# Alexandre Odesskii

## Professor of Mathematics

Office: Mackenzie Chown J432
905 688 5550 x3297
aodesski@brocku.ca

#### Research Interests

My main research interests are in Mathematical Physics in the sense of Mathematics inspired by ideas that come from Theoretical Physics. More precisely, I am interested in algebraic and geometric structures which come from quantum field theory, statistical mechanics and the theory of integrable systems.

#### Publications and collections

[1] M. Kontsevich, A. Odesskii, Multiplication kernels, Lett. Math. Phys. 111 (2021), no. 6, Paper No. 152, 59 pp.,

[2] M. Kontsevich, A. Odesskii, p-Determinants and monodromy of differential operators, Selecta Math. (N.S.) 28 (2022), no. 3, Paper No. 52,

[3] A. Odesskii, V. Sokolov, Nonabelian elliptic Poisson structures on projective spaces, J. Geom. Phys. 169 (2021), Paper No. 104330, 15 pp.,

[4] A. Odesskii, Poisson structures on loop spaces of CP^n and an r-matrix associated with the universal elliptic curve, Journal of Geometry and Physics Volume 140, June 2019, Pages 152-160,

[5] A. Odesskii, B. Feigin, “Flat deformations of algebras and functional equations”, Journal of Combinatorial Algebra, Швейцария. 2019. Vol. 3. No. 3. P. 215-236,

[6] A. Odesskii, “Poisson structures on loop spaces of CP^n and an r-matrix associated with the universal elliptic curve”, Journal of Geometry and Physics, Volume 140, June 2019, pp 152-160,

[7] A. Odesskii, “K-projectors”, Journal of Algebra, Volume 511, 1 October 2018, pp 1-15,

[8] A. V. Odesskii, “Integrable structures of dispersionless systems and differential geometry”, Theoretical and Mathematical Physics, May 2017, Volume 191, Issue 2, pp 692-709,

[9] M. Babela, A. Odesskii, “A family of integrable evolution equations of third order”, J. Nonlinear Math. Phys. 24 (2017), no. 1, 73-78

[10] A. Odesskii, “A simple construction of integrable Whitham type hierarchies”, Geometric methods in physics, 195-208, Trends Math., Birkhauser/Springer, Cham, 2014

[11] V. E. Zakharov, A. V. Odesskii, A. V., M. Cisternino, M. Onorato, “Five-wave classical scattering matrix and integrable equations”, Theoret. and Math. Phys. 180 (2014), no. 1, 759-764

[12] A. V. Odesskii, V. N. Rubtsov, V. V. Sokolov, “Parameter-dependent associative Yang-Baxter equations and Poisson brackets”, Int. J. Geom. Methods Mod. Phys. 11 (2014), no. 9

[13] A. V. Odesskii, V. N. Rubtsov, V. V. Sokolov, “Double Poisson brackets on free associative algebras”, Noncommutative birational geometry, representations and combinatorics, 225-239, Contemp. Math., 592, Amer. Math. Soc., Providence, RI, 2013

[14] A. Odesskii, V. Sokolov, “Non-homogeneous systems of hydrodynamic type possessing Lax representations”, Comm. Math. Phys. 324 (2013), no. 1, 47-62

[15] A. Odesskii, T. Wolf, “Compatible quadratic Poisson brackets related to a family of elliptic curves”, J. Geom. And Physics (63), 2013, 107-117

[16] A. V. Odesskii, V. N. Rubtsov, V. V. Sokolov, “Bi-Hamiltonian ordinary differential equations with matrix variables”, Teoret. Mat. Fiz., 171:1 (2012), 26-32

[17] E.V. Ferapontov, Alexander Odesskii, and N.M. Stoilov, Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2+1 dimensions, Journal of Mathematical Physics 52, 2011

[18] A. Odesskii, V. Sokolov, Integrable (2+1)-dimensional systems of hydrodynamic type, Theoretical and Mathematical Physics, 163:2 (2010), 179-221

[19] A. Odesskii, V. Sokolov, Integrable pseudopotentials related to generalized hypergeometric functions, Selecta Mathematica (2010) 16:145.

[20] E. V. Ferapontov, A. V. Odesskii, Integrable Lagrangians and modular forms, Journal of Geometry and Physics 60 (2010) 896-906.

[21] A. Odesskii, V. Sokolov, Classification of integrable hydrodynamic chains, Journal of Physics A: Math. Theor. 43(2010) 434027 (15pp.).

[22] A. Odesskii, V. Sokolov, Integrable elliptic pseudopotentials, Theoretical and Mathematical Physics, 161(1): 1338-1350 (2009).

[23] Odesskii, Alexander; Rubtsov, Vladimir, Integrable systems associated with elliptic algebras. Quantum groups, 81-105, IRMA Lect. Math. Theor. Phys., 12, Eur. Math. Soc., Zurich, 2008.

[24] Odesskii, A. V.; Sokolov, V. V., On (2+1)-dimensional systems of hydrodynamic type possessing a pseudopotential with movable singularities. (Russian) Funktsional. Anal. i Prilozhen. 42 (2008), no. 3, 53–62, 96; translation in Funct. Anal. Appl. 42 (2008), no. 3, 205–212

[25] Odesskii, A. V.; Pavlov, M. V.; Sokolov, V. V., Classification of integrable Vlasov-type equations. (Russian) Teoret. Mat. Fiz. 154 (2008), no. 2, 249–260. (Reviewer: Yuri B. Suris)

[26] Odesskii, Alexander, A family of (2+1)-dimensional hydrodynamic type systems possessing a pseudopotential. Selecta Math. (N.S.) 13 (2008), no. 4, 727–742. (Reviewer: Ian A. B. Strachan)

[27] Odesskii, Alexander; Sokolov, Vladimir, Pairs of compatible associative algebras, classical Yang-Baxter equation and quiver representations. Comm. Math. Phys. 278 (2008), no. 1, 83–99. (Reviewer: Zhong Qi Ma) 17B62 (81R12)

[28] Odesskii, Alexander; Sokolov, Vladimir, Algebraic structures connected with pairs of compatible associative algebras. Int. Math. Res. Not. 2006, Art. ID 43734, 35 pp. (Reviewer: Thierry Dana-Picard)

[29] Odesskii, A. V.; Sokolov, V. V., Integrable matrix equations related to pairs of compatible associative algebras. J. Phys. A 39 (2006), no. 40, 12447–12456. (Reviewer: Stanislav Z. Pakuliak)

[30] Odesskii, A. V.; Sokolov, V. V., Compatible Lie brackets related to elliptic curve, J. Math. Phys. 47 (2006), no. 1, 013506, 14 pp. (Reviewer: Olivier G. Schiffmann)

[31] Odesskii, Alexander, Bihamiltonian elliptic structures. Mosc. Math. J. 4 (2004), no. 4, 941–946, 982.

[32] Odesskii, Alexander, Set-theoretical solutions to the Yang-Baxter relation from factorization of matrix polynomials and $\theta$-functions. Mosc. Math. J. 3 (2003), no. 1, 97–103, 259. (Reviewer: Gigel Militaru)

[33] Odesskii, A. V.; Rubtsov, V. N., Polynomial Poisson algebras with a regular structure of symplectic leaves. (Russian) Teoret. Mat. Fiz. 133 (2002), no. 1, 3–23; translation in Theoret. and Math. Phys. 133 (2002), no. 1, 1321–1337 (Reviewer: Zakaria Giunashvili)

[34] Enriquez, B.; Odesskii, A., Quantization of canonical cones of algebraic curves. Ann. Inst. Fourier (Grenoble) 52 (2002), no. 6, 1629–1663. (Reviewer: Olivier G. Schiffmann)

[35] Belavin, A. A.; Odesskii, A. V.; Usmanov, R. A., New relations in the algebra of the Baxter $Q$-operators. (Russian) Teoret. Mat. Fiz. 130 (2002), no. 3, 383–413; translation in Theoret. and Math. Phys. 130 (2002), no. 3, 323–350 (Reviewer: Andrey V. Tsiganov)

[36] Braden, H. W.; Gorsky, A.; Odesskii, A.; Rubtsov, V., Double-elliptic dynamical systems from generalized Mukai-Sklyanin algebras. Nuclear Phys. B 633 (2002), no. 3, 414–442. (Reviewer: Alexander V. Shapovalov)

[37] Odesskii, A. V., Belavin elliptic R-matrices and exchange algebras. (Russian) Funktsional. Anal. i Prilozhen. 36 (2002), no. 1, 59–74, 96; translation in Funct. Anal. Appl. 36 (2002), no. 1, 49–61 32G34

[38] Feigin, B. L.; Odesskii, A. V., Quantized moduli spaces of the bundles on the elliptic curve and their applications. Integrable structures of exactly solvable two-dimensional models of quantum field theory (Kiev, 2000), 123–137, NATO Sci. Ser. II Math. Phys. Chem., 35, Kluwer Acad. Publ., Dordrecht, 2001. (Reviewer: Zhenbo Qin)

[39] Feigin, B. L.; Odesskii, A. V., Functional realization of some elliptic Hamiltonian structures and bosonization of the corresponding quantum algebras. Integrable structures of exactly solvable two-dimensional models of quantum field theory (Kiev, 2000), 109–122, NATO Sci. Ser. II Math. Phys. Chem., 35, Kluwer Acad. Publ., Dordrecht, 2001. (Reviewer: Shao-Ming Fei)

[40] Feigin, B. L.; Odesskii, A. V., Vector bundles on an elliptic curve and Sklyanin algebras. Topics in quantum groups and finite-type invariants, 65–84, Amer. Math. Soc. Transl. Ser. 2, 185, Amer. Math. Soc., Providence, RI, 1998. (Reviewer: Michel Van den Bergh)

[41] Feigin, B. L.; Odesskii, A. V., Coordinate ring of the quantum Grassmannian and intertwiners for the representations of Sklyanin algebras. Topics in quantum groups and finite-type invariants, 55–64, Amer. Math. Soc. Transl. Ser. 2, 185, Amer. Math. Soc., Providence, RI, 1998. (Reviewer: Michel Van den Bergh)

[42] Feigin, Boris; Jimbo, Michio; Miwa, Tetsuji; Odesskii, Alexandr; Pugai, Yaroslav, Algebra of screening operators for the deformed $W\sb n$ algebra. Comm. Math. Phys. 191 (1998), no. 3, 501–541. (Reviewer: Jun’ichi Shiraishi)

[43] Odesskii, A. V.; Feigin, B. L., Elliptic deformations of current algebras and their representations by difference operators. (Russian) Funktsional. Anal. i Prilozhen. 31 (1997), no. 3, 57–70, 96; translation in Funct. Anal. Appl. 31 (1997), no. 3, 193–203 (1998) (Reviewer: V. Leksin)

[44] Feigin, Boris; Odesskii, Alexander, A family of elliptic algebras. Internat. Math. Res. Notices 1997, no. 11, 531–539. (Reviewer: Michel Van den Bergh)

[45] Odesskii, A. V.; Feigin, B. L., Sklyanin’s elliptic algebras. The case of a point of finite order. (Russian) Funktsional. Anal. i Prilozhen. 29 (1995), no. 2, 9–21, 95; translation in Funct. Anal. Appl. 29 (1995), no. 2, 81–90

[46] Odesskii, A. V.; Feigin, B. L., Constructions of elliptic Sklyanin algebras and of quantum $R$-matrices. (Russian) Funktsional. Anal. i Prilozhen. 27 (1993), no. 1, 37–45; translation in Funct. Anal. Appl. 27 (1993), no. 1, 31–38 (Reviewer: Tomasz Brzeziiski)

[47] Odesski, Aleksandr V., Rational degeneration of elliptic quadratic algebras. Infinite analysis, Part A, B (Kyoto, 1991), 773–779, Adv. Ser. Math. Phys., 16, World Sci. Publ., River Edge, NJ, 1992. (Reviewer: Michel Van den Bergh)

[48] Odesskii, A. V.; Feigin, B. L., Sklyanin’s elliptic algebras. (Russian) Funktsional. Anal. i Prilozhen. 23 (1989), no. 3, 45–54, 96; translation in Funct. Anal. Appl. 23 (1989), no. 3, 207–214 (1990) (Reviewer: S. Paul Smith)

[49] Odesskii, A. V., An analogue of the Sklyanin algebra. (Russian) Funktsional. Anal. i Prilozhen. 20 (1986), no. 2, 78–79

### Review Articles, refereed

[50] Odesskii, A. V., Elliptic algebras. (Russian) Uspekhi Mat. Nauk 57 (2002), no. 6(348), 87–122; translation in Russian Math. Surveys 57 (2002), no. 6, 1127–1162 (Reviewer: Dmitriy A. Rumynin)

### Conference contributions, refereed

[51] A.V.Odesskii and B.L.Feigin, Quantized moduli spaces of the bundles on the elliptic curve and their applications. Integrable structures of exactly solvable two-dimensional models of quantum field theory (Kiev, 2000), 123–137, NATO Sci. Ser. II Math. Phys. Chem., 35, Kluwer Acad. Publ., Dordrecht , 2001. (Reviewer: Zhenbo Qin), math.QA/9812059

[52] A.V.Odesskii and B.L.Feigin, Functional realization of some elliptic Hamiltonian structures and bosonization of the corresponding quantum algebras. Integrable structures of exactly solvable two-dimensional models of quantum field theory ( Kiev , 2000), 109–122, NATO Sci. Ser. II Math. Phys. Chem., 35, Kluwer Acad. Publ., Dordrecht , 2001. (Reviewer: Shao-Ming Fei), math.QA/9912037

#### Courses

• MATH 3P09 Partial Differential Equations
• MATH 2P08 Ordinary Differential Equations
• MATH 3P08 Advanced Differential Equations