Abstract
This article describes recent work on new classes of non-commutative algebras which have been dubbed quantum algebras, quantum groups, and quantized enveloping algebras. The basic ring theoretic properties are described, and a number of questions and problems are raised.
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References
H.H. Andersen, The Linkage Principle and the sum formula for quantum groups. Preprint, 1989.
H. H. Anderson and P. Polo Representations of quantum algebras. Preprint, 1990.
M. Artin and W. Schelter Graded algebras of global dimension 3n Adv. Math. 66 (1987), 171–216.
M. Artin, J. Tate and M. van den Bergh Some algebras associated to automorphisms of curves. Preprint, 1989.
M. Artin, J. Tate and M. van den Bergh Modules over regular algebras of dimension 3. Preprint, 1989.
M. Artin and M. van den Bergh Twisted Homogeneous Coordinate Rings. Preprint, 1990.
J. Backelin and R. Froberg Koszul algebras, Veronese subrings and rings with linear resolution Revue Roumaine de Math. Pures et Appl. 30 (1985), 85–97.
R. J. Baxter, “Exactly solved models in statistical mechanics,” Academic Press, New York, 1982.
A. Beilinson and V. Ginsburg, Mixed categories, Ext-duality and Representations (results and conjectures). Preprint.
A. A. Belavin Discrete Groups and the Integrability of Quantum Systems Func. Anal, and its Appl. 14 (1980), 260–267.
A. A. Belavin and V. G. Drinfeld On the solutions of the classical Yang- Baxter equations, Func. Anal, and Appl. 16 (1982), 159–180.
A. A. Belavin and V. G. Drinfeld Triangle Equations and simple Lie algebras, Sov. Sci. Rev. C4 (1984).
G. Bergman Diamond Lemma for Ring Theory, Adv. Math. 29 (1978), 178–218.
K. Bragiel Twisted SU(3) group to appear.
I. V. Cherednik Some finite dimensional representations of generalized Sklyanin algebras, Func. Anal, and Appl 19 (1985), 77–79.
R. Dipper and S. Donkin Quantum GL. Preprint.
P. Doubilet and G. C. Rota Skew-symmetric Invariant Theory Adv. Math. 21 (1976), 196–203.
V. G. Drinfeld Hamiltonian structures on Lie groups, Lie bialgebras, and the geometric meaning of the Yang-Baxter equations, Sov. Math. Dokl. 32 (1985), 254–258.
V. G. Drinfeld Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985), 254–258.
V. G. Drinfeld Degenerate affine Hecke algebras and Yangians, Func. Anal, and Appl. 20 (1986), 58–60.
V. G. Drinfeld Quantum Groups Proc. Int. Cong. Math.; Berkeley, 1 (1986), 798–820.
V. G. Drinfeld A new realisation of Yangians and quantised affine algebras, Sov. Math. Dokl. 36/296 (1988), 212–216.
V. G. Drinfeld On quadratic commutation relations in the quasi-classic limit, Mat. Fizika Funkc. Analiz. Kiev, 25–33. (in Russian).
J. Du, B. Parshall and J-P. Wang, Two parameter quantum linear groups and the Hyperbolic invariance of q-Schur algebra. Preprint, 1990.
L. D. Faddeev, N. Y. Reshetikhin and L. A. Takhtajan Quantization of Lie groups and Lie algebras. Preprint LOMI
L. D. Faddeev, and L. A. Takhtajan Liouville model on the lattice, in “Lecture Notes in Physics, No. 246,” Springer-Verlag, 1986, pp. 166–178.
M. Gerstenhaber On the deformation of rings and algebras Ann. Math. 79 (1964), 59–103.
M. Gerstenhaber and S. D. Schack Quantum groups as deformations of Hopf algebras Proc. Nat. Acad. Sci. 87 (1990), 478–481.
T. Hayashi Q-analogues of Clifford and Weyl algebras. Spinor and oscillator representations of quantum enveloping algebras. Preprint, Nagoya, 1990.
M. Jimbo A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Mat. Phys. 10 (1985), 63–69.
M. Jimbo, Quantum R matrix for the generalised Toda system, Commun. Math. Phys. 102 (1986), 537–547.
M. Jimbo A q-analogue of U(&l(n + 1)), Hecke algebra and the Yang-Baxter equation Lett. Math. Phys. 11 (1986), 247–252.
Naihuan Jing, Mo-Lin Ge, Yong-Shi Wu New quantum group associated with a “non-standard” Braid group representation. Preprint, 1990.
V. F. R. Jones Polynomial invariants of knots via von Neumann algebras Bull. Amer. Math. Soc. 12 (1985), 103–111.
V. F. R. Jones Hecke algebra representations of braid groups, and link polynomials, Ann. Math. 126 (1987), 335–388.
V. F. R. JonesOn knot invariants related to some statistical mechanical models.
A. A. Kirillov and N. Yu. ReshetikhinThe Yangians, Bethe Ansatz and combinatorics Lett. Math. Phys. 12 (1986), 199–208.
A. A. Kirillov and N. Yu. ReshetikhinRepresentations of the algebra U q (sl(2)), q-orthogonal polynomials and invariants of links. LOMI preprint (1988).
T. KohnoMonodromy representations of Braid groups and Yang-Baxter equations, Ann. Inst. Fourier 37 (1987), 139–160.
Y. Kosmann-SchwarzbachPoisson-Drinfeld groups; Proc. Oberwolfach Conf. on non-linear evolution equations; M. Albowitz, B. Fuchsteiner, M. Kruskal ed. World Scientific Publ. (1986).
P. P. Kulish and N. ReshetikhinQuantum linear problem for the Sine-Gordon equation and higher representations, J. Sov. Math. 23 (1983), 2435–2441.
P. P. Kulish, N. Reshetikhin, and E. K. SklyaninYang-Baxter equation and representation theory Lett. Math. Phys. 5 (1981), 393–403.
P. P. Kulish and E. K. SklyaninSolutions of the Yang-Baxter equation J. Sov. Math. 19 (1982), 1596–1620.
C. LofwallOn the subalgebra generated by the 1-dimensional elements in the Yoneda Ext-algebra, Springer LNM 1183 (1988), 291–338.
G. LusztigQuantum deformations of certain simple modules over enveloping algebras, Adv. Math. 70 (1988), 237–249.
G. LusztigModular representations and quantum groups, Contemp. Math. 82 (1989), 59–78.
G. Lusztig, Finite dimensional Hopf algebras arising from quantum groups. Preprint.
G. Lusztig, Quantum groups at roots of 1. Preprint.
G. LusztigCanonical bases arising from quantized enveloping algebras. Preprint, 1990.
S. MajidQuasitriangular Hopf Algebras and Yang-Baxter equations, Inter. J. Mod. Phys. 5 (1990), 1–91.
S. Majid and Ya. S. SoibelmanRank of quantized enveloping algebras and modular functions. Preprint, 1990
Yu. I. ManinSome remarks on Koszul algebras and Quantum groups Ann. Inst. Fourier 37 (1987), 191–205.
Yu. I. Manin, “Quantum groups and Non-commutative geometry,” Les Publ, du Centre de Récherches Math., Université de Montreal, 1988.
T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi, M. UenoRepresentations of Quantum groups and a q-analogue of orthogonal polynomials, C. R. Acad. Sci. Paris 307 (1988), 559–564.
J. C. Monnell and J. J. PettitCrossed products and multiplicative analogues of Weyl algebras J. Lond. Math. Soc. 38 (1988), 47–55.
S. Montgomery and S. P. SmithSkew derivations and U q (sl(2)) Israel J. Math., to appear.
M. Noumi, H. Yamada and K. MimachiFinite dimensional representations of the quantum group GL q (n + 1,C) and the zonal spherical functions on U q (n)\U q (n + 1). Preprint, 1989.
A. V. Odesskii and B. L. FeiginSklyanin algebras associated with an elliptic curve. Preprint (1988).
A. V. Odesskii and. B. L. FeiginElliptic Sklyanin Algebras Fune. Anal. Appl. 23 (1989), 45–54.
G. I. OlshanskiiYangians and universal enveloping algebras LOMI 164 (1987), 142–150.
B. Parshall and J-P. WangQuantum Linear Groups I. Preprint. University of Virginia (1989).
B. Parshall and J-P. WangQuantum Linear Groups II. Preprint.University of Virginia (1989).
P. PodlesQuantum Spheres, Lett. Math. Phys. 14 (1987), 193–202.
S. B. PriddyKoszul resolutions Trans. Amer. Math. Soc. 152 (1970), 39–60.
N. Reshetikhin, Theoret. Math. Phys. 63 (1985).
N. ReshetikhinQuantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I., LOMI (1988,), E-4–87. Preprint. II., LOMI (1988), E-17–87. Preprint.
N. Reshetikhin and V. G. TuraevInvariants of 3-manifolds via link polynomials and quantum groups, MSRI (1989). Preprint.
C. M. RingelHall Algebras and Quantum Groups. Preprint. Bielefeld (1989).
M. RossoComparaison des groupes SU(2) quantiques de Drinfeld et de Woronowiez, C. R. Acad. Sci. Paris 304 (1987), 323–326.
M. RossoRepresentations irreducibles de dimension finie du q-analogue de V algebre enveloppante d’une algebre de Lie semisimple, C. R. Acad. Sci. Paris 305, 587–590.
M. RossoFinite dimensional representations of the quantum analog of the enveloping algebra of a complex semisimple Lie algebra, Comm. Math. Phys. 117, 581–593.
M. RossoGroupes quantiques et modeles à vertex de V. Jones en theorie des noeuds C. R. Acad.Sci. Paris 307 (1988), 207–210.
M. RossoAn analogue of the PBW theorem and the universal R-matrix for U h (sl(n + 1)). Preprint. Palaiseau (1989).
M. RossoAnalogues de la forme de Killing et du théorème d’Harish-Chandra pour les groupes quantiques. Preprint. Palaiseau (1989).
A. N. Rudakov and I. R. ShafarevichIrreducible representations of a simple three dimensional Lie algebra over a field of finite characteristic, Math. Notes 2 (1968), 760–767.
E. K. SklyaninSome algebraic structures connected with the Yang-Bctxter equation, Fune. Anal, and Appl. 16 (1982), 263–270.
E. K. SklyaninSome algebraic structures connected with the Yang-Baxter equation. Representations of quantum algebras, Func. Anal, and Appl. 17 (1983), 273–284.
E. K. SklyaninAn algebra generated by quadratic relations, Usp. Mat. Nauk. 40 (1985), 214.
S. P. Smith and J. T. StaffordRegularity of the 4-dimensional Sklyanin algebra. in preparation.
Ya. S. Soibelman, Irreducible representations of the functional algebra of quantised SU(n) and the Schubert cells, Func. Anal, and Appl. (to appear).
E. Taft and J. TowberQuantum deformation of flag schemes and Grassman schemes I, a q-deformation of the shape algebra for GL(n). Preprint (1989).
M. TakeuchiQuantum Orthogonal and Symplectic Groups and their embedding into Quantum GL(n), Proc. Japan Acad. 65 (1989), 55–58.
M. TakeuchiMatrix Bialgebras and Quantum Groups. Preprint. Tsukuba (1989).
M. Takeuchi, A two parameter quanitization of GL(n); Summary.
L. A. TakhtajanQuantum Groups and Integrable Models. Preprint. Steklov Inst. (1988).
L. A. TakhtajanNoncommutative homology of quantum tori, Func. Anal. Appl. 23 (1989), 147–149.
T. TanisakiHarish-Chandra isomorphisms for Quantum algebras. Preprint, Osaka Univ. (1989).
T. TanisakiFinite dimensional representations of quantum groups. Preprint, Osaka Univ. (1989).
V. G. TuraevThe Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), 527–553.
L. L. Vaksman and S. Ya SoibelmanThe algebra of functions on quantised SU(2), Func. Anal, and Appl. 22 (1988), 170–181.
J-L. VerdierGroupes Quantiques Seminaire Bourbaki (1986–87,). No. 685.
H. WenzlBraid group representations and the quantum Yang-Baxter equation. Preprint.
S. L. WoronowiczTwisted SU(2)-group. An example of a non-commutative differential calculus, Publ. R.I.M.S., Kyoto Univ. 23 (1987), 117–181.
S. L. WoronowiczCompact Matrix pseudogroups, Comm. Math. Phys. III (1987), 613–615.
S. L. WoronowiczTannaka-Krein duality for compact Matrix pseudogroups, Invent. Math. 93 (1988), 35–76.
Xi, NanhuaRepresentations of Finite Dimensional Hopf algebras arising from Quantum Groups. Preprint (1989).
H. YamaneA Poincare-Birkhoff-Witt Theorem for the quantum group of type A, Proc. Japan Acad. 64 (1988), 385–386.
H. YamaneA PBW Theorem for quantized universal enveloping algebras of type A N Publ. RIMS Kyoto 25 (1989), 503–520.
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Smith, S.P. (1992). Quantum Groups: An Introduction and Survey for Ring Theorists. In: Montgomery, S., Small, L. (eds) Noncommutative Rings. Mathematical Sciences Research Institute Publications, vol 24. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9736-6_6
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