Large commutative subalgebras of quantum algebras. (English. Russian original) Zbl 1100.16039
J. Math. Sci., New York 134, No. 1, 1879-1909 (2006); translation from Sovrem. Mat. Prilozh. 18, 98-127 (2004).
Let \(Q_{n,r}\) be an associative algebra generated over a field \(k\) by elements \(x_1^{\pm 1},\dots,x_r^{\pm 1},x_{r+1},\dots,x_n\) with defining relations \(x_ix_j=q_{ij}x_jx_i\), where \(q_{ij}\in k^*\). Suppose that \(A\) is an \(\mathbb{N}^n_0\)-filtered associative algebra with respect to some semigroup order on \(\mathbb{N}^n_0\) such that the associated graded algebra is isomorphic to \(Q_{n,0}\). Then the transcendence degree of \(A\) does not exceed the maximal cardinality of a system of independent elements of a specific base of \(A\) whose images commute in \(Q_{n,0}\). It follows that the maximal cardinalities of commuting independent elements in \(Q_{n,0}\) and of similar monomials coincide. The center of \(Q_{n,0}\) is generated by finitely many monomials. There are given applications to the estimation of the transcendence degrees of some quantum algebras.
Reviewer: Vyacheslav A. Artamonov (Moskva)
MSC:
| 16W35 | Ring-theoretic aspects of quantum groups (MSC2000) |
| 16W70 | Filtered associative rings; filtrational and graded techniques |
| 16S32 | Rings of differential operators (associative algebraic aspects) |
| 16S15 | Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) |
Keywords:
quantum algebras; filtered algebras; associated graded algebras; transcendence degrees; commuting independent elements; centersReferences:
| [1] | J. Alev and F. Dumas, ”Automorphismes de certaines completes de corps de Weyl quantiques,” Collect. Math., 46, Nos. 1–2, 1–9 (1995). · Zbl 0842.16022 |
| [2] | J. Alev and F. Dumas, ”Corps de Weyl mixtes,” Bol. Acad. Nac. Cien., 65, 29–43 (2000). · Zbl 1006.16019 |
| [3] | J. Alev and F. Dumas, ”Sur le corps des fractions de certaines algebres quantiques,” J. Algebra, 170, No.1, 229–265. · Zbl 0820.17015 |
| [4] | V. A. Artamonov and P. M. Cohn, ”The skew field of rational functions on the quantum plane,” J. Math. Sci., 93, No.6, 824–829 (1999). · Zbl 0928.16029 · doi:10.1007/BF02366342 |
| [5] | V. A. Artamonov, ”General quantum polynomials: irreducible modules and Morita-equivalence,” Izv. Ross. Akad. Nauk, Ser. Mat., 63, No.5, 847–880 (1999). · Zbl 0974.16019 |
| [6] | V. A. Artamonov, ”The skew field of rational quantum functions,” Russ. Math. Surv., 54, No.4, 825–827 (1999). · Zbl 0976.16016 · doi:10.1070/RM1999v054n04ABEH000181 |
| [7] | V. A. Artamonov and R. Wisbauer, ”Homological properties of quantum polynomials,” Algebras Represent. Theory, 4, No.3, 219–247 (2001). · Zbl 1019.16014 · doi:10.1023/A:1011458821831 |
| [8] | V. A. Artamonov, ”Pointed Hopf algebras acting on quantum polynomials,” J. Algebra, 259, 323–352 (2003). · Zbl 1023.16028 · doi:10.1016/S0021-8693(02)00571-9 |
| [9] | V. A. Artamonov, ”Actions of quantum groups on quantum spaces,” Vestn. Mosk. Univ., Ser. 1, Mat., Mekh., No. 3, 13–17 (2003). · Zbl 1055.16040 |
| [10] | M. Artin, W. Schelter, and J. Tate, ”Quantum deformations of GL n ,” Commun. Pure Appl. Math., 44, 879–895 (1991). · Zbl 0753.17015 · doi:10.1002/cpa.3160440804 |
| [11] | K. A. Brown and K. R. Goodearl, Lectures on Algebraic Quantum Groups, Adv. Cour. Math., Birkhauser, Barcelona (2002). · Zbl 1027.17010 |
| [12] | C. J. B. Brookes, ”Crossed products and finitely presented groups,” J. Group Theory, 3, 433–444 (2000). · Zbl 0978.16028 · doi:10.1515/jgth.2000.034 |
| [13] | C. J. B. Brookes and J. R. J. Groves, ”Modules over crossed products of a division ring by a free Abelian group, I,” J. Algebra, 229, 24–54 (2000). · Zbl 0958.16028 · doi:10.1006/jabr.1999.8194 |
| [14] | P. M. Cohn, ”Centralisateur dans les corps libres,” in: Series formelles en variables non commutatives et applications (ed. J. Berstel), Vieux-Boucau les Bains (Landes), Ecole de Printemps d’Informatique Theorique (1978), pp. 45–54. · Zbl 0402.16003 |
| [15] | E. E. Demidov, ”Modules over a quantum Weyl algebra,” Mosc. Univ. Math. Bull., 48, 49–51 (1993). · Zbl 0831.17005 |
| [16] | E. E. Demidov, ”Some aspects of the theory of quantum groups,” Russ. Math. Surv., 48, No.6, 41–79 (1993). · Zbl 0807.52017 · doi:10.1070/RM1993v048n06ABEH001091 |
| [17] | J. Dixmier, Enveloping Algebras, North Holland, Amsterdam (1977). · Zbl 0346.17010 |
| [18] | V. G. Drinfel’d, ”Quantum groups,” J. Sov. Math., 41, No.2, 898–915 (1988). · Zbl 0641.16006 · doi:10.1007/BF01247086 |
| [19] | L. D. Faddeev and L. A. Takhtajan, ”A Liouville model on the lattice,” Lect. Notes Math. Phys., 24, 166–179 (1986). · doi:10.1007/3-540-16452-9_10 |
| [20] | L. D. Faddeev, N. Yu. Reshetikhin, and L. A. Takhtajan, ”Quantization of Lie groups and Lie algebras,” Leningr. Math. J., 1, 193–225 (1990). · Zbl 0715.17015 |
| [21] | I. M. Gelfand and A. A. Kirillov, ”Sur les corps lies aux algebres enveloppantes des algebres de Lie,” Publ. Math. IHES, 31, 5–19 (1966). · Zbl 0144.02104 |
| [22] | K. R. Goodearl, ”Quantized coordinate rings and related Noetherian algebras,” Expository paper for Proc. 35th Symp. Ring Theory and Representation Theory, Okayama (2002). |
| [23] | A. Joseph, ”On the prime and primitive spectra of the algebra of functions on a quantum group,” J. Algebra, 169, 441–511 (1994). · Zbl 0814.17013 · doi:10.1006/jabr.1994.1294 |
| [24] | A. Joseph, Quantum Groups and Their Primitive Ideals, Erg. Math. (3), 29, Springer-Verlag, Berlin (1995). · Zbl 0808.17004 |
| [25] | P. P. Kulish and N. Yu. Reshetikhin, ”Quantum linear problem for the sine-Gordon equation and higher representations,” Zap. Nauch. Sem. St. Peterb. Otd. Mat. Inst. Steklova, 101, 101–110 (1981). |
| [26] | S. Lang, Algebra, Springer-Verlag, New York (1986). |
| [27] | G. Maltsiniotis, Calcul differenciel quantique, Group de travail, Univ. Paris VII (1992). |
| [28] | Yu. I. Manin, ”Some remarks on Koszul complex and quantum groups,” Ann. Inst. Fourier, 37, 191–205 (1987). · Zbl 0625.58040 |
| [29] | Yu. I. Manin, Quantum Groups and Non-Commutative Geometry, Publ. Centre de Recherches Math., Univ. de Montreal (1988). · Zbl 0724.17006 |
| [30] | Yu. I. Manin, ”Multiparametric quantum deformation of the general linear supergroup,” Commun. Math. Phis., 123, 163–175 (1989). · Zbl 0673.16004 · doi:10.1007/BF01244022 |
| [31] | Yu. I. Manin, Topics in Non-Commutative Geometry, Princeton Univ. Press (1991). |
| [32] | J. C. McConnell and J. J. Pettit, ”Crossed products and multiplicative analogues of Weyl algebras,” J. London Math. Soc., 38, No.2, 47–55 (1988). · Zbl 0652.16007 · doi:10.1112/jlms/s2-38.1.47 |
| [33] | J. C. McConnel and J. C. Robson, Noncommutative Noetherian Rings, Wiley Interscience Publ. (1987). |
| [34] | A. V. Odesskii, ”Elliptic algebras,” Russ. Math. Surv., 57, No.6, 1127–1162 (2002). · Zbl 1062.16035 · doi:10.1070/RM2002v057n06ABEH000573 |
| [35] | A. V. Odesskii and B. L. Feigin, ”Elliptic Sklyanin algebras,” Funct. Anal. Appl., 23, No.3, 207–214 (1989). · Zbl 0713.17009 · doi:10.1007/BF01079526 |
| [36] | A. N. Panov, ”Skew fields of twisted rational functions and the skew field of rational functions on GL q (n, K),” St. Petersburg Math. J., 7, 129–143 (1996). |
| [37] | A. N. Panov, ”Skew fields of fractions of quantum solvable algebras,” J. Algebra, 236, 110–121 (2001). · Zbl 0966.17008 · doi:10.1006/jabr.2000.8463 |
| [38] | A. N. Panov, ”Stratification of the prime spectrum of quantum solvable algebras,” Commun. Algebra, 29, No.9, 3809–3827 (2001). · Zbl 1056.16038 · doi:10.1081/AGB-100105976 |
| [39] | A. N. Panov, ”Quantum solvable algebras, ideals, and representations at root of 1,” Transformation Groups, 7, No.4, 379–402 (2002). · Zbl 1039.17018 · doi:10.1007/s00031-002-0018-x |
| [40] | R. Resco, ”Transcendental division algebras and simple noetherian rings,” Isr. J. Math., 32, Nos. 2–3, 236–256 (1979). · Zbl 0404.16012 · doi:10.1007/BF02764919 |
| [41] | A. Sadbery, ”Consistent multiparameter quantization of GL(n),” J. Phys., 23, 697–704 (1990). |
| [42] | E. K. Sklyanin, ”Some algebraic structures connected to the Yang-Baxter equation,” Funct. Anal. Appl., 16, No.4, 27–34 (1982). |
| [43] | E. K. Sklyanin, ”Some algebraic structures connected to the Yang-Baxter equation. Representations of quantum algebras,” Funct. Anal. Appl., 17, No.4, 273–284 (1983). · Zbl 0536.58007 · doi:10.1007/BF01076718 |
| [44] | Y. S. Soibelman and L. L. Vaksman, ”Algebra of functions on the quantum group SU(2),” Funct. Anal. Appl., 22, 170–181 (1988). · Zbl 0679.43006 |
| [45] | Ya. S. Soibelman, ”Irreducible representations of the function algebra on the quantum group SU(n),” Sov. Math. Dokl., 40, 34–38 (1990). |
| [46] | Ya. S. Soibelman, ”The algebra of functions on a compact quantum group and its representations,” Leningr. Math. J., 2, 161–178 (1991). |
| [47] | F. van Oystaeyen, Algebraic Geometry for Associative Algebras, Marcel Dekker, New York (2000). · Zbl 0986.16014 |
| [48] | S. L. Woronovicz, ”Compact matrix pseudogroups,” Commun. Math. Phys., 111, 613–665 (1987). · Zbl 0627.58034 · doi:10.1007/BF01219077 |
| [49] | S. A. Zelenova, ”Commutative subalgebras of the ring of quantum polynomials and of the skew field of quantum Laurent series,” Mat. Sb., 192, No.3, 375–384 (2001). · Zbl 1014.16041 · doi:10.1070/SM2001v192n03ABEH000550 |
| [50] | S. A. Zelenova, ”Transcendence degree of commutative subalgebras of quantum algebras,” in: V Int. Conf. ”Algebra and Number Theory: Modern Problems and Applications,” Abstracts, Russia, Tula, May, 19–24 (2003), pp. 116–117. |
| [51] | S. A. Zelenova, ”Certain commutative subalgebras of a quantum polynomial algebra are finitely generated,” Russ. Math. Surv., 58, No3, 622–623 (2003). · Zbl 1064.16044 · doi:10.1070/RM2003v058n03ABEH000630 |
| [52] | S. A. Zelenova, ”Commutative subalgebras of quantum algebras,” Mat. Zametki, 75, No.2, 208–221 (2004). · Zbl 1111.16042 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
