List of Publications

Uwe Franz

• Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras, (with Isabelle Baraquin, Malte Gerhold, Anna Kula, and Mariusz Tobolski), Preprint arxiv:2309.07767.

Books

• Probability on Real Lie Algebra, (with Nicolas Privault), Cambridge Tracts in Mathematics, vol 206, Cambridge University Press, 2016

• Noncommutative Mathematics for Quantum Systems, (with Adam Skalski), Cambridge IISc series, Cambridge University Press, 2016.

• Stochastic processes and Operator Calculus on Quantum Groups, (with R. Schott), Kluwer Academic Publishers (now Springer Verlag), Mathematics and Its Applications
  Vol. 490, ISBN 978-90-481-5290-2 (Print), 978-94-015-9277-2 (Online), DOI 10.1007/978-94-015-9277-2, 236 pp., July 1999.

Books edited

• G. Aubrun, A. Skalski, R. Speicher, Quantum Symmetries, Lecture Notes of a School held in Metabief, France 2014, (Uwe Franz, ed.). Springer Lecture Notes in Mathematics, Vol. 2189, 2017, DOI 10.1007/978-3-319-63206-3.

• Topological Quantum Groups, (with Adam Skalski and Piotr Sołtan), lecture notes of the school Topological Quantum Groups, Banach Center Publications, Vol. 111, 2017.

• Ph. Biane, L. Bouten, F. Cipriani, N. Konno, N. Privault, and Q. Xu, Quantum Potential Theory, (U. Franz and M. Schürmann, eds.), Lecture Notes of a School held in Greifswald in Greifswald in Februar-March 2007, Springer Lecture Notes in Mathematics, Vol. 1954, 2008, DOI 10.1007/978-3-540-69365-9.

• O.E. Barndorff-Nielsen, U. Franz, R. Gohm, B. Kümmerer, S. Thorbjørnsen, Quantum Independent Increment Processes II : Quantum Lévy Processes, Classical Probability, and Applications, (U. Franz and M. Schürmann, eds.), Lecture Notes of a School held in Greifswald in Greifswald in March 2003, Springer Lecture Notes in Mathematics, Vol. 1866, 2006, DOI 10.1007/11376637.

• D. Applebaum, B.V.R. Bhat, J. Kustermans, J.M. Lindsay, Quantum Independent Increment Processes I : From Classical Probability to Quantum Stochastic Calculus, (U. Franz and M. Schürmann, eds.), Lecture Notes of a School held in Greifswald in Greifswald in March 2003, Springer Lecture Notes in Mathematics, Vol. 1865, 2005, DOI 10.1007/b105131.

• Quantum Probability and Infinite Dimensional Analysis : From Foundations to Applications, (with M. Schürmann), Proceedings of a Conference organized in Greifswald in June 2003, Quantum Probability and White Noise Calculus, Volume XVIII, World Scientific, 2005.

Articles in Journals

• Error Basis and Quantum Channel, (with Purbayan Chakraborty and B.V.R. Bhat), Preprint arXiv:2305.14274, accepted for publication in Open Systems & Information Dynamics.

• Schoenberg Correspondence for k-(super)Positive Maps on Matrix Algebras, (with Purbayan Chakraborty and B.V.R. Bhat, Positivity 27, 51 (2023), DOI 10.1007/s11117-023-01003-6.

• Hunt’s Formula for SUq(N) and Uq(N), (with Anna Kula, J. Martin Lindsay, and Michael Skeide), Indiana Univ. Math. J. 72 (2023), 1717-1748, DOI 10.1512/iumj.2023.72.9485, hal-03116703.

• Brownian motion on involutive braided spaces, (with Michael Schürmann and Monika Varšo), Revista de la Unión Matemática Argentina Vol. 65, no. 2 (2023), pp. 495–532, DOI 10.33044/revuma.2574, hal-03664273.

• Second cohomology groups of the Hopf*-algebras associated to universal unitary quantum groups, (with Biswarup Das, Anna Kula and Adam Skalski), Annales de l’Institut Fourier, Volume 73 (2023) no. 2, pp. 479-509, DOI 10.5802/aif.3527.

• Connectedness and Gaussian Parts for Compact Quantum Groups, (with Amaury Freslon and Adam Skalski), [Journal of Geometry and Physics 184 (2023) 104710, DOI 10.1016/j.geomphys.2022.104710.

• De Finetti Theorems for the unitary dual group, (with Isabelle Baraquin, Guillaume Cébron, Laura Maaßen, and Moritz Weber), SIGMA 18 (2022), 067, 29 pages, DOI 10.3842/SIGMA.2022.067.

• The RFD and Kac quotients of the Hopf*-algebras of universal orthogonal quantum groups, (with Biswarup Das and Adam Skalski), Annales mathématiques Blaise Pascal, Volume 28 (2021) no. 2, pp. 141-155, DOI 10.5802/ambp.402.

• Invariant Markov semigroups on quantum homogeneous spaces, with Biswarup Das and Xumin Wang, Journal of Noncommutative Geometry 15 (2), 531-580 (2021), DOI 10.4171/JNCG/404.

• Beurling-Fourier algebras of compact quantum groups : characters and finite dimensional representations, with Hunhee Lee, Indiana Univ. Math. J. 70 No. 2 (2021), 605–637, DOI 10.1512/iumj.2021.70.8405.

• Monotone Increment Processes, Classical Markov Processes and Loewner Chains, with Takahiro Hasebe and Sebastian Schleißinger, Dissertationes Mathematicae 552 (2020), 1-119, DOI 10.4064/dm808-1-2020.

• Lévy-Khintchine decompositions for generating functionals on universal CQG-algebras, (with Biswarup Das, Anna Kula, and Adam Skalski), Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 21, No. 03, 1850017 (2018) DOI 10.1142/S0219025718500170

• Homological properties of quantum permutation algebras, (with Julien Bichon and Malte Gerhold), New York J. Math. 23 (2017) 1671–1695, http://nyjm.albany.edu/j/2017/23-74.html.

• Convolution, subordination and characterization problems in noncommutative probability, (with W. Ejsmont, K. Szpojankowski), Indiana Univ. Math. J. 66 No. 1 (2017), 237–257.

• Hypercontractivity of heat semigroups on free quantum groups, (with Guixiang Hong, François Lemeux, Michaël Ulrich, and Haonan Zhang), J. of Operator Theory Vol. 77, No. 1, Winter 2017, pp. 61-76.

• Integration over the quantum diagonal subgroup and associated Fourier-like algebras, (with Hun Hee Lee and Adam Skalski), International Journal of Mathematics Vol. 27, 1650073 (2016) [37 pages], 10.1142/S0129167X16500737.

• On the Lévy-Khinchin decomposition of generating functionals, (with M. Gerhold, A. Thom), Communications in Stochastic Analysis,Vol. 9, No. 4 (2015) 529-544.

• One-to-one correspondence between generating functionals and cocycles on quantum groups in presence of symmetry, (with B. Das, A. Kula, A. Skalski), Mathematische Zeitschrift Volume 281, Issue 3, pp 949-965, 2015.

• Stochastic Analysis for Obtuse Random Walks, (with T. Hamdi), J. Theoret. Probab. 28, Issue 2, 619-649, 2015, DOI 10.1007/s10959-013-0522-z.

• Conic structure of the non-negative operator convex functions on (0, ∞), (with F. Hiai), Ann. Funct. Anal. 5, 158-175, 2014.

• Higher order extension of Löwner’s theory : Operator k-tone functions, (with F. Hiai, É. Ricard), Trans. Amer. Math. Soc. 366, 3043-3074, 2014.

• Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory, (with F. Cipriani, A. Kula), J. Funct. Anal. 266, 2789-294, 2014.

• Classication of Idempotent States on the compact quantum groups U q (2), SU q (2), and SO q (3), (with A. Skalski, R. Tomatsu), J. Noncom. Geom. 7, 221-254, 2013.

• On square roots of the Haar state on compact quantum groups, (with A. Skalski and R. Tomatsu), J. Pure and Applied Alg. 216, 2079-2093, 2012.

• Idempotent states and the inner linearity property, (with T. Banica and A. Skalski),
  Bulletin Polish Academy Sciences Mathematics, vol. 60, 123-132, 2012.

• Piecewise deterministic Markov processes as limits of Markov jump processes, (with V. Liebscher und S. Zeiser), Adv. In Appl. Proabab. 44 (2012), 729-748.

• Computing Coordinates on Lie Groups, (with Ph. Feinsilver and R. Schott) to appear in International Journal of Pure and Applied Mathematics, vol. 60, no. 4, 2010.

• On idempotent states on quantum groups, (with A. Skalski), Journal of Algebra, Vol. 322, no. 5, pp. 1774-1802, 2009.

• Monotone and Boolean Convolutions for Non-compactly Supported Probability Measures, Indiana University Mathematics Journal, Vol. 58, no. 3, pp. 1151-1186, 2009.

• Autocatalytic genetic networks modeled by piecewise-deterministic Markov processes, (with Z. Zeiser, V. Liebscher), Journal of Mathematical Biology, Vol. 60, no. 2, pp. 207-246, 2009.

• Hybrid Modeling of Noise Reduction by a Negatively Autoregulated System, with S.Zeiser, J. Müller, V. Liebscher, Bulletin of Mathematical Biology 71, 1006-1024, 2009.

• Boolean convolution of probability measures on the unit circle, in Analyse et probabilités'', P. Biane, J. Faraut, and H. Ouerdiane (eds.), Séminaires et Congrès 16, 83-93, 2009. -* {Simulation of genetic networks modelled by piecewise deterministic Markov processes}, with S. Zeiser, O. Wittich, V. Liebscher, IET Syst. Biol. 2, no. 3, 112-1135, 2008. -* {On ergodic properties of convolution operators associated with compact quantum groups}, (with A.\ Skalski), Colloq. Math. 113, no. 1, 13-23, 2008. -* {Approximation of quantum Lévy processes by quantum random walks}, (with A. Skalski), Proc. Indian Acad. Sci. Math. Sci. 118, no. 2, 281-288, 2008. -* {Multiplicative monotone convolutions, in "Quantum Probability''}, Marek Bozejko, Wojciech Mlotkowski and Janusz Wysoczanski (eds.), Banach Center Publications Volume 73, p. 153-166, 2006. -* {Renormalized powers of quantum white noise}, (with L. Accardi and A. Boukas), [Inf. Dim. Anal., Quantum Prob. and Rel. Topics->http://journals.wspc.com.sg/idaqp/idaqp.shtml], Vol. 9, No. 1, pp. 129-147, 2006. -* {Non-Gaussian Malliavin Calculus on Real Lie Algebras}, (with N. Privault and R. Schott), Journal of Functional Analysis, Vol. 218, No. 2, pp. 347-371, 2005 [Link to the article in J. Funct. Analysis.->http://dx.doi.org/10.1016/j.jfa.2004.02.004] -* {Second Quantized Automorphisms of the Renormalized Square of White Noise}, (with L. Accardi and G. Amosov), [Inf. Dim. Anal., Quantum Prob. and Rel. Topics->http://journals.wspc.com.sg/idaqp/idaqp.shtml], Vol. 7, No. 2, pp. 183-194, 2004. -* {Quasi-Invariance Formulas for Components of Quantum Lévy Processes}, (with N. Privault), Inf. Dim. Anal., [Quantum Prob. and Rel. Topics->http://journals.wspc.com.sg/idaqp/idaqp.shtml], Vol. 7, No. 1, pp. 131-145, 2004. -* {Unification of Boolean, Monotone, Anti-Monotone, and Tensor Independence and Lévy Processes}, [Mathematische Zeitschrift->http://www.link.springer.de/link/service/journals/00209/index.htm] Vol. 243, No. 4, pp. 779-816, 2003. [Link to the article in Math. Zeitschrift->http://link.springer.de/link/service/journals/00209/bibs/3243004/32430779.htm] -* {Renormalized Squares of White Noise and other Non-Gaussian Noises as Lévy Processes on Real Lie Algebras}, (with L. Accardi and M. Skeide), [Communications in Mathematical Physics->http://link.springer.de/link/service/journals/00220/index.htm] Vol. 228, No. 1, pp. 123-150, 2002. [Link to the article in Comm. Math. Phys.->http://link.springer.de/link/service/journals/00220/bibs/2228001/22280123.htm] -* {Monotone Independence is Associative}, [Inf. Dim. Anal., Quantum Prob. and Rel. Topics,->http://journals.wspc.com.sg/idaqp/idaqp.shtml] Vol. 4, No. 3, pp. 401-407, 2001. -* {Malliavin Calculus and Skorohod Integration for Quantum Stochastic Calculus,} (with R. Léandre and R. Schott), [Inf. Dim. Anal., Quantum Prob. and Rel. Topics->http://journals.wspc.com.sg/idaqp/idaqp.shtml] Vol. 4, No. 1, pp. 11-38, 2001. -* {The Affine Group and Generalized Gegenbauer Polynomials}, (with Ph. Feinsilver) Computers and Mathematics with Applications Vol. 41, 1173-1182, 2001. -* {Solving Evolution Equations on Lie Groups,} (with Ph. Feinsilver and R. Schott), Journal of Physics A: Math. and Gen. 33, pp. 2777-2798, 2000. -* {Classical Markov Processes from Quantum Lévy Processes}, [Inf. Dim. Anal., Quantum Prob. and Rel. Topics->http://journals.wspc.com.sg/idaqp/idaqp.shtml], Vol. 2, No.1, pp. 105-129, 1999. -* {Limit Theorems for the Hierarchy of Freeness,} (with R. Lenczewski), Probab. Math. Stat. Vol. 19, pp. 23-41, 1999. -* {Phase Retrieval for Probability Distributions on Quantum Groups and Braided Groups}, (with D. Neuenschwander and R. Schott), Journal of Theoretical Probability, Vol. 12, No. 2, pp. 585-594, 1999. -* {Diffusions on Braided Spaces}, (with R. Schott), Journal of Math. Phys., Vol. 39, No. 5, pp. 2748-2762, 1998. -* {Evolution Equations and Lévy Processes on Quantum Groups}, (with R. Schott), Journal of Physics A: Math. and Gen. 31, pp. 1395-1404, 1998. -* {Gauss Laws in the Sense of Bernstein and Uniqueness of Embedding into Convolution Semigroups on Quantum Groups and Braided Groups}, (with D. Neuenschwander and R. Schott), Probab. Theory Relat. Fields 109, pp. 101-127, 1997. -* {Duality and Multiplicative Processes on Quantum Groups,} (with Ph. Feinsilver and R. Schott), Journal of Theoretical Probability, Vol. 10, No. 3, pp. 795-818, 1997. {{Communications to the Comptes Rendus de l'Académie des Sciences Paris}}<br/> -* A New Characterisation of Idempotent States on Finite and Compact Quantum Groups, (with A. Skalski), Comptes Rendus Mathematique, Vol. 347, Issue 17-18, pp. 991-996, 2009. -* Smoothness of Wigner densities on the affine algebra, (with N. Privault and R. Schott), Comptes Rendus Mathématiques, Vol. 337, Issue 9, pp. 609-614, 2003. -* Malliavin Calculus for Quantum Stochastic Processes, (with R. Léandre and R. Schott), C. R. Acad. Sci. Paris, t. 328, Série I, p. 1061-1066, 1999. -* Gauss Laws in the Sense of Bernstein and Uniqueness of Embedding into Covolution Semigroups on Quantum Groups and Braided Groups, (with D. Neuenschwander and R. Schott), C. R. Acad. Sci. Paris, t. 324, Série I, p. 827-832, 1997. -* Feynman-Kac Formula and Appell Systems on Quantum Groups, (with Ph. Feinsilver and R. Schott), C. R. Acad. Sci. Paris, t. 321, Série I, p. 1615-1619, 1995. {{Conferences with Committee and Proceedings}}<br/> -* Lévy processes on the Lorentz-Lie algebra, (with Ameur Dhahri), presented at the [«International Conference on Infinite Dimensional Analysis, Quantum Probability and Related Topics, QP38» -> https://www.rs.noda.tus.ac.jp/qbic/QP38html/indexQP38.html] held at Tokyo University of Science in October 2017, and submitted for publication in the proceedings of that meeting. Availabe at [hal -> https://hal.archives-ouvertes.fr/] as [hal-03103901v1 -> https://hal.archives-ouvertes.fr/hal-03103901v1]. -* Introduction to compact and discrete quantum groups, (with [Adam Skalski -> http://www.impan.pl/~skalski/] and [Piotr  Sołtan -> http://www.fuw.edu.pl/~psoltan/], Banach Center Publications volume `Topological Quantum Groups', see also [Lecture Notes -> http://bcc.impan.pl/15TQG/index.php/lecture-notes.html] or [arxiv:1703.10766 -> http://arxiv.org/abs/1703.10766], 2017. -* {Lévy Processes on Quantum Permutation Groups}, (with A. Kula and A. Skalski), In [“Noncommutative Analysis, Operator Theory and Applications” -> http://www.springer.com/fr/book/9783319291147], Fabio Cipriani, Fabrizio Colombo, Irene Sabadini (eds.), pp. 193-259, Birkhäuser, 2016. -* {Markov property of monotone Lévy processes,} (with N. Muraki), in [Infinite Dimensional Harmonic Analysis III->http://www.worldscibooks.com/mathematics/5981.html], Proceedings of the Third German-Japanese Symposium, University of Tübingen, Germany, 15-20 September 2003, H. Heyer, T. Hirai, T. Kawazoe, K. Saito (Eds.), p. 37-57, 2005, see also [math.PR/0401390.->http://arxiv.org/abs/math.PR/0401390] -* Lévy Processes on Real Lie Algebras, in [Recent Developments in Stochastic Analysis and Related Topics->http://www.worldscibooks.com/mathematics/5710.html], Proceedings of the First Sino-German conference on Stochastic Analysis (A Satellite Conference of ICM 2002), Beijing, China, 29 August - 3 September 2002, World Scientific, 2004.<br/> -* {What is Stochastic Independence?} in [QP-PQ: Quantum Probability and White Noise Analysis, Vol. 16: NON-COMMUTATIVITY, INFINITE-DIMENSIONALITY AND PROBABILITY AT THE CROSSROADS->http://www.worldscientific.com/books/mathematics/5215.html], Proceedings of the RIMS Workshop on Infinite-Dimensional Analysis and Quantum Probability, Kyoto, Japan, November 20-22, 2001, N. Obata, T. Matsui, A. Hora (Eds.) World Scientific, 2003.<br/> -* {Free Lévy processes on dual groups}, second MaPhySto Conference on Lévy Processes - Theory and Applications at the [MaPhySto Centre for Mathematical Physics and Stochastics,->http://www.maphysto.dk/] Aarhus, Denmark, January 21-25, 2002.<br/> Contribution to the [Miniprocedings->http://www.maphysto.dk/cgi-bin/w3-msql/publications/genericpublication.html?publ=411]: [Postscript->http://www.math-inf.uni-greifswald.de/algebra/franz/papers/aarhus.ps], [DVI ->http://www.math-inf.uni-greifswald.de/algebra/franz/papers/aarhus.dvi] -* {Lévy Processes on Quantum Hypergroups}, (with M. Schürmann), in Infinite Dimensional Harmonic Analysis, Transactions of a Japanese-German Symposium held from Sept. 20th to 24th, 1999, at the University of Kyoto, H. Heyer, T. Hida, and N. Obata (Eds.), p. 93-114, 2000.<br/> -* {Infinitely divisible representations of Uq(g),} (with V.K. Dobrev, H.-D. Doebner, and R.Schott), Proceedings of the International Symposium Quantum Theory and Symmetries, 18-22 July 1999, Goslar, Germany, H.-D. Doebner, V.K. Dobrev, J.-D. Hennig, W. Lücke (Eds.), World Scientific, Singapore, p. 464-469, 2000. -* {Lévy Processes on Uq(g)}, (with V.K. Dobrev, H.-D. Doebner, and R. Schott), Proceedings of the Third International Workshop Lie Theory and its Applications in Physics III, 11-14 July 1999, Clausthal, Germany, H.-D. Doebner, V.K. Dobrev, J. Hilgert (Eds.), World Scientific, Singapore, p. 280-292, 2000. -* {Lévy Processes on Quantum Groups}, inProbability on Algebraic Structures’’, G. Budzban, Ph. Feinsilver, and A. Mukherjea (Eds.), Contemporary Mathematics Vol. 261, American Mathematical Society, p. 161-179, 2000
  

• Lévy Processes on U_q(g) as Infinitely Divisible Representations, (with V.K. Dobrev, H.-D. Doebner, and R.Schott), in Probability on Algebraic Structures'', G. Budzban, Ph. Feinsilver, and A. Mukherjea (Eds.), Contemporary Mathematics Vol. 261, American Mathematical Society, p. 181-192, 2000. -* {A Mixed Classical-Quantum Limit Theorem on Bialgebras,} Workshop on Quantum Probability, Gdansk, Poland, 1-6. July 1997, published in Banach Center Publications, Vol. 43, pp. 183-189, 1998. -* {On the Computations of Polynomial Representations of Nilpotent Lie Groups: A Symbolic Mathematical Approach}, (with Ph. Feinsilver and R. Schott), Proceedings of the 1997 ACM Symposium on Applied Computing, ACM Publ., p. 537-539, 1997. -* {A Stochastic Approach to Evolution Equations on Nilpotent Quantum Groups,} (with R. Schott), ["Quantum Group Symposium at Group21"->http://www.pt.tu-clausthal.de/%7Egroup21/qg-proc.html](Proceedings of the Quantum Group Symposium at the XXI International Colloquium on Group Theoretical Methods in Physics, Goslar, Germany, 15-20. July 1996), H.-D. Doebner and V.K. Dobrev (Eds.), Heron Press, Sofia, p. 152-157, 1997; ISBN 954-590-027-5. -* {Operator Calculus and Symbolic Computation on Lie Groups}, (with P. Feinsilver and R. Schott), in [Group21: Physical Applications and Mathematical Aspects of Geometry, Groups and Algebras->http://www.pt.tu-clausthal.de/%7Egroup21/proceedings.html], Proceeding of the XXI International Colloquium on Group Theoretical Methods in Physics, 15-20 July 1996, Goslar, Germany, H.-D. Doebner, P. Nattermann, W. Scherer and C. Schulte (Eds.), World Scientific, Singapore, Vol. 1, p. 157-161, 1997. -* {Diffusions on Braided Spaces,} (with R. Schott), [Proceedings of IV Wigner Symposium on Group Theory and its Applications->http://sunflower.singnet.com.sg/%7Ewspclib/Books/physics/3103.html], Guadalajara, Mexico, August 7-11, 1995, N.M. Atakishiyev, T.H. Seligman, K.B. Wolf, editors, World Scientific, p. 239-242, 1996. -* {Duality and Multiplicative Processes on Quantum Groups}, (with Ph. Feinsilver and R. Schott), Proceedings of the International Conference onNonlinear, Deformed and Irreversible Quantum Systems’’, held in Clausthal, Germany, August 15-19, 1994. H.D. Doebner, V.K. Dobrev, P. Nattermann, editors, World Scientific, p. 462-468, 1995.

• Generalized Lie algebras, (with B. Gruber), in Symmetries in Science VI : From the rotation group to quantum algebras, Proceedings of a symposium held in Bregenz, Austria, August 2-7, 1992. Plenum Press, New York, p. 247-253, 1993.

Habilitation Thesis

• The Theory of Quantum Lévy Processes, Habilitation thesis EMAU Greifswald, 204 pages, math.PR/0407488, 2004.

Ph.D. Thesis

• Contribution to the Study of Stochastic Processes on Quantum Groups, Unversité Henri Poincaré - Nancy 1, June 1997.