Geodesic
Arens--Michael envelopes of nilpotent Lie algebras, holomorphic functions of exponential type and homological epimorphisms
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 81 (2020) no. 1, pp. 117-136.

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Our aim is to give an explicit description of the Arens–Michael envelope for the universal enveloping algebra of a finite-dimensional nilpotent complex Lie algebra. It turns out that the Arens–Michael envelope belongs to a class of completions introduced by R. Goodman in 1970s. To find a precise form of this algebra we characterize preliminary the set of holomorphic functions of exponential type on a simply connected nilpotent complex Lie group. This approach leads to unexpected connections to Riemannian geometry and the theory of order and type for entire functions. As a corollary, it is shown that the Arens–Michael envelope considered above is a homological epimorphism. So we get a positive answer to a question investigated earlier by Dosi and Pirkovskii. References: 36 entries. 
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O. Yu. Aristov. Arens--Michael envelopes of nilpotent Lie algebras, holomorphic functions of exponential type and homological epimorphisms. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 81 (2020) no. 1, pp. 117-136. http://geodesic.mathdoc.fr/item/MMO_2020_81_1_a4/

[1] S. S. Akbarov, “Golomorfnye funktsii eksponentsialnogo tipa i dvoistvennost dlya grupp Shteina s algebraicheskoi svyaznoi komponentoi edinitsy”, Fundament. i prikl. matem., 14:1 (2008), 3–178

[2] O. Yu. Aristov, “Holomorphic functions of exponential type on connected complex Lie groups”, J. Lie Theory, 29:4 (2019), 1045–1070, arXiv: 1903.08080 [math.RT] | MR | Zbl

[3] A. Bellaïche, “The tangent space in sub-Riemannian geometry”, Sub-Riemannian Geometry, Progr. Math., 144, Birkhäuser, Basel, 1996, 1–78 | MR | Zbl

[4] J. Bonet, S. Dierolf, “On distinguished Fréchet spaces”, Progr. Funct. Analysis, Elsevier Sci. Publ. BV, 1992, 201–214 | MR

[5] Y. Cornulier, P. de la Harpe, Metric geometry of locally compact groups, EMS Tracts in mathematics, 25, Zürich, 2016 | MR | Zbl

[6] H. G. Dales, Banach algebras, automatic continuity, London Math. Society Monographs, 24, The Clarendon Press, Oxford University Press, New York, 2000 | MR

[7] A. A. Dosiev, “Gomologicheskie razmernosti algebry tselykh funktsii ot elementov nilpotentnoi algebry Li”, Funkts. analiz i ego pril., 37:1 (2003), 73–77 | MR | Zbl

[8] A. Dosiev, “Local left invertibility for operator tuples and noncommutative localizations”, J. K-Theory, 4:1 (2009), 163–191 | DOI | MR | Zbl

[9] A. Dosi, “Taylor functional calculus for supernilpotent Lie algebra of operators”, J. Operator Theory, 63:1 (2010), 191–216 | MR | Zbl

[10] A. Dosiev, “Formally-radical functions in elements of a nilpotent Lie algebra and noncommutative localizations”, Algebra Colloq., 17:1, Sp. iss. (2010), 749–788 | MR

[11] N. Dungey, A. F. M. ter Elst, D. W. Robinson, Analysis on Lie groups with polynomial growth, Progr. Math., 214, Birkhäuser Boston Inc, Boston, MA, 2003 | MR | Zbl

[12] H. A. M. Dzinotyiweyi, “Weighted function algebras on groups and semigroups”, Bull. Austral. Math. Soc., 33 (1986), 307–318 | DOI | MR | Zbl

[13] G. Favre, “Une algèbre de Lie caractéristiquement nilpotente de dimension 7”, C. R. Acad. Sci. Paris. Sér. A-B, 274 (1972), A1338–A1339 | MR

[14] R. Goodman, “Filtrations and canonical coordinates on nilpotent Lie groups”, Trans. Amer. Math. Soc., 237 (1978), 189–204 | DOI | MR | Zbl

[15] R. Goodman, “Holomorphic representations of nilpotent Lie groups”, J. Funct. Anal., 31:1 (1979), 115–137 | DOI | MR | Zbl

[16] M. Goze, Y. Khakimdjanov, Nilpotent Lie algebras, Mathematics and Its Applications, 361, Springer, Dordrecht, 1996 | MR

[17] A. Ya. Khelemskii, Gomologiya v banakhovykh i topologicheskikh algebrakh, Izd-vo MGU, M., 1986 | MR

[18] A. Ya. Khelemskii, Banakhovy i polinormirovannye algebry: obschaya teoriya, predstavleniya, gomologii, Nauka, M., 1989

[19] A. Ya. Helemskii, “Homology for the algebras of analysis”, Handbook of algebra, v. 2, North-Holland, Amsterdam, 2000, 151–274 | DOI | MR | Zbl

[20] E. Hewitt, K. A. Ross, Abstract harmonic analysis, v. I, Springer-Verlag, Berlin–Heidelberg–New York, 1979 | MR | Zbl

[21] R. Karidi, “Geometry of balls in nilpotent Lie groups”, Duke Math. J., 74:2 (1994), 301–317 | DOI | MR | Zbl

[22] G. Köthe, Topological vector spaces, v. I, Grundlehren der mathematischen Wissenschaften, 159, Springer-Verlag, Berlin–Heidelberg, 1983 | DOI | MR

[23] P. Lelong, L. Gruman, Entire functions of several complex variables, Grundlehren der mathematischen Wissenschaften, 282, Springer-Verlag, Berlin–Heidelberg, 1986 | DOI | MR | Zbl

[24] G. L. Litvinov, “O vpolne neprivodimykh predstavleniyakh kompleksnykh i veschestvennykh nilpotentnykh grupp Li”, Funkts. analiz i ego pril., 3:4 (1969), 87–88 | MR | Zbl

[25] K. H. Neeb, Holomorphy and convexity in Lie theory, de Gruyter Expositions in Mathematics, 28, de Gruyter, Berlin–New York, 2000 | MR

[26] A. Yu. Pirkovskii, “Stably flat completions of universal enveloping algebras”, Dissertationes Math., 441 (2006), 1–60 | DOI | MR

[27] A. Yu. Pirkovskii, “Arens-Michael enveloping algebras and analytic smash products”, Proc. Amer. Math. Soc., 134 (2006), 2621–2631 | DOI | MR | Zbl

[28] A. Yu. Pirkovskii, “Obolochki Arensa-Maikla, gomologicheskie epimorfizmy i otnositelno kvazisvobodnye algebry”, Trudy MMO, 69, 2008, 34–125

[29] H. H. Schaefer, M. P. Wolff, Topological vector spaces, Springer-Verlag, 1999 | MR | Zbl

[30] L. B. Schweitzer, “Dense $m$-convex Fréchet subalgebras of operator algebra crossed products by Lie groups”, Internat. J. Math., 4:4 (1993), 601–673 | DOI | MR | Zbl

[31] J. L. Taylor, “Homology and cohomology for topological algebras”, Adv. Math., 9 (1972), 137–182 | DOI | MR | Zbl

[32] J. L. Taylor, “A general framework for a multi-operator functional calculus”, Adv. Math., 9 (1972), 183–252 | DOI | MR | Zbl

[33] Yu. V. Turovskii, “Spektralnye svoistva elementov normirovannykh algebr i invariantnye podprostranstva”, Funkts. analiz i ego pril., 18:2 (1984), 77–78 | MR

[34] N. T. Varopoulos, L. Saloff-Coste, T. Coulhon, Analysis, geometry on groups, Cambridge Tracts in Mathematics, 100, Cambridge University Press, Cambridge, 1992 | MR

[35] G. Warner, Harmonic analysis on semi-simple Lie groups, v. I, Grundlehren der mathematischen Wissenschaften, 188, Springer-Verlag, Berlin–Heidelberg, 1972 | MR | Zbl

[36] G. Willis, “Conjugation weights and weighted convolution algebras on totally disconnected, locally compact groups”, AMSI International Conference on Harmonic Analysis and Applications (Austral. Nat. Univ., Canberra, 2013), Proc. Centre Math. Appl. Austral. Nat. Univ., 45, 136–147 | MR | Zbl