Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

 
 

 

Arens-Michael envelopes, homological epimorphisms, and relatively quasi-free algebras


Author: Alexei Yul'evich Pirkovskii
Translated by: Alex Martsinkovsky
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 69 (2008).
Journal: Trans. Moscow Math. Soc. 2008, 27-104
MSC (2000): Primary 46M18
DOI: https://doi.org/10.1090/S0077-1554-08-00169-6
Published electronically: November 19, 2008
MathSciNet review: 2549445
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We describe and investigate Arens-Michael envelopes of associative algebras and their homological properties. We also introduce and study analytic analogs of some classical ring-theoretic constructs: Ore extensions, Laurent extensions, and tensor algebras. For some finitely generated algebras, we explicitly describe their Arens-Michael envelopes as certain algebras of noncommutative power series, and we also show that the embeddings of such algebras in their Arens-Michael envelopes are homological epimorphisms (i.e., localizations in the sense of J. Taylor). For that purpose we introduce and study the concepts of relative homological epimorphism and relatively quasi-free algebra. The above results hold for multiparameter quantum affine spaces and quantum tori, quantum Weyl algebras, algebras of quantum $ (2\times 2)$-matrices, and universal enveloping algebras of some Lie algebras of small dimensions.


References [Enhancements On Off] (What's this?)

  • 1. V.A. Artamonov, Algebras of quantum polynomials, Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz. vol. 26, pp. 5-34, VINITI. Moscow, 2002. (Russian)
  • 2. W.B. Arveson, Subalgebras of $ C^*$-algebras, Acta Math. 123 (1969), 141-224. MR 0253059 (40:6274)
  • 3. W.B. Arveson, Subalgebras of $ C^*$-algebras, II, Acta Math. 128 (1972), no. 3-4, 271-308. MR 0394232 (52:15035)
  • 4. B. Blackadar and J. Cuntz, Differential Banach algebra norms and smooth subalgebras of $ C^*$-algebras , J. Operator Theory 26 (1991), no. 2, 255-282. MR 1225517 (94f:46094)
  • 5. P. Bonneau, M. Flato, M. Gerstenhaber, and G. Pinczon, The hidden group structure of quantum groups: strong duality, rigidity and preferred deformations, Comm. Math. Phys. 161 (1994), 125-156. MR 1266072 (95b:17016)
  • 6. N. Bourbaki, Éléments de mathématique, Algèbre, Chapitre 10, Algèbre homologique, Masson, Paris, 1980. MR 610795 (82j:18022)
  • 7. J. Cuntz and D. Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995), 251-289. MR 1303029 (96c:19002)
  • 8. J. Cuntz, Bivariante $ K$-Theorie für lokalkonvexe Algebren und der Chern-Connes-Charakter, Doc. Math. 2 (1997), 139-182. MR 1456322 (98h:19006)
  • 9. J. Cuntz, Cyclic theory and the bivariant Chern-Connes character, Noncommutative geometry, Lecture Notes in Math., vol. 1831, pp. 73-135, Springer, Berlin, 2004. MR 2058473 (2005e:58007)
  • 10. H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs, New Series, vol. 24, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 2000. MR 1816726 (2002e:46001)
  • 11. K. R. Davidson, Free semigroup algebras, A survey, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000). Oper. Theory Adv. Appl. 129, pp. 209-240, Birkhäuser, Basel, 2001. MR 1882697 (2003d:47104)
  • 12. K. R. Davidson and D. R. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras, Math. Ann. 311 (1998), 275-303. MR 1625750 (2001c:47082)
  • 13. K. R. Davidson and G. Popescu, Noncommutative disc algebras for semigroups, Canad. J. Math. 50 (1998), 290-311. MR 1618326 (99f:46105)
  • 14. E. E. Demidov, Modules over a quantum Weyl algebra, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1993, no. 1, 53-56, MR 1293939 (95e:17015)
  • 15. A. A. Dosiev, Algebras of holomorphic functions in elements of a nilpotent Lie algbera and Taylor localizations, Preprint.
  • 16. A. A. Dosiev, Homological dimensions of the algebra of entire functions of elements of a nilpotent Lie algebra, Funktsional. Anal. i Prilozhen. 37(1) (2003), 73-77. MR 1988010 (2004d:46054)
  • 17. W. Geigle and H. Lenzing, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991), 273-343. MR 1140607 (93b:16011)
  • 18. S. I. Gelfand and Y. I. Manin, Methods of homological algebra, Springer Monographs in Mathematics, second edition, Springer-Verlag, Berlin, 2003. MR 1950475 (2003m:18001)
  • 19. M. Gerstenhaber and S. D. Schack, Algebraic cohomology and deformation theory, Deformation theory of algebras and structures and applications (Il Ciocco, 1986). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 247, pp. 11-264, Kluwer Acad. Publ., Dordrecht, 1988. MR 981619 (90c:16016)
  • 20. K. R. Goodearl, Quantized coordinate rings and related Noetherian algebras, Proceedings of the 35th Symposium on Ring Theory and Representation Theory (Okayama, 2002), pp. 19-45, Symp. Ring Theory Represent. Theory Organ. Comm., Okayama, 2003. MR 1969455
  • 21. K. R. Goodearl and E. S. Letzter, Quantum $ n$-space as a quotient of classical $ n$-space, Trans. Amer. Math. Soc. 352 (2000), 5855-5876. MR 1781280 (2001e:16061)
  • 22. R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 65-222. MR 656198 (83j:58014)
  • 23. R. Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne, Lecture Notes in Mathematics, vol. 20, Springer-Verlag, Berlin, 1966. MR 0222093 (36:5145)
  • 24. A. Ya. Helemskii, Banach and locally convex algebras, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 1993. MR 1231796 (94f:46001)
  • 25. A. Ya. Helemskii, The homology of Banach and topological algebras, Mathematics and its Applications (Soviet Series), vol. 41, Kluwer Academic Publishers Group, Dordrecht, 1989. MR 1093462 (92d:46178)
  • 26. A. Ya. Helemskii, Homological methods in the holomorphic calculus of several operators in Banach space, after Taylor, Uspekhi Mat. Nauk 36 (1981), 127-172. MR 608943 (82e:47020)
  • 27. A. Ya. Helemskii, A certain class of flat Banach modules and its applications. Vestnik Moskov. Univ. Ser. I Mat. Meh. 27 (1972), no. 1, 29-36. MR 0295080 (45:4148)
  • 28. G. Hochschild, Relative homological algebra, Trans. Amer. Math. Soc. 82 (1956), 246-269. MR 0080654 (18:278a)
  • 29. D. A. Jordan, A simple localization of the quantized Weyl algebra, J. Algebra 174, (1995), 267-281. MR 1332871 (96m:16035)
  • 30. M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften, vol. 292, Corrected reprint of the 1990 original, Springer-Verlag, Berlin, 1994. MR 1074006 (92a:58132)
  • 31. C. Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR 1321145 (96e:17041)
  • 32. B. Keller, Derived categories and their uses, Handbook of algebra, M. Hazewinkel, ed., Amsterdam, North-Holland, 1996, vol. 1, pp. 671-701. MR 1421815 (98h:18013)
  • 33. E. Kissin and V. S. Shul'man, Differential properties of some dense subalgebras of $ C^*$-algebras, Proc. Edinburgh Math. Soc. (2) 37 (1994), 399-422. MR 1297311 (95j:46088)
  • 34. Y. A. Kopylov and V. I. Kuz'minov, Exactness of the cohomology sequence for a short exact sequence of complexes in a semiabelian category, Siberian Adv. Math. 13(3) (2003), 72-80. MR 2028407
  • 35. Lectures on $ q$-analogues of Cartan domains and associated Harish-Chandra modules, L. Vaksman (ed.), Kharkov, 2001. Preprint arXiv:math.QA/0109198.
  • 36. G. L. Litvinov, On double topological algebras and Hopf topological algebras, Trudy Sem. Vektor. Tenzor. Anal. 18 (1978), 372-375; English transl., Selecta Math. Soviet. 10(4) (1991), 339-343. MR 0504538 (80a:46024)
  • 37. D. Luminet, A functional calculus for Banach PI-algebras, Pacific J. Math. 125 (1986), 127-160. MR 860755 (88c:46060)
  • 38. D. Luminet, Functions of several matrices, Boll. Un. Mat. Ital. B. (7) 11 (1997), 563-586. MR 1479512 (98i:32007)
  • 39. S. Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, second edition, Springer-Verlag, New York, 1998. MR 1712872 (2001j:18001)
  • 40. Yu. I. Manin, Some remarks on Koszul algebras and quantum groups, Ann. Inst. Fourier (Grenoble) 37 (1987), 191-205. MR 927397 (89e:16022)
  • 41. J. C. McConnell and J. J. Pettit, Crossed products and multiplicative analogues of Weyl algebras, J. London Math. Soc. (2) 38 (1988), 47-55. MR 949080 (90c:16011)
  • 42. A. I. Markushevich, Theory of functions of a complex variable, vols. I, II, III, Chelsea Publishing Co., New York, 1977. MR 0444912 (56:3258)
  • 43. R. Meyer, Analytic cyclic cohomology, Ph.D. Thesis, Westfälische Wilhelms-Universität Münster, 1999. Preprint arXiv.org:math.KT/9906205.
  • 44. R. Meyer, Embeddings of derived categories of bornological modules, Preprint arXiv.org:math.FA/0410596.
  • 45. B. Mitiagin, S. Rolewicz, and W. Zelazko W., Entire functions in $ B_0$-algebras, Studia Math. 21 (1961/1962), 291-306. MR 0144222 (26:1769)
  • 46. A. Neeman and A. Ranicki, Noncommutative localization and chain complexes, I. Algebraic $ K$- and $ L$-theory, Preprint arXiv.org:math.RA/0109118.
  • 47. M. Pflaum and M. Schottenloher, Holomorphic deformations of Hopf algebras and applications to quantum groups, J. Geometry and Physics 28 (1988), 31-44. MR 1653122 (99k:58018)
  • 48. A. Yu. Pirkovskii, Stably flat completions of universal enveloping algebras, Dissertationes Math. (Rozprawy Math.) 441 (2006), 1-60. MR 2283621 (2007k:46131)
  • 49. A. Yu. Pirkovskii, Arens-Michael enveloping algebras and analytic smash products, Proc. Amer. Math. Soc. 134 (2006), 2621-2631. MR 2213741 (2007b:46135)
  • 50. G. Popescu, Non-commutative disc algebras and their representations, Proc. Amer. Math. Soc. 124 (1996), 2137-2148. MR 1343719 (96k:47077)
  • 51. G. Popescu, Noncommutative joint dilations and free product operator algebras, Pacific J. Math. 186 (1998), 111-140. MR 1665059 (2000k:46082)
  • 52. D. Proskurin and Yu. Samoĭlenko, Stability of the $ C\sp *$-algebra associated with twisted CCR, Algebr. Represent. Theory 5 (2002), 433-444. MR 1930972 (2003i:46059)
  • 53. F. Prosmans, Algèbre homologique quasi-abelienne, Mem. DEA, Univ. Paris 13, Juin 1995.
  • 54. F. Prosmans, Derived categories for Functional Analysis, Publ. Res. Inst. Math. Sci. 36 (2000), 19-83. MR 1749013 (2001g:46156)
  • 55. W. Pusz and S. L. Woronowicz, Twisted second quantization, Rep. Math. Phys. 27 (1989), 231-257. MR 1067498 (93c:81082)
  • 56. D. Quillen, Higher Algebraic $ K$-theory I, Lecture Notes in Math., vol. 341, pp. 85-147, Berlin, Springer, 1973. MR 0338129 (49:2895)
  • 57. Y. V. Radyno, Linear equations and bornology, Beloruss. Gos. Univ., Minsk, 1982. (Russian) MR 685429 (85d:46006)
  • 58. D. A. Raĭkov, Semiabelian categories, Dokl. Akad. Nauk SSSR 188 (1969), 1006-1009. MR 0255639 (41:299)
  • 59. W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics, second edition, McGraw-Hill, New York, 1991. MR 1157815 (92k:46001)
  • 60. H. H. Schaefer. Topological vector spaces, Third printing corrected, Graduate Texts in Mathematics, vol. 3, Springer-Verlag, New York, 1971. MR 0342978 (49:7722)
  • 61. J.-P. Schneiders, Quasi-abelian categories and sheaves, Mem. Soc. Math. Fr. (N.S.) No. 76 (1999). MR 1779315 (2001i:18023)
  • 62. J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955-1956), 1-42. MR 0082175 (18:511a)
  • 63. S. Sinel'shchikov and L. Vaksman, On $ q$-analogues of bounded symmetric domains and Dolbeault complexes, Math. Phys. Anal. Geom. 1 (1998), 75-100. MR 1687517 (2000f:58016)
  • 64. B. Stenström, Rings of quotients, Die Grundlehren der Mathematischen Wissenschaften, Band 217. Springer-Verlag, New York, 1975. MR 0389953 (52:10782)
  • 65. M. E. Sweedler, Hopf algebras, Benjamin, New York, 1969. MR 0252485 (40:5705)
  • 66. J. L. Taylor, Homology and cohomology for topological algebras, Adv. Math. 9 (1972), 137-182. MR 0328624 (48:6966)
  • 67. J. L. Taylor, A general framework for a multi-operator functional calculus, Adv. Math. 9 (1972), 183-252. MR 0328625 (48:6967)
  • 68. J. L. Taylor, Functions of several noncommuting variables, Bull. Amer. Math. Soc. 79 (1973), 1-34. MR 0315446 (47:3995)
  • 69. J. L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, Graduate Studies in Mathematics, vol. 46. American Mathematical Society, Providence, RI, 2002. MR 1900941 (2004b:32001)
  • 70. L. Vaksman, The maximum principle for ``holomorphic functions'' in the quantum ball, Mat. Fiz. Anal. Geom. 10(1) (2003), 12-28. MR 1937043 (2004h:46088)
  • 71. L. L. Vaksman and D. L. Shklyarov, Integral representations of functions in the quantum disk I, Mat. Fiz. Anal. Geom. 4(3) (1997), 286-308. MR 1615820 (99g:81098)
  • 72. L. Waelbroeck, Topological vector spaces and algebras, Lecture Notes in Mathematics, vol. 230, Springer-Verlag, Berlin, 1971. MR 0467234 (57:7098)
  • 73. W. Żelazko, Operator algebras on locally convex spaces, Proc. of the 5th International Conference on Topological Algebras and Applications, Athens, 2005, Contemporary Math. 427 (2007), 431-442. MR 2326378

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2000): 46M18

Retrieve articles in all journals with MSC (2000): 46M18


Additional Information

Alexei Yul'evich Pirkovskii
Affiliation: Russian Peoples’ Friendship University, 117198 Moscow, Russia
Email: pirkosha@online.ru; pirkosha@sci.pfu.edu.ru

DOI: https://doi.org/10.1090/S0077-1554-08-00169-6
Published electronically: November 19, 2008
Additional Notes: The author was supported by the RFFI grant No. 05-01-00982 and No. 05-01-00001 and the President of Russia’s grant MK-2049.2004.1
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society