, 1988) Progr. Math., vol. 84, Birkhäuser Boston, Boston, MA, 1990, pp. 141-182. · Zbl 0749.22003
[26] Marc A. Rieffel, Deformation quantization for actions of \({\mathbf R}^d\), Mem. Amer. Math. Soc. 106 (1993), no. 506, x+93. · Zbl 0798.46053
[27] Marc A. Rieffel, \(K\)-groups of \(C^*\)-algebras deformed by actions of \({\mathbf R}^d\), J. Funct. Anal. 116 (1993), no. 1, 199-214. · Zbl 0803.46077
[28] Marc A. Rieffel, Non-compact quantum groups associated with abelian subgroups, Comm. Math. Phys. 171 (1995), no. 1, 181-201. · Zbl 0857.17014
[29] H. H. Schaefer and M. P. Wolff, Topological vector spaces, 2nd ed., Graduate Texts in Mathematics, vol. 3, Springer-Verlag, New York, 1999. · Zbl 0983.46002
[30] Larry B. Schweitzer, Dense \(m\)-convex Fréchet subalgebras of operator algebra crossed products by Lie groups, Internat. J. Math. 4 (1993), no. 4, 601-673. · Zbl 0802.46040
[31] Shlomo Sternberg, Symplectic homogeneous spaces, Trans. Amer. Math. Soc. 212 (1975), 113-130. · Zbl 0317.22013
[32] François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. · Zbl 0171.10402
[33] André Unterberger, The calculus of pseudodifferential operators of Fuchs type, Comm. Partial Differential Equations 9 (1984), no. 12, 1179-1236. · Zbl 0561.35081
[34] Y. Voglaire, Quantization of Solvable Symplectic Symmetric Spaces, Ph.D. thesis, UCLouvain (2011).
[35] Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 188. · Zbl 0265.22020
[36] Alan Weinstein, Traces and triangles in symmetric symplectic spaces, Symplectic geometry and quantization (Sanda and Yokohama, 1993) Contemp. Math., vol. 179, Amer. Math. Soc., Providence, RI, 1994, pp. 261-270. · Zbl 0820.58024
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