Revista de la
Unión Matemática Argentina
Brownian motion on involutive braided spaces
Uwe Franz, Michael Schürmann, and Monika Varšo

Volume 65, no. 2 (2023), pp. 495–532    

Published online: December 28, 2023

https://doi.org/10.33044/revuma.2574

Download PDF

Abstract

We study (quantum) stochastic processes with independent and stationary increments (i.e., Lévy processes), and in particular Brownian motions in braided monoidal categories. The notion of increments is based on a bialgebra or Hopf algebra structure, and positivity is taken w.r.t. an involution. We show that involutive bialgebras and Hopf algebras in the Yetter–Drinfeld categories of a quasi- or coquasi-triangular $\ast$-bialgebra admit a symmetrization (or bosonization) and that their Lévy processes are in one-to-one correspondence with a certain class of Lévy processes on their symmetrization. We classify Lévy processes with quadratic generators, i.e., Brownian motions, on several braided Hopf-$\ast$-algebras that are generated by their primitive elements (also called braided $\ast$-spaces), and on the braided $\mathrm{SU}(2)$-quantum groups.

References

  1. L. Accardi, M. Schürmann, and W. von Waldenfels, Quantum independent increment processes on superalgebras, Math. Z. 198 no. 4 (1988), 451–477.  DOI  MR  Zbl
  2. D. Ellinas and I. Tsohantjis, Random walk and diffusion on a smash line algebra, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 no. 2 (2003), 245–264.  DOI  MR  Zbl
  3. M. Émery, On the Azéma martingales, in Séminaire de Probabilités, XXIII, Lecture Notes in Math. 1372, Springer, Berlin, 1989, pp. 66–87.  DOI  MR  Zbl
  4. U. Franz and R. Schott, Diffusion on braided spaces, J. Math. Phys. 39 no. 5 (1998), 2748–2762.  DOI  MR  Zbl
  5. U. Franz and R. Schott, Stochastic Processes and Operator Calculus on Quantum Groups, Mathematics and its Applications 490, Kluwer, Dordrecht, 1999.  DOI  MR  Zbl
  6. U. Franz, R. Schott, and M. Schürmann, Lévy processes and Brownian motion on braided spaces, available as chapter 5 in U. Franz, The Theory of Quantum Lévy Processes, Habilitation thesis, Greifswald University, 2004. arXiv:math/0407488 [math.PR].
  7. U. Franz and A. Skalski, Noncommutative Mathematics for Quantum Systems, Cambridge-IISc Series, Cambridge University Press, Delhi, 2016.  DOI  MR  Zbl
  8. M. Gerhold, S. Kietzmann, and S. Lachs, Additive deformations of braided Hopf algebras, in Noncommutative Harmonic Analysis With Applications to Probability III, Banach Center Publ. 96, Polish Acad. Sci. Inst. Math., Warsaw, 2012, pp. 175–191.  DOI  MR  Zbl
  9. U. Grenander, Probabilities on Algebraic Structures, John Wiley & Sons, New York-London; Almqvist & Wiksell, Stockholm-Göteborg-Uppsala, 1963.  MR  Zbl
  10. I. Heckenberger and H.-J. Schneider, Hopf Algebras and Root Systems, Mathematical Surveys and Monographs 247, American Mathematical Society, Providence, RI, 2020.  DOI  MR  Zbl
  11. R. L. Hudson and K. R. Parthasarathy, Quantum Ito's formula and stochastic evolutions, Comm. Math. Phys. 93 no. 3 (1984), 301–323.  DOI  MR  Zbl
  12. P. Kasprzak, R. Meyer, S. Roy, and S. L. Woronowicz, Braided quantum $\mathrm{SU}(2)$ groups, J. Noncommut. Geom. 10 no. 4 (2016), 1611–1625.  DOI  MR  Zbl
  13. C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Springer-Verlag, New York, 1995.  DOI  MR  Zbl
  14. A. Klimyk and K. Schmüdgen, Quantum Groups and Their Representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.  DOI  MR  Zbl
  15. S. MacLane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, New York-Berlin, 1971.  MR  Zbl
  16. S. Majid, Quantum groups and quantum probability, in Quantum Probability & Related Topics, QP-PQ, VI, World Scientific, River Edge, NJ, 1991, pp. 333–358.  MR  Zbl
  17. S. Majid, Quantum random walks and time reversal, Internat. J. Modern Phys. A 8 no. 25 (1993), 4521–4545.  DOI  MR  Zbl
  18. S. Majid, $\ast$-structures on braided spaces, J. Math. Phys. 36 no. 8 (1995), 4436–4449.  DOI  MR  Zbl
  19. S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cambridge, 1995.  DOI  MR  Zbl
  20. M. Malczak, Realisierung von ,,braided“ Quanten-Lévy-Prozessen, Master's thesis, Greifswald University, 2017. Available at https://web.archive.org/web/20230106074918/https://math-inf.uni-greifswald.de/storages/uni-greifswald/fakultaet/mnf/mathinf/gerhold/Abschlussarbeiten/Malczak_Masterarbeit.pdf.
  21. P.-A. Meyer, Quantum Probability for Probabilists, Lecture Notes in Mathematics 1538, Springer-Verlag, Berlin, 1993.  DOI  MR  Zbl
  22. S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conference Series in Mathematics 82, American Mathematical Society, Providence, RI, 1993.  DOI  MR  Zbl
  23. K. R. Parthasarathy, Azéma martingales and quantum stochastic calculus, in Proc. R. C. Bose Memorial Symposium, Wiley Eastern, New Delhi, 1990, pp. 551–569.
  24. K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Monographs in Mathematics 85, Birkhäuser Verlag, Basel, 1992.  DOI  MR  Zbl
  25. N. Y. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev, Quantization of Lie groups and Lie algebras, Algebra i Analiz 1 no. 1 (1989), 178–206, translation in Leningrad Math. J. 1 (1990), no. 1, 193–225.  MR  Zbl
  26. M. Schürmann, The Azéma martingales as components of quantum independent increment processes, in Séminaire de Probabilités, XXV, Lecture Notes in Math. 1485, Springer, Berlin, 1991, pp. 24–30.  DOI  MR  Zbl
  27. M. Schürmann, White Noise on Bialgebras, Lecture Notes in Mathematics 1544, Springer-Verlag, Berlin, 1993.  DOI  MR  Zbl
  28. M. Schürmann and M. Skeide, Infinitesimal generators on the quantum group ${\mathrm{SU}}_q(2)$, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 no. 4 (1998), 573–598.  DOI  MR  Zbl
  29. D. N. Yetter, Quantum groups and representations of monoidal categories, Math. Proc. Cambridge Philos. Soc. 108 no. 2 (1990), 261–290.  DOI  MR  Zbl