Representation theory of (modified) reflection equation algebra of 𝐺𝐿(𝑚|𝑛) type

Gurevich, D.; Pyatov, P.; Saponov, P.

doi:10.1090/S1061-0022-09-01045-0

Representation theory of (modified) reflection equation algebra of $GL(m|n)$ type
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by D. Gurevich, P. Pyatov and P. Saponov
Translated by: the authors

St. Petersburg Math. J. 20 (2009), 213-253

DOI: https://doi.org/10.1090/S1061-0022-09-01045-0

Published electronically: January 30, 2009

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Abstract:

Let $R:V^{\otimes 2}\to V^{\otimes 2}$ be a Hecke type solution of the quantum Yang–Baxter equation (a Hecke symmetry). Then, the Hilbert–Poincaré series of the associated $R$-exterior algebra of the space $V$ is the ratio of two polynomials of degrees $m$ (numerator) and $n$ (denominator).

Under the assumption that $R$ is skew-invertible, a rigid quasitensor category $\textrm {SW}(V_{(m|n)})$ of vector spaces is defined, generated by the space $V$ and its dual $V^*$, and certain numerical characteristics of its objects are computed. Moreover, a braided bialgebra structure is introduced in the modified reflection equation algebra associated with $R$, and the objects of the category $\textrm {SW}(V_{(m|n)})$ are equipped with an action of this algebra. In the case related to the quantum group $U_q(sl(m))$, the Poisson counterpart of the modified reflection equation algebra is considered and the semiclassical term of the pairing defined via the categorical (or quantum) trace is computed.

References

P. Akueson and D. Gurevich, Dual quasitriangular structures related to the Temperley-Lieb algebra, Lie groups and Lie algebras, Math. Appl., vol. 433, Kluwer Acad. Publ., Dordrecht, 1998, pp. 1–16. MR 1628806
A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. in Math. 64 (1987), no. 2, 118–175. MR 884183, DOI 10.1016/0001-8708(87)90007-7
Alexander Braverman and Dennis Gaitsgory, Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type, J. Algebra 181 (1996), no. 2, 315–328. MR 1383469, DOI 10.1006/jabr.1996.0122
Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 1300632
I. V. Cherednik, Factorizing particles on a half line, and root systems, Teoret. Mat. Fiz. 61 (1984), no. 1, 35–44 (Russian, with English summary). MR 774205
Ludwik Dąbrowski, Harald Grosse, and Piotr M. Hajac, Strong connections and Chern-Connes pairing in the Hopf-Galois theory, Comm. Math. Phys. 220 (2001), no. 2, 301–331. MR 1844628, DOI 10.1007/s002200100433
A. A. Davydov, Totally positive sequences and $R$-matrix quadratic algebras, J. Math. Sci. (New York) 100 (2000), no. 1, 1871–1876. Algebra, 12. MR 1774356, DOI 10.1007/BF02677498
Gustav W. Delius, Christopher Gardner, and Mark D. Gould, The structure of quantum Lie algebras for the classical series $B_l,C_l$ and $D_l$, J. Phys. A 31 (1998), no. 8, 1995–2019. MR 1628724, DOI 10.1088/0305-4470/31/8/012
J. Donin, Double quantization on coadjoint representations of semisimple Lie groups and their orbits, ArXiv: QA/9909160.
J. Donin, D. Gurevich, and S. Shnider, Double quantization on some orbits in the coadjoint representations of simple Lie groups, Comm. Math. Phys. 204 (1999), no. 1, 39–60. MR 1705663, DOI 10.1007/s002200050636
J. Donin and A. Mudrov, Explicit equivariant quantization on coadjoint orbits of $\textrm {GL}(n,\Bbb C)$, Lett. Math. Phys. 62 (2002), no. 1, 17–32. MR 1952112, DOI 10.1023/A:1021677725539
V. G. Drinfel′d, On quadratic commutation relations in the quasiclassical case, Mathematical physics, functional analysis (Russian), “Naukova Dumka”, Kiev, 1986, pp. 25–34, 143 (Russian). MR 906075
Nguyễn Phuong Dung and Phùng H\grcf{o} Hai, On the Poincaré series of quadratic algebras associated to Hecke symmetries, Int. Math. Res. Not. 40 (2003), 2193–2203. MR 1997298, DOI 10.1155/S107379280313022X
L. D. Faddeev and P. N. Pyatov, The differential calculus on quantum linear groups, Contemporary mathematical physics, Amer. Math. Soc. Transl. Ser. 2, vol. 175, Amer. Math. Soc., Providence, RI, 1996, pp. 35–47. MR 1402914, DOI 10.1090/trans2/175/03
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev, Quantization of Lie groups and Lie algebras, Algebra i Analiz 1 (1989), no. 1, 178–206 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 1, 193–225. MR 1015339
X. Gomez and S. Majid, Braided Lie algebras and bicovariant differential calculi over co-quasitriangular Hopf algebras, J. Algebra 261 (2003), no. 2, 334–388. MR 1966634, DOI 10.1016/S0021-8693(02)00580-X
D. I. Gurevich, Generalized translation operators on Lie groups, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 18 (1983), no. 4, 305–317 (Russian, with English and Armenian summaries). MR 723563
Dimitri Gourevitch, Équation de Yang-Baxter et quantification des cocycles, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 12, 845–848 (French, with English summary). MR 1058509
D. I. Gurevich, Algebraic aspects of the quantum Yang-Baxter equation, Algebra i Analiz 2 (1990), no. 4, 119–148 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 4, 801–828. MR 1080202
Dimitri Gurevich, Braided modules and reflection equations, Quantum groups and quantum spaces (Warsaw, 1995) Banach Center Publ., vol. 40, Polish Acad. Sci. Inst. Math., Warsaw, 1997, pp. 99–110. MR 1481738
D. Gurevich, R. Leclercq, and P. Saponov, Traces in braided categories, J. Geom. Phys. 44 (2002), no. 2-3, 251–278. MR 1969784, DOI 10.1016/S0393-0440(02)00076-1
D. Gurevich, R. Leclercq, and P. Saponov, $q$-index on braided non-commutative spheres, J. Geom. Phys. 53 (2005), no. 4, 392–420. MR 2125400, DOI 10.1016/j.geomphys.2004.07.007
D. I. Gurevich, P. N. Pyatov, and P. A. Saponov, The Cayley-Hamilton theorem for quantum matrix algebras of $\textrm {GL}(m|n)$ type, Algebra i Analiz 17 (2005), no. 1, 160–182 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 17 (2006), no. 1, 119–135. MR 2140677, DOI 10.1090/S1061-0022-06-00895-8
D. I. Gurevich, P. N. Pyatov, and P. A. Saponov, Quantum matrix algebras of $\textrm {GL}(m|n)$-type: the structure of the characteristic subalgebra and its spectral parametrization, Teoret. Mat. Fiz. 147 (2006), no. 1, 14–46 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 147 (2006), no. 1, 460–485. MR 2254713, DOI 10.1007/s11232-006-0054-0
D. Gurevich and P. Saponov, Quantum line bundles via Cayley-Hamilton identity, J. Phys. A 34 (2001), no. 21, 4553–4569. MR 1835952, DOI 10.1088/0305-4470/34/21/309
D. I. Gurevich and P. A. Saponov, On non-one-dimensional representations of the reflection equation algebra, Teoret. Mat. Fiz. 139 (2004), no. 1, 45–61 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 139 (2004), no. 1, 486–499. MR 2076908, DOI 10.1023/B:TAMP.0000022741.80096.b3
D. Gurevich and P. Saponov, Geometry of non-commutative orbits related to Hecke symmetries, Quantum groups, Contemp. Math., vol. 433, Amer. Math. Soc., Providence, RI, 2007, pp. 209–250. MR 2349624, DOI 10.1090/conm/433/08328
Simone Gutt and John Rawnsley, Traces for star products on symplectic manifolds, J. Geom. Phys. 42 (2002), no. 1-2, 12–18. MR 1894072, DOI 10.1016/S0393-0440(01)00053-5
Phung Ho Hai, Poincaré series of quantum spaces associated to Hecke operators, Acta Math. Vietnam. 24 (1999), no. 2, 235–246. MR 1710780
A. Isaev, Quantum groups and Yang–Baxter equations, Fiz. Èlementar. Chastits i Atom. Yadra 26 (1995), 1204–1263; English transl., Phys. Particles Nuclei 26 (1995), 501–526.
A. Isaev, O. Ogievetsky, and P. Pyatov, On quantum matrix algebras satisfying the Cayley-Hamilton-Newton identities, J. Phys. A 32 (1999), no. 9, L115–L121. MR 1678401, DOI 10.1088/0305-4470/32/9/002
A. P. Isaev and P. N. Pyatov, Covariant differential complexes on quantum linear groups, J. Phys. A 28 (1995), no. 8, 2227–2246. MR 1338072
S. M. Khoroshkin and V. N. Tolstoy, Universal $R$-matrix for quantized (super)algebras, Comm. Math. Phys. 141 (1991), no. 3, 599–617. MR 1134942
P. P. Kulish, Representations of $q$-Minkowski space algebra, Algebra i Analiz 6 (1994), no. 2, 195–205; English transl., St. Petersburg Math. J. 6 (1995), no. 2, 365–374. MR 1290824
P. P. Kulish and E. K. Sklyanin, Algebraic structures related to reflection equations, J. Phys. A 25 (1992), no. 22, 5963–5975. MR 1193836
Jiang-Hua Lu and Alan Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), no. 2, 501–526. MR 1037412
Volodymyr Lyubashenko and Anthony Sudbery, Quantum supergroups of $\textrm {GL}(n|m)$ type: differential forms, Koszul complexes, and Berezinians, Duke Math. J. 90 (1997), no. 1, 1–62. MR 1478542, DOI 10.1215/S0012-7094-97-09001-3
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Shahn Majid, Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995. MR 1381692, DOI 10.1017/CBO9780511613104
A. Mudrov, Characters of $\scr U_q(\textrm {gl}(n))$-reflection equation algebra, Lett. Math. Phys. 60 (2002), no. 3, 283–291. MR 1917138, DOI 10.1023/A:1016283527111
—, On quantization of Semenov-Tian-Shansky Poisson bracket on simple algebraic groups, ArXiv: QA/0412360.
O. Ogievetsky, Uses of quantum spaces, Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000) Contemp. Math., vol. 294, Amer. Math. Soc., Providence, RI, 2002, pp. 161–232. MR 1907189, DOI 10.1090/conm/294/04973
O. Ogievetsky and P. Pyatov, Lecture on Hecke algebras, Symmetries and Integrable Systems (Dubna, Russia, June 8–11, 1999), JINR, Dubna, D2,5-2000,218, pp. 39–88. (English); Preprint CPT-2000/P.4076 and MPI 01–40.
—, Orthogonal and symplectic quantum matrix algebras and Cayley–Hamilton theorem for them, ArXiv: QA/0511618.
P. Podleś, Quantum spheres, Lett. Math. Phys. 14 (1987), no. 3, 193–202. MR 919322, DOI 10.1007/BF00416848
Alexander Polishchuk and Leonid Positselski, Quadratic algebras, University Lecture Series, vol. 37, American Mathematical Society, Providence, RI, 2005. MR 2177131, DOI 10.1090/ulect/037
Marc A. Rieffel, Deformation quantization of Heisenberg manifolds, Comm. Math. Phys. 122 (1989), no. 4, 531–562. MR 1002830
P. A. Saponov, The Weyl approach to the representation theory of reflection equation algebra, J. Phys. A 37 (2004), no. 18, 5021–5046. MR 2065220, DOI 10.1088/0305-4470/37/18/008
Michael A. Semenov-Tian-Shansky, Dressing transformations and Poisson group actions, Publ. Res. Inst. Math. Sci. 21 (1985), no. 6, 1237–1260. MR 842417, DOI 10.2977/prims/1195178514
Albert Jeu-Liang Sheu, Quantization of the Poisson $\textrm {SU}(2)$ and its Poisson homogeneous space—the $2$-sphere, Comm. Math. Phys. 135 (1991), no. 2, 217–232. With an appendix by Jiang-Hua Lu and Alan Weinstein. MR 1087382
John R. Stembridge, A characterization of supersymmetric polynomials, J. Algebra 95 (1985), no. 2, 439–444. MR 801279, DOI 10.1016/0021-8693(85)90115-2
V. G. Turaev, Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673
Hans Wenzl, Hecke algebras of type $A_n$ and subfactors, Invent. Math. 92 (1988), no. 2, 349–383. MR 936086, DOI 10.1007/BF01404457
S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989), no. 1, 125–170. MR 994499
R. B. Zhang, Structure and representations of the quantum general linear supergroup, Comm. Math. Phys. 195 (1998), no. 3, 525–547. MR 1640999, DOI 10.1007/s002200050401