Abstract
We suggest a generalization of Pontryagin duality from the category of commutative, complex Lie groups to the category of (not necessarily commutative) Stein groups with algebraic connected component of identity. In contrast to the other similar generalizations, in our approach the enveloping category consists of Hopf algebras (in a proper symmetrical monoidal category).
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References
S. S. Akbarov, “Stereotype spaces, algebras, homologies: An outline,” in: A. Ya. Helemskii, ed., Topological Homology, Nova Science Publishers (2000), pp. 1–29.
S. S. Akbarov, “Pontryagin duality in the theory of topological vector spaces and in topological algebra,” J. Math. Sci., 113, No. 2, 179–349 (2003).
S. S. Akbarov, “Pontryagin duality and topological algebras,” in: K. Jarosz and A. Soltysiak, eds., Topological Algebras, Their Applications and Related Topics, Vol. 67, Banach Center Publications (2005), pp. 55–71.
V. A. Artamonov, V. N. Salii, L. A. Skornyakov, L. N. Shevrin, and E. G. Shul’geifer, General Algebra [in Russian], Vol. 2, Nauka, Moscow (1991).
N. Bourbaki, Espaces Vectoriels Topologiques. Chaps. I–V, Hermann, Paris.
N. Bourbaki, Topologie Générale. Chaps. III–VIII, Hermann, Paris.
K. Brauner, “Duals of Fréchet spaces and a generalization of the Banach–Dieudonné theorem,” Duke Math. J., 40, No. 4, 845–855 (1973).
V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press (1995).
A. Van Daele, “Dual pairs of Hopf ∗-algebras,” Bull. London Math. Soc., 25, 209–230 (1993).
A. Van Daele, “Multiplier Hopf algebras,” Trans. Amer. Math. Soc., 342, 917–932 (1994).
A. Van Daele, “An algebraic framework for group duality,” Adv. Math., 140, 323–366 (1998).
A. Van Daele, The Haar Measure on Some Locally Compact Quantum Groups, http://arXiv.org/abs/math/0109004v1 (2001).
S. Dăscălescu, C. Năstăsescu, and Ş. Raianu, Hopf Algebras, Marcel Dekker (2001).
M. Enock and J.-M. Schwartz, “Une dualité dans les algèbres de von Neumann,” Note C. R. Acad. Sci. Paris, 277, 683–685 (1973).
M. Enock and J.-M. Schwartz, “Une catégorie d’algèbres de Kac,” Note C. R. Acad. Sci. Paris, 279, 643–645 (1974).
M. Enock and J.-M. Schwartz, “Une dualité dans les algèbres de von Neumann,” Supp. Bull. Soc. Math. France Mém., 44, 1–144 (1975).
M. Enock and J.-M. Schwartz, Kac Algebras and Duality of Locally Compact Groups, Springer (1992).
H. Grauert and R. Remmert, Theory of Stein Spaces, Springer (1977).
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 2, Springer (1963).
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press (1985).
H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart (1981).
S. Kakutani and V. Klee, “The finite topology of a linear space,” Arch. Math., 14, No. 1, 55–58 (1963).
C. Kassel, Quantum Groups, Springer (1995).
M. Krein, “A principle of duality for bicompact groups and quadratic block algebras,” Dokl. Akad. Nauk SSSR, 69, 725–728 (1949).
J. Kustermans, W. Pusz, P. M. So_ltan, S. Vaes, A. Van Daele, L. Vainerman, and S. L. Woronowicz, “Locally compact quantum groups,” in: P. M. Hajak, ed., Quantum Symmetry in Noncommutative Geometry, Locally Compact Quantum Groups. Lect. Notes School/Conf. on Noncommutative Geometry and Quantum Groups, Warsaw, 2001, Banach Center Publications, to appear.
J. Kustermans and S. Vaes, “Locally compact quantum groups,” Ann. Sci. École Norm. Sup., 4è sér., 33, 837–934 (2000).
B. Ya. Levin, Lectures on Entire Functions, Amer. Math. Soc. (1996).
S. MacLane, Categories for the Working Mathematician, Springer, Berlin (1971).
K.-H. Neeb, Holomorphy and Convexity in Lie Theory, Walter de Gruyter (2000).
A. Pietsch, Nukleare Lokalkonvexe Raume, Akademie, Berlin (1965).
A. Yu. Pirkovskii, “Arens–Michael envelopes, homological epimorphisms, and relatively quasifree algebras,” Tr. Mosk. Mat. Obshch., 69, 34–123 (2008).
L. Pontrjagin, “The theory of topological commutative groups,” Ann. Math., 35, No. 2, 361–388 (1934).
W. Rudin, Functional Analysis, McGraw-Hill (1973).
N. Saavedra Rivano, Catègories Tannakiennes, Lect. Notes Math., Vol. 265, Springer (1972).
P. Schauenburg, “On the braiding on a Hopf algebra in a braided category,” New York J. Math., 4, 259–263 (1998).
B. V. Shabat, Introduction to Complex Analysis [in Russian], Vol. I, Nauka, Moscow (1985).
B. V. Shabat, Introduction to Complex Analysis [in Russian], Vol. II, Nauka, Moscow (1985).
H. H. Shaeffer, Topological Vector Spaces, Macmillan (1966).
M. F. Smith, “The Pontrjagin duality theorem in linear spaces,” Ann. Math., 56, No. 2, 248–253 (1952).
R. Street, Quantum Groups: A Path to Current Algebra, Australian Math. Soc. Lect. Ser., No. 19, Cambridge (2007).
M. E. Sweedler, “Cocommutative Hopf algebras with antipode,” Bull. Amer. Math. Soc., 73, No. 1, 126–128 (1967).
J. L. Taylor, Several Complex Variables with Connections to Algebraic Geometry and Lie Groups, Grad. Stud. Math., Vol. 46, Amer. Math. Soc., Providence (2002).
L. I. Vainerman, “A characterization of objects that are dual to locally compact groups,” Funkts. Anal. Prilozen., 8, No. 1, 75–76 (1974).
L. I. Vainerman and G. I. Kac, “Nonunimodular ring groups, and Hopf–von Neumann algebras,” Dokl. Akad. Nauk SSSR, 211, 1031–1034 (1973).
L. I. Vainerman and G. I. Kac, “Nonunimodular ring groups and Hopf–von Neumann algebras,” Mat. Sb., 94 (136), 194–225, 335 (1974).
E. B. Vinberg and A. L. Onishchik, A Seminar on Lie Groups and Algebraic Groups [in Russian], URSS, Moscow (1995).
S. Wang, Quantum ‘ax + b’ group as quantum automorphism group of k[x], http://arXiv.org/abs/math/9807094v2 (1998).
S. L. Woronowicz, “Quantum ‘az + b’ group on complex plane,” Int. J. Math., 12, No. 4, 461–503 (2001).
S. Zhang, Braided Hopf Algebras, arXiv:math.RA/0511251, v8 25 May 2006.
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 14, No. 1, pp. 3–178, 2008.
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Akbarov, S.S. Holomorphic functions of exponential type and duality for stein groups with algebraic connected component of identity. J Math Sci 162, 459–586 (2009). https://doi.org/10.1007/s10958-009-9646-1
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DOI: https://doi.org/10.1007/s10958-009-9646-1