Приложения квантования в физике и некоммутативных алгебрах» / «Applications of the quantization in Physics and Noncommutative algebras тема диссертации и автореферата по ВАК РФ 01.01.06, кандидат наук Разавиниа Фаррох

• Разавиниа Фаррох
• кандидат наук
• 2021

Оглавление диссертации кандидат наук Разавиниа Фаррох

Table of Contents

Abstract I

1 Conventions and the organization of the thesis

1.1 Conventions and notation

1.2 Organization of the thesis

2 Introduction

2.1 The Basics: From groups to algebras

2.2 Ore extensions

2.3 Ambiskew polynomial rings

2.4 Generalized Weyl algebras

2.5 Down-up algebras

2.6 Generalized Heisenberg algebras and weak generalized Weyl algebras

2.7 Lie algebras

2.7.1 Adjoints and the Commutator

2.7.2 The Lie algebra sl(2)

2.7.3 Semisimple Lie algebras and the Cartan subalgebra

2.7.4 Root systems

2.7.5 Kac-Moody Lie algebras

2.7.6 Kac-Moody presentation of affine sl2

2.8 Hopf algebras

2.8.1 Algebras

2.8.2 Coalgebras

2.8.3 Morphisms, tensor products, and bialgebras

2.8.4 Antipodes and Hopf algebras

2.8.5 Commutativity, cocommutativity

2.9 Quantum group Uq(sl(2))

2.9.1 q-Analysis and quantum group Uq(sl(2))

2.9.2 Quantum affine algebra Uq(sl2)

2.9.3 Uq (0) and the quantum Serre relations

2.10 Quantization and algebra problems

2.10.1 Free algebras

2.10.2 Matrix representations of algebras

2.10.3 Algebra of generic matrices

2.10.4 The Amitsur-Levitzki theorem

2.10.5 Deformation quantization

2.10.6 Algebraically closed skew field

2.11 Algebras automorphisms and quantization

2.11.1 Jacobian Conjecture

2.11.2 Some results related to the Jacobian Conjecture

2.11.3 Ind-schemes and varieties of automorphisms

2.11.4 Conjecture of Dixmier and quantization

2.11.5 Tame automorphisms

2.11.6 Approximation by tame automorphisms

2.11.7 Holonomic D-modules, Lagrangian submanifolds

2.11.8 Tame automorphisms and the Quantization Conjectures

2.11.9 Quantization of classical algebras

2.12 Torus actions on free associative algebras and the Bialynicki-Birula theorem

3 Local coordinate systems on quantum flag manifolds

3.1 Feigin's homomorphisms on Uq (n)

3.2 The contribution between Quantum Serre relations and screening operators

3.2.1 sl(3) case

3.2.2 affinized Lie algebra sl(2)

3.3 Local integral of motions; Volkov's scheme

3.3.1 Example Uq(sl2); two point invariants

3.3.2 Example Uq (sl2); three point invariants

3.4 Lattice Virasoro algebra

3.4.1 Lattice Virasoro algebra connected to sl2

3.4.2 q-binomial for positive and negative exponent

3.4.3 Formulation for to extend to four and more invariant points134

3.4.4 Generators of lattice Virasoro algebra coming from 2-dimensional representation of sl2

3.4.5 Results; Generators of lattice Virasoro algebra coming from

3 and 4-dimensional representation of sl2

3.5 Conclusion

3.6 Weak Faddeev-Takhtajan-Volkov algebras; Lattice Wn algebras

3.7 Weak Faddeev-Takhtajan-Volkov algebras

3.7.1 Lattice W2 algebra

3.7.2 Lattice W3 algebra

3.7.3 Lattice W4 algebra; main generator

3.7.4 Lattice W5 algebra; main generator

3.7.5 Lattice Wn algebra; main generator

4 Generalized Heisenberg Algebras and their quantum analogs

5 Quantum generalized Heisenberg algebras

5.1 Quantum generalized Heisenberg algebras

5.2 The structure of quantum generalized Heisenberg algebras

5.2.1 Constructing Hq(f, g) as an ambiskew polynomial ring

5.2.2 Constructing Hq(f, g) as a weak generalized Weyl algebra

5.2.3 Some useful equations

5.2.4 Center of quantum generalized Heisenberg algebras

5.2.5 Primality of Hq(f,g)

5.3 Classification of quantum generalized Heisenberg algebras

5.4 The finite-dimensional simple Hq(f, g)-modules

5.4.1 Doubly-infinite weight Hq(f, g)-modules

5.4.2 Finite-dimensional simple Hq(f, g)-modules

5.4.3 Isomorphisms between finite-dimensional simple Hq (f, g)-modules

5.5 Locally finite derivations of Hq(f, g) when deg f >

5.6 Automorphisms of quantum generalized Heisenberg algebras when deg f >

5.7 Gelfand-Kirillov dimension of the quantum generalized Heisenberg algebras

5.8 Hopf quantum generalized Heisenberg algebras

5.8.1 Simple modules over Hopf quantum generalized Heisenberg algebras

5.9 Conclusion/Future Work

5.9.1 Quantum generalized Heisenberg algebras

5.9.2 Generalized Heisenberg algebras

6 Quantization proof of Bergman's centralizer theorem

6.1 Centralizer theorems

6.1.1 Cohn's centralizer theorem

6.1.2 Bergman's centralizer theorem

6.1.3 Centralizer theorem in free group algebras

6.2 Reduction to generic matrix

6.3 Quantization proof of rank one

6.4 Centralizers are integrally closed

6.4.1 Invariant theory of generic matrices

6.4.2 Algebra of generic matrices with traces is integrally closed

6.4.3 Trace algebras

6.4.4 Proof of centralizers are integrally closed

6.4.5 Completion of the proof

6.4.6 On the rationality of subfields of generic matrices

7 Torus actions on free associative algebras, lifting and Bialynicki-Birula type theorems

7.1 Actions of algebraic tori

7.2 Maximal torus action on the free algebra

7.3 Action of F* on F(zi, z2)

7.4 Positive-root torus actions

7.5 Non-linearizable torus actions

7.6 Discussion

Bibliography

Введение диссертации (часть автореферата) на тему «Приложения квантования в физике и некоммутативных алгебрах» / «Applications of the quantization in Physics and Noncommutative algebras»

Abstract

This thesis is dedicated to the study of noncommutative algebras and (broadly speaking) their applications in physics. We introduce and study a new class of algebras, which we name quantum generalized Heisenberg algebras and denote them by Hq(f, g), related to generalized Heisenberg algebras, but allowing more parameters of freedom, so as to encompass a wider range of applications and include previously studied algebras, such as (generalized) down-up algebras and generalized Heisenberg algebras. In particular, our class now includes the enveloping algebra of the 3-dimensional Heisenberg Lie algebra and its q-deformation, neither of which can be realized as a generalized Heisenberg algebra. Here we will classify the finite-dimensional irreducible representations of these algebras, and we will determine their automorphism groups when deg f > 1 and we also will solve the isomorphism problem for this class of algebras. We will study ring-theoretical properties like Gelfand-Kirillov dimension and being Noetherian. In another direction, we will investigate Hopf quantum generalized Heisenberg algebras and their simple module theory. We will try to construct a new Poisson bracket on our simplest example sl2 and then we will try to give a universal construction based on our universal variables and then will try to construct lattice W2 algebras which will play a key role in our other constructions on lattice W3 algebras and finally we will try to find the only nontrivial dependent generator of our lattice W4 algebras and so on for lattice Wn algebras.

We find a quantization proof of Bergman's centralizer theorem by showing that the free associative algebra has no commutative subalgebra of rank greater than or equal to two. In order to achieve this, we study the algebra of generic matrices and prove that if there is a commutative subalgebra generated by two nonsingular elements in the algebra of generic matrices, then quantized images have a non-zero commutator which contradicts the fact that they are commutative. Thereafter, we use generic matrices reduction to provide the proof of the fact that the centralizer is integrally closed. These two results supply the whole quantization proof of Bergman's centralizer theorem.

We examine several instances of algebraic torus action on the free associative algebra. We prove the free algebra analogue of a classical theorem of A. Bialynicki-Birula, which establishes linearity of maximal torus action. We also formulate and prove linearity theorems in a few specific situations, as well as provide a framework for construction of non-linearizable torus actions.

Список литературы диссертационного исследования кандидат наук Разавиниа Фаррох, 2021 год

Bibliography

[1] Eiichi Abe. Hopf algebras. Cambridge Tracts in Mathematics 74. Cambridge University Press, Cambridge-New York, 1980.

[2] Faddeev L. Alekseev, A. and M. Semenov-Tian-Shansky. Hidden quantum groups inside kac-moody algebra. Communications in mathematical physics, 149(2):335-345,1992.

[3] J. Alev and L. Le Bruyn. Automorphisms of generic 2 by 2 matrices. In Perspectives in Ring Theory, pages 69-83. Springer, 1988.

[4] A. S. Amitsur and J. Levitzki. Minimal identities for algebras. Proceedings of the American Mathematical Society, 1(4):449-463,1950.

[5] D. J. Anick. Limits of tame automorphisms of k[x\,... ,xn]. Journal of Algebra, 82(2):459-468,1983.

[6] Alexander A. Belov Antonov, Alexander and Karén Chaltikian. Lattice conformal theories and their integrable perturbations. Journal of Geometry and Physics, 22(4):298-318,1997.

[7] V. A. Artamonov. Projective metabelian groups and Lie algebras. Izvestiya: Mathematics, 12(2):213-223,1978.

[8] V. A. Artamonov. Projective modules over universal enveloping algebras. Mathematics of the USSR-Izvestiya, 25(3):429,1985.

[9] V. A. Artamonov. Nilpotence, projectivity, decomposability. Siberian Mathematical Journal, 32(6):901-909, 1991.

[10] V. A. Artamonov. The quantum Serre problem. Russian Math. Surveys, 53(4):3-77,1998.

[11] V. A. Artamonov. Quantum polynomials advances in algebra and combinatorics, vol. 19-34. World Science Publications, Hackensack, NJ, 2008.

339

[12] M. Artin. Noncommutative Rings. Class notes, University of California, Berkeley, 1999.

[13] T. Asanuma. Non-linearizable algebraic k*-actions on affine spaces. In-ventiones mathematicae, 138(2):281-306,1999.

[14] E. Backelin. Endomorphisms of quantized Weyl algebras. Letters in Mathematical Physics, 97(3):317-338, 2011.

[15] H. Bass. A non-triangular action of ga on a3. Journal of Pure and Applied Algebra, 33(1):1-5,1984.

[16] H. Bass, E. H. Connell, and D. Wright. The Jacobian conjecture: reduction of degree and formal expansion of the inverse. Bulletin of the American Mathematical Society, 7(2):287-330,1982.

[17] V. Bavula and F. van Oystaeyen. The simple modules of certain generalized crossed products. J. Algebra, 2(194):521-566,1997.

[18] V. V. Bavula. Finite-dimensionality of Extn and Torn of simple modules over a class of algebras. Funktsional. Anal. i Prilozhen, 3(25):80-82,1991.

[19] V. V. Bavula and D. A. Jordan. Isomorphism problems and groups of automorphisms for generalized Weyl algebras. Trans. Amer. Math. Soc, 2(353):769-794, 2001.

[20] Vladimir V. Bavula. Generalized weyl algebras and their representations. St. Petersburg Mathematical Journal, 4(1):71-92,1993.

[21] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer. Deformation theory and quantization. I. deformations of symplectic structures. Annals of Physics, 111(1):61-110,1978.

[22] A. Belov. Linear recurrence equations on a tree. Mathematical Notes, 78(5):603-609, 2005.

[23] A. Belov, L. Bokut, L. Rowen, and J.-T. Yu. The Jacobian conjecture, together with Specht and Burnside-type problems. In Automorphisms in Birational and Affine Geometry, pages 249-285. Springer, 2014.

[24] A. Ya. Belov. The local finite basis property and local representability of varieties of associative rings. Izvestiya: Mathematics, 74(1):1-126,2010.

[25] Alexander A. Belov and Karen D. Chaltikian. Lattice analogues of w-algebras and classical integrable equations. Physics Letters B, 309(3-4):268-274,1993.

[26] A. Belov-Kanel and A. Elishev. On planar algebraic curves and holonomic D-modules in positive characteristic. Journal of Algebra and Its Applications, 15(08):1650155, 2016.

[27] A. Belov-Kanel and M. Kontsevich. Automorphisms of the Weyl algebra. Letters in Mathematical Physics, 74(2):181-199,2005.

[28] A. Belov-Kanel and M. Kontsevich. The Jacobian conjecture is stably equivalent to the Dixmier conjecture. Moscow Mathematical Journal, 7(2):209-218, 2007.

[29] A. Ya. Belov-Kanel, A. M. Elishev, F. Razavinia, Jie-Tai Yu, and Wen-chao Zhang. Noncommutative Bialynicki-Birula theorem. Chebyshevskii sbornik, 21(1):51-61, 2020.

[30] G. Benkart and T. Roby. Down-up algebras. J. Algebra, 1(209):305-344, 1998.

[31] Georgia Benkart and Sarah Witherspoon. A Hopf structure for down-up algebras. Math. Z, 3(238):523-553, 2001.

[32] Arkady. Berenstein. Group-like elements in quantum groups, and feigin's conjecture. arXiv preprint q-alg/9605016,1996.

[33] G. M. Bergman. Centralizers in free associative algebras. Transactions of the American Mathematical Society, 137:327-344,1969.

[34] G. M. Bergman. The diamond lemma for ring theory. Advances in Mathematics, 29(2):178-218,1978.

[35] A. Bialynicki-Birula. Remarks on the action of an algebraic torus on kn. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys, 14:177-181,1966.

[36] A. Bialynicki-Birula. Remarks on the action of an algebraic torus on kn, II. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys, 15:123-125,1967.

[37] A. Bialynicki-Birula. Some theorems on actions of algebraic groups. Annals of Mathematics, 98:480-497,1973.

[38] Lawrence C. Biedenharn and Max A. Lohe. Quantum group symmetry and q-tensor algebras. World Scientific, 1995.

[39] T. Bitoun. The p-support of a holonomic D-module is Lagrangian, for p large enough. PhD thesis, University of Orsay, 2010.

[40] Richard E. Block. The irreducible representations of the Lie algebra si (2) and of the Weyl algebra. Adv. in Math, 1(39):69-110,1981.

[41] L. Bokut and E. Zelmanov. Selected works of A. I. Shirshov. Springer, 2009.

[42] L. A. Bokut. Embedding Lie algebras into algebraically closed Lie algebras. Algebra Logika, 1:47-53,1962.

[43] L. A. Bokut. Embedding of algebras into algebraically closed algebras. In Doklady Akademii Nauk, volume 145(5), pages 963-964. Russian Academy of Sciences, 1962.

[44] L. A. Bokut. Theorems of embedding in the theory of algebras. In Colloq. Math, volume 14, pages 349-353, 1966.

[45] M. Bresar, C. Procesi, and S. Spenko. Functional identities on matrices and the Cayley-Hamilton polynomial. arXiv preprint arXiv:1212.4597, 2013.

[46] Lieven Le Bruyn. Conformal si2 enveloping algebras. Comm. Algebra, 4(23):1325-1362,1995.

[47] L. A. Campbell. A condition for a polynomial map to be invertible. Mathematische Annalen, 205(3):243-248,1973.

[48] Paolo. Caressa. The algebra of poisson brackets. Young Algebra Seminar, Roma Tor Vergata, 2000.

[49] Roger Carter. Lie algebras of finite and affine type. Cambridge university press, 2005.

[50] Paula A. A. B. Carvalho and Samuel A. Lopes. Automorphism of generalized down-up algebra. Comm. Algebra, 2(3):1308-1319,2008.

[51] Paula A. A. B. Carvalho and Ian M. Musson. Down-up algebra and their representation theory. J. Algebra, 3(118):256-280, 2001.

[52] Thomas Cassidy and Brad Shelton. Basic properties of generalized down-up algebras. Journal of Algebra, 1(279):402-421, 2004.

[53] P. M. Cohn. Subalgebras of free associative algebras. Proceedings of the London Mathematical Society, 3(4):618-632,1964.

[54] P. M. Cohn. Progress in free associative algebras. Israel Journal of Mathematics, 19(1-2):109-151,1974.

[55] P. M. Cohn. A brief history of infinite-dimensional skew fields. Math. Scient, 17:1-14,1992.

[56] E M F Curado and M A Rego-Monteiro. Multi-parametric deformed heisenberg algebras: a route to complexity. Journal of Physics A: Mathematical and General, 15(34):3253, 2001.

[57] A. J. Czerniakiewicz. Automorphisms of a free associative algebra of rank 2. i. Transactions of the American Mathematical Society, 160:393-401,1971.

[58] A. J. Czerniakiewicz. Automorphisms of a free associative algebra of rank 2. ii. Transactions of the American Mathematical Society, 171:309-315,1972.

[59] M. De Bondt and A. Van den Essen. The Jacobian conjecture for symmetric Druzkowski mappings. University of Nijmegen, Department of Mathematics, 2004.

[60] M. De Bondt and A. Van den Essen. A reduction of the Jacobian conjecture to the symmetric case. Proceedings of the American Mathematical Society, 133(8):2201-2205, 2005.

[61] Manuel Jose Ribeiro de Castro Silva Martins. Hopf algebras and ore extensions. Master thesis, 2016.

[62] C. De Concini and C. Procesi. A characteristic free approach to invariant theory. In Young Tableaux in Combinatorics, Invariant Theory, and Algebra, pages 169-193. Elsevier, 1982.

[63] C. De Concini and C. Procesi. The invariant theory of matrices. American Mathematical Society Providence, RI, 2017.

[64] W. Dicks. Automorphisms of the free algebra of rank two, group actions on rings.(brunswick, maine, 1984). Contemp. Math, 43:63-68,1985.

[65] W. Dicks and J. Lewin. A Jacobian conjecture for free associative algebras. Communications in Algebra, 10(12):1285-1306,1982.

[66] J. Dixmier. Sur les algebres de Weyl. Bulletin de la Société Mathématique de France, 96:209-242,1968.

[67] C. Dodd. The p-cycle of holonomic D-modules and auto-equivalences of the Weyl algebra. arXiv preprint arXiv:1510.05734,2015.

[68] S. Donkin. Invariants of several matrices. Inventiones Mathematicae, 110(1):389-401, 1992.

[69] S. Donkin. Invariant functions on matrices. Mathematical Proceedings of the Cambridge Philosophical Society, 113(1):23-43,1993.

[70] V. Drensky and J.-T. Yu. A cancellation conjecture for free associative algebras. Proceedings of the American Mathematical Society, 136(10):3391-3394, 2008.

[71] Vladimir Gershonovich Drinfeld. Quantum groups. Lett. Math. Phys, (155):18-49,1986.

[72] L. DruZkowski. An effective approach to Keller's Jacobian conjecture. Mathematische Annalen, 264(3):303-313,1983.

[73] L. DruZkowski. The Jacobian conjecture: symmetric reduction and solution in the symmetric cubic linear case. Annales Polonici Mathematici, 1(87):83-92, 2005.

[74] A. Elishev. Automorphisms of polynomial algebras, quantization and Kontse-vich conjecture. PhD thesis, Moscow Institute of Physics and Technology (in Russian), 2019.

[75] A. Elishev, A. Kanel-Belov, F. Razavinia, J.-T. Yu, and W. Zhang. Torus actions on free associative algebras, lifting and Bialynicki-Birula type theorems. arXiv preprint arXiv:1901.01385,2019.

[76] Ian M. Musson Ellen Kirkman and D. S. Passman. Noetherian down-up algebras. Proc. Amer. Math. Soc, 11(127):3161-3167,1999.

[77] B. Fedosov. A simple geometrical construction of deformation quantization. Journal of Differential Geometry, 40(2):213-238,1994.

[78] B.L. Feigin. talk at RIMS, 1992.

[79] Davide Gaiotto and Edward Witten. Probing quantization via branes. arXiv preprint arXiv:2107.12251, 2021.

[80] K. R. Goodearl. Semiclassical limits of quantized coordinate rings. Advances in ring theory, Trends Math, Birkhauser/Springer Basel AG, Basel, pages 165-204, 2010.

[81] Kenneth R. Goodearl and Robert Breckenridge Warfield Jr. An introduction to noncommutative noetherian rings. Cambridge University Press, 61, 2004.

[82] Edouard. Goursat. A Course in Mathematical Analysis: pt. 2. Differential equations, volume 2. Dover Publications, 1916.

[83] A. Gutwirth. The action of an algebraic torus on the affine plane. Transactions of the American Mathematical Society, 105(3):407-414,1962.

[84] Albert Heinle. Factorization of polynomials in a class of noncommutative algebras. Bachelor thesis, 2010.

[85] Kazuhiro. Hikami. Lattice wn algebra and its quantization. Nuclear Physics B, 505(3):749-770, 1997.

[86] Kazuhiro Hikami and Rei Inoue. Classical lattice w algebras and integrable systems. Journal ofPhysics A: Mathematical and General, 30(19):6911, 1997.

[87] Timothy J. Hodges. Noncommutative deformations of type-a kleinian singularities. Journal of Algebra, 2(161):271-290,1993.

[88] Chan Huh. Gk-dimensions of extension algebras. East Asian Math. J, 14(1):149-155, 1998.

[89] K. Iohara and F. Malikov. Rings of skew polynomials and gel'fand-kirillov conjecture for quantum groups. Communications in mathematical physics, 164(2):217-237, 1994.

[90] David A Jordan. down-up algebras and ambiskew polynomial rings. Journal of Algebra, (228):311-346, 2000.

[91] David A. Jordan. Simple ambiskew polynomial rings II: non-bijective endomorphisms. Rings, modules and codes, Contemp. Math, Amer. Math. Soc, 3:177-200, 2019.

[92] David A. Jordan and Imogen E. Wells. Invariants for automorphisms of certain iterated skew polynomial rings. Proc. Edinburgh Math. Soc. (2), 3(39):461-472,1996.

[93] H. W. Jung. Über ganze birationale Transformationen der Ebene. Journal für die reine und angewandte Mathematik, 184:161-174,1942.

[94] Victor G. Kac. Infinite dimensional lie algebras. Cambridge university press, 1990.

[95] S. Kaliman, M. Koras, L. Makar-Limanov, and P. Russell. C*-actions on C3 are linearizable. Electron. Res. Announc. Amer. Math. Soc, 3:63-71,1997.

[96] T. Kambayashi and P. Russell. On linearizing algebraic torus actions. Journal of Pure and Applied Algebra, 23(3):243-250,1982.

[97] A. Kanel-Belov, V. Borisenko, and V. Latysev. Monomial algebras. Journal ofMathematical Sciences, 87:3463-3575, 1997.

[98] A. Kanel-Belov, A. Elishev, and J.-T. Yu. Independence of the B-KK isomorphism of infinite prime. arXiv preprint arXiv:1512.06533, 2015.

[99] A. Kanel-Belov, A. Elishev, and J.-T. Yu. Augmented polynomial symplec-tomorphisms and quantization. arXiv preprint arXiv:1812.02859, 2018.

[100] A. Kanel-Belov, S. Grigoriev, A. Elishev, J.-T. Yu, and W. Zhang. Lifting of polynomial symplectomorphisms and deformation quantization. Communications in Algebra, 46(9):3926-3938, 2018.

[101] A. Kanel-Belov, F. Razavinia, and W. Zhang. Bergman's centralizer theorem and quantization. Communications in Algebra, 46(5):2123-2129, 2018.

[102] A. Kanel-Belov, F. Razavinia, and W. Zhang. Centralizers in free associative algebras and generic matrices. arXiv preprint arXiv:1812.03307,2018.

[103] A. Kanel-Belov, L. H. Rowen, and Ü. Vishne. Structure of Zariski-closed algebras. Trans. Amer. Math. Soc., 362(9):4695-4734,2010.

[104] A. Kanel-Belov, J.-T. Yu, and A. Elishev. On the augmentation topology of automorphism groups of affine spaces and algebras. International Journal of Algebra and Computation, 28(08):1449-1485,2018.

[105] Christian Kassel. Quantum groups. Springer Science & Business Media Vol. 155, 2012.

[106] B. Keller. Notes for an introduction to Kontsevich's quantization theorem, preprint, 2003.

[107] O.-H. Keller. Ganze cremona-transformationen. Monatshefte filr Mathematik, 47(1):299-306,1939.

[108] A. Kemer. Multilinear identities of the algebras over a field of characteristic p. Internat. J. Algebra Comput., 5(2):189-198,1995.

[109] E.E. Kirkman and L.W. Small. q-analogs of harmonic oscillators and related rings. Israel J. Math, 1-2(81):111-127,1993.

[110] Ian Musson Kirkman, Ellen and D. Passman. Noetherian down-up algebras. Proceedings of the American Mathematical Society, 127(11):3161-3167, 1999.

[111] Anatoli Klimyk and Konrad Schmudgen. Quantum groups and their representations. Springer Science & Business Media, 2012.

[112] P. S. Kolesnikov. The Makar-Limanov algebraically closed skew field. Algebra and Logic, 39(6):378-395, 2000.

[113] P. S. Kolesnikov. Different definitions of algebraically closed skew fields. Algebra and Logic, 40(4):219-230, 2001.

[114] M. Kontsevich. Deformation quantization of Poisson manifolds. Letters in Mathematical Physics, 66(3):157-216,2003.

[115] M. Kontsevich. Holonomic D-modules and positive characteristic. Japanese Journal of Mathematics, 4(1):1-25,2009.

[116] M. Koras and P. Russell. C*-actions on C3: The smooth locus of the quotient is not of hyperbolic type. Journal of Algebraic Geometry, 8(4):603-694, 1999.

[117] Günter R. Krause and Thomas H. Lenagan. Growth of algebras and Gelfand-Kirillov dimension, volume 22 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, revised edition, 2000.

[118] V. S. Kulikov. Generalized and local Jacobian problems. Russian Academy of Sciences. Izvestiya Mathematics, 41(2):351,1993.

[119] V. S. Kulikov. The Jacobian conjecture and nilpotent maps. Journal of Mathematical Sciences, 106(5):3312-3319,2001.

[120] Samuel A. Lopes. Non-Noetherian generalized Heisenberg algebra. J. Algebra Appl, 13(1):1431754, 2016.

[121] Samuel A. Lopes and Farrokh Razavinia. Quantum generalized Heisenberg algebra and their representation. arXiv preprint arXiv:2001.04198, accepted for publication in J. Communications in Algebra., 2020.

[122] Samuel A. Lopes and Farrokh Razavinia. Structure and isomorphism of quantum generalized Heisenberg algebra. arXiv preprint arXiv:2006.01459, accepted for publication in J. Algebra Appl., 2020.

[123] M. Lothaire. Combinatorics on words, volume 17. Cambridge University Press, 1997.

[124] Rencai Lu and Kaiming Zhao. Finite-dimensional simple modules over generalized Heisenberg algebras. Linear Algebra Appl, 475:276-291, 2015.

[125] Shahn Majid. Foundations of quantum group theory. Cambridge university press, 2000.

[126] L. Makar-Limanov. Automorphisms of a free algebra with two generators. Functional Analysis and Its Applications, 4(3):262-264,1970.

[127] L. Makar-Limanov. On automorphisms of Weyl algebra. Bulletin de la Société Mathématique de France, 112:359-363,1984.

[128] L. Makar-Limanov. Algebraically closed skew fields. Journal of Algebra, 93(1):117-135,1985.

[129] L. Makar-Limanov, Ü. Turusbekova, and Ü. Umirbaev. Automorphisms and derivations of free Poisson algebras in two variables. Journal ofAlge-bra, 322(9):3318-3330, 2009.

[130] Leonid Makar-Limanov. Locally nilpotent derivations, a new ring invariant and applications. preprint, pages 1-60, 1998.

[131] James Christopher Robson McConnell, John C. and Lance W. Small. Non-commutative noetherian rings. American Mathematical Soc, 30,2001.

[132] N. Miasnikov. Centralizers in free group algebras and nonsingular curves. Journal of Algebra, 516(15):490-513,12 2018.

[133] M. N. Mnatsakanova and Yu S. Vernov. Properties of the algebra of q-deformed commutators in indefinite-metric space. Theoretical and mathematical physics, 3(113):1497-1507,1997.

[134] T.-T. Moh. On the Jacobian Conjecture and the configurations of roots. J. Reine Angew. Math., 340:140-212,1983.

[135] Susan Montgomery. Hopf algebras and their actions on rings. American Mathematical Soc., No. 82,1993.

[136] J. E. Moyal. Quantum mechanics as a statistical theory. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 45(1), pages 99124. Cambridge University Press, 1949.

[137] Jonathan Nilsson. Simple s/n+i-module structures on U(h). J. Algebra, (424):294-329, 2015.

[138] Oystein Ore. Theory of non-commutative polynomials. Annals ofmathe-matics, pages 480-508, 1933.

[139] S. Y. Orevkov. The commutant of the fundamental group of the complement of a plane algebraic curve. Russian Mathematical Surveys, 45(1):221, 1990.

[140] S. Y. Orevkov. An example in connection with the Jacobian conjecture. Mathematical Notes, 47(1):82-88,1990.

[141] S. Y. Orevkov. The fundamental group of the complement of a plane algebraic curve. Mathematics of the USSR-Sbornik, 65(1):267,1990.

[142] V. L. Popov. Around the Abhyankar-Sathaye conjecture. arXiv preprint arXiv:1409.6330, 2014.

[143] Iwan Praton. Simple modules and primitive ideals of non-Noetherian generalized down-up algebras. Comm. Algebra, 3(37):811-839,2009.

[144] Iwan Praton. Primitive ideals of Noetherian generalized down-up algebras. Comm. Algebra, 11(39):4289-4318, 2011.

[145] C. Procesi. Rings with polynomial identities, volume 17. M. Dekker, 1973.

[146] C. Procesi. The invariant theory of nxn matrices. Advances in Mathematics, 19:306-381, 1976.

[147] Robert A. Proctor. Representations of si(2, C) on posets and the Sperner property. SIAM J. Algebraic Discrete Methods, 2(3):275-280,1982.

[148] Y.P. Pugay. Lattice w algebras and quantum groups. Theoretical and Mathematical Physics, 100(1):900-911,1994.

[149] M. Razar. Polynomial maps with constant Jacobian. Israel Journal of Mathematics, 32(2-3):97-106,1979.

[150] Farrokh. Razavinia. Local coordinate systems on quantum flag manifolds. Chebyshevskii Sbornik; ISSN: 2226-8383, 21(4):171-195, 2020.

[151] Farrokh. Razavinia. Weak Faddeev-Takhtajan-Volkov algebras; Lattice wn algebras. Chebyshevskii Sbornik; ISSN: 2226-8383,22(1):273-291,2020.

[152] Yu. P. Razmyslov. Identities of Algebras and their Representations, volume 138. American Mathematical Society, Providence, RI, 1994.

[153] M. A Rego-Monteiro. Logistic algebras. No. CBPF-NF-99-23. SCAN-9908034,1999.

[154] Volodymyr Mazorchuk Rencai Lu and Kaiming Zhao. Simple weight modules over weak generalized Weyl algebras. J. Pure Appl. Algebra, 219(8):3427-3444, 2015.

[155] S. Rosset. A new proof of the Amitsur-Levitski identity. Israel Journal of Mathematics, 23(2):187-188,1976.

[156] L. Rowen, R. Yavich, S. Malev, and A. Kanel-Belov. Evaluations of non-commutative polynomials on algebras: methods and problems, and the L'vov-Kaplansky Conjecture. SIGMA, 16:071, 61 pages, 2020.

[157] L. H. Rowen. Graduate Algebra: Noncommutative View, volume 9. American Mathematical Society, Providence, RI, 2008.

[158] Sonia Rueda. Some algebras similar to the enveloping algebra of sl(2). Comm. Algebra, 3(30):1127-1152, 2002.

[159] G. Schwarz. Exotic algebraic group actions. CRAcad. Sci. Paris, 309:89-94, 1989.

[160] Estanislao Herscovich Sergio Chouhy and Andrea Solotar. Hochschild homology and cohomology of down-up algebras. J. Algebra, (498):102-128, 2018.

[161] I. R. Shafarevich. On some infinite-dimensional groups. ii. Izvestiya Rossi-iskoi Akademii Nauk. Seriya Matematicheskaya, 45(1):214-226, 1981.

[162] Y. Sharifi. Centralizers in Associative Algebras. PhD thesis, Simon Fraser University, 2013.

[163] I. Shestakov and U. Umirbaev. The Nagata automorphism is wild. Proceedings of the National Academy of Sciences, 100(22):12561-12563,2003.

[164] I. Shestakov and U. Umirbaev. Poisson brackets and two-generated sub-algebras of rings of polynomials. Journal of the American Mathematical Society, 17(1):181-196, 2004.

[165] I. Shestakov and U. Umirbaev. The tame and the wild automorphisms of polynomial rings in three variables. Journal of the American Mathematical Society, 17(1):197-227, 2004.

[166] S. P. Smith. A class of algebras similar to the enveloping algebra of sl(2). Trans. Amer. Math. Soc, 1(322):285-314,1990.

[167] Richard P. Stanley. Differential posets. J. Amer. Math. Soc, 4(1):919-961, 1988.

[168] Mariano Suarez-Alvarez and Quimey Vivas. Automorphisms and isomorphisms of quantum generalized weyl algebras. Journal of Algebra, (424):540-552, 2015.

[169] Moss E. Sweedler. Hopf algebras. W.A. Benjamin, New York, 1969.

[170] Matthew Towers. Poisson and Hochschild cohomology and the semiclas-sical limit. J. Noncommut. Geom, 3(9):665-696, 2015.

[171] Y. Tsuchimoto. Preliminaries on Dixmier conjecture. Mem. Fac. Sci. Kochi Univ. Ser. A Math, 24:43-59, 2003.

[172] Y. Tsuchimoto et al. Endomorphisms of Weyl algebra and p-curvatures. Osaka Journal of Mathematics, 42(2):435-452,2005.

[173] U. Umirbaev. On the extension of automorphisms of polynomial rings. Siberian Mathematical Journal, 36(4):787-791,1995.

[174] U. Umirbaev. The Anick automorphism of free associative algebras. Journalfür die reine und angewandte Mathematik (Crelles Journal), 2007(605):165-178, 2007.

[175] Michel Van den Bergh. On involutivity of p-support. Int. Math. Res. Not. IMRN, (15):6295-6304, 2015.

[176] A. Van den Essen. Polynomial Automorphisms: and the Jacobian Conjecture, volume 190. Birkhauser, 2012.

[177] W. Van der Kulk. On polynomial rings in two variables. Nieuw Arch. Wisk.(3), 1:33-41,1953.

[178] Yu S. Vernov and Melita Nikolaevna Mnatsakanova. Regular representations of ther-deformed heisenberg algebra. Theoretical and Mathematical Physics, 2(125):1531-1538, 2000.

[179] A. G. Vitushkin. On the homology of a ramified covering over C2. Mathematical Notes-New York, 64(5):726-731,1998.

[180] A. G. Vitushkin. Evaluation of the Jacobian of a rational transformation of C2 and some applications. Matematicheskie Zametki, 66(2):308-312,1999.

[181] S. Wang. A Jacobian criterion for separability. Journal of Algebra, 65(2):453-494, 1980.

[182] D. Wright et al. On the Jacobian conjecture. Illinois Journal of Mathematics, 25(3):423-440, 1981.

[183] A. V. Yagzhev. Algorithmic problem of recognizing automorphisms among endomorphisms of free associative algebras of finite rank. Siberian Mathematical Journal, 21(1):142-146, 1980.

[184] A. V. Yagzhev. Endomorphisms of free algebras. Siberian Mathematical Journal, 21(1):133-141, 1980.

[185] A. V. Yagzhev. Keller's problem. Siberian Mathematical Journal, 21(5):747-754, 1980.

[186] A. V. Yagzhev. Invertibility of endomorphism of free associative algebras. Mathematical Notes, 49(4):426-430, 1991.

[187] M. G. Zaidenberg. On exotic algebraic structures on affine spaces. In Geometric Complex Analysis, pages 691-714. World Scientific, 1996.

[188] James J. Zhang. A note on GK dimension of skew polynomial extensions. Proc. Amer. Math. Soc, 2(125):363-373, 1997.

[189] W. Zhang. Alternative proof of Bergman's centralizer theorem by quantization. Master's Thesis, Bar-Ilan University, 2017.

[190] A. N. Zubkov. Matrix invariants over an infinite field of finite characteristic. Siberian Mathematical Journal, 34(6):1059-1065,1993.

[191] A. N. Zubkov. A generalization of the Razmyslov-Procesi theorem. Algebra and Logic, 35(4):241-254,1996.