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Nilpotence, radicals and monoidal structures. With an appendix by Peter O’Sullivan. (Nilpotence, radicaux et structures monoïdales.) (French) Zbl 1165.18300
Rend. Semin. Mat. Univ. Padova 108, 107-291 (2002); erratum ibid. 113, 125-128 (2005).
Authors’ abstract: For \(K\) a field, a Wedderburn \(K\)-linear category is a \(K\)-linear category \(\mathcal A\) whose radical \(\mathcal R\) is locally nilpotent and such that \(\overline{A}:={\mathcal A}/{\mathcal R}\) is semi-simple and remains so after any extension of scalars. We prove existence and uniqueness results for sections of the projection \(A\to\overline{A}\), in the vein of the theorems of Wedderburn. There are two such results: one in the general case and one when \(A\) has a monoidal structure for which \(R\) is a monoidal ideal. The latter applies notably to Tannakian categories over a field of characteristic zero, and we get a generalisation of the Jacobson–Morozov theorem: the existence of a pro-reductive envelope \(^p\)Red\((G)\) associated to any affine group scheme \(G\) over \(K(^p\text{Red} (G_a)=\text{SL}_2\), and \(^p\)Red\((G)\) is infinite-dimensional for any bigger unipotent group). Other applications are given in this paper as well as in the authors’ note [C. R., Math., Acad. Sci. Paris 334, No. 11, 989–994 (2002; Zbl 1052.14021)] on motives.

MSC:
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
14F42 Motivic cohomology; motivic homotopy theory
14L15 Group schemes
16N99 Radicals and radical properties of associative rings
18E40 Torsion theories, radicals
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References:
[1] Y. ANDRÉ - B. KAHN, Construction inconditionnelle de groupes de Galois motiviques, C. R. Acad. Sci. Paris, Sér I., 331 (2002), pp. 989-994. Zbl1052.14021 MR1913723 · Zbl 1052.14021 · doi:10.1016/S1631-073X(02)02384-1
[2] A. BEILINSON, Height pairing between algebraic cycles, in: K-theory, Arithmetic and Geometry, Lect. notes in Math. 1289, Springer (1987), pp. 27-41. Zbl0652.14008 MR923131 · Zbl 0652.14008
[3] D. BENSON, Representations and cohomology I, Cambridge studies 30, Cambridge Univ. Press, 1995. Zbl0908.20001 MR1110581 · Zbl 0908.20001
[4] F. BEUKERS - D. BROWNAWELL - G. HECKMAN, Siegel normality, Annals of Math., 127 (1988), pp. 279-308. Zbl0652.10027 MR932298 · Zbl 0652.10027 · doi:10.2307/2007054
[5] N. BOURBAKI, Algèbre, chapitre VIII, Hermann, 1958.
[6] N. BOURBAKI, Groupes et algèbres de Lie, chapitre VI, Hermann/CCLS, 1975. MR453824
[7] N. BOURBAKI, Groupes et algèbres de Lie, chapitre VIII, Hermann/CCLS, 1975. MR453824
[8] L. BREEN, Tannakian categories, in: Motives, Proc. Symposia pure Math., 55 (I), AMS (1994), pp. 337-376. Zbl0810.18008 MR1265536 · Zbl 0810.18008
[9] A. BRUGUIÈRES, Théorie tannakienne non commutative, Comm. in Algebra, 22 (14) (1994), pp. 5817-5860. Zbl0808.18005 MR1298753 · Zbl 0808.18005 · doi:10.1080/00927879408825165
[10] A. BRUGUIÈRES, Tresses et structure entière sur la catégorie des représentations de SLN quantique, Comm. in Algebra, 28 (2000), pp. 1989-2028. Zbl0951.18003 MR1747368 · Zbl 0951.18003 · doi:10.1080/00927870008826941
[11] H. CARTAN - S. EILENBERG, Homological algebra, Princeton Univ. Press, 1956. Zbl0075.24305 MR77480 · Zbl 0075.24305
[12] P. CARTIER, Construction combinatoire des invariants de Vassiliev-Kontsevich des noeuds, C. R. Acad. Sci. Paris, 316 (1993), pp. 1205-1210. Zbl0791.57006 MR1221650 · Zbl 0791.57006
[13] J. CUNTZ - D. QUILLEN, Algebra extensions and nonsingularity, Journal A.M.S., 8 2 (1995), pp. 251-289. Zbl0838.19001 MR1303029 · Zbl 0838.19001 · doi:10.2307/2152819
[14] P. DELIGNE, Catégories tannakiennes, in: The Grothendieck Festschrift, vol. 2, Birkhäuser P.M., 87 (1990), pp. 111-198. Zbl0727.14010 MR1106898 · Zbl 0727.14010
[15] P. DELIGNE et al., Quantum fields and strings: a Course for Mathematicians, AMS, 1999.
[16] P. GABRIEL, Des catégories abéliennes, Bull. Soc. Math. France, 90 (1962), pp. 323-448. Zbl0201.35602 MR232821 · Zbl 0201.35602 · numdam:BSMF_1962__90__323_0 · eudml:87023
[17] P. GABRIEL, Problèmes actuels de théorie des représentations, L’Ens. Math., 20 (1974), pp. 323-332. Zbl0302.16028 MR366984 · Zbl 0302.16028
[18] J. GIRAUD, Cohomologie non abélienne, Springer, 1971. Zbl0226.14011 MR344253 · Zbl 0226.14011
[19] A. GROTHENDIECK, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2), 9 (1957), pp. 119-221. Zbl0118.26104 MR102537 · Zbl 0118.26104
[20] V. GINZBURG, Principal nilpotent pairs in a semisimple Lie algebra. I, Invent. Math., 140 (2000), pp. 511-561. Zbl0984.17007 MR1760750 · Zbl 0984.17007 · doi:10.1007/s002220000060
[21] V. GULETSKII - C. PEDRINI, The Chow motive of the Godeaux surface, prépublication (2001). Zbl1054.14009 MR1954064 · Zbl 1054.14009
[22] D. HAPPEL, Triangulated categories in the representation theory of finite dimensional algebras, London Math. Soc. Lect. notes, 119 (1988), Cambridge Univ. Press. Zbl0635.16017 MR935124 · Zbl 0635.16017
[23] G. HIGMAN, On a conjecture of Nagata, Proc. Camb. Philos. Soc., 52 (1956), pp. 1-4. Zbl0072.02502 MR73581 · Zbl 0072.02502
[24] N. JACOBSON, Rational methods in Lie theory, Ann. of Math., 36 (1935), pp. 875-881. MR1503258 JFM61.1030.03 · JFM 61.1030.03
[25] U. JANNSEN, Motives, numerical equivalence and semi-simplicity, Invent. Math., 107 (1992), pp. 447-452. Zbl0762.14003 MR1150598 · Zbl 0762.14003 · doi:10.1007/BF01231898 · eudml:143974
[26] U. JANNSEN, Equivalence relations on algebraic cycles, in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), pp. 225-260, NATO Sci. Ser. C Math. Phys. Sci., 548, Kluwer Acad. Publ., Dordrecht, 2000. Zbl0988.14003 MR1744947 · Zbl 0988.14003
[27] A. JOYAL - R. STREET, Braided monoidal categories, Adv. in Math., 102 (1993), pp. 20-78. Zbl0817.18007 MR1250465 · Zbl 0817.18007 · doi:10.1006/aima.1993.1055
[28] C. KASSEL - M. ROSSO - V. TURAEV, Quantum groups and knot invariants, S.M.F. Panoramas et synthèses, 5 (1997). Zbl0878.17013 MR1470954 · Zbl 0878.17013
[29] N. KATZ - W. MESSING, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math., 23 (1974), pp. 73-77. Zbl0275.14011 MR332791 · Zbl 0275.14011 · doi:10.1007/BF01405203 · eudml:142251
[30] G. M. KELLY, On the radical of a category, J. Australian Math. Soc., 4 (1964), pp. 299-307. Zbl0124.01501 MR170922 · Zbl 0124.01501 · doi:10.1017/S1446788700024071
[31] O. KERNER - A. SKOWROŃSKI, On module categories with nilpotent infinite radical, Compos. Math., 77 3 (1991), pp. 313-333. Zbl0717.16012 MR1092772 · Zbl 0717.16012 · numdam:CM_1991__77_3_313_0 · eudml:90076
[32] S. I. KIMURA, Chow motives can be finite-dimensional, in some sense, à paraître au J. of Alg. Geom.
[33] B. KOSTANT, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math., 81 (1959), pp. 973-1032. Zbl0099.25603 MR114875 · Zbl 0099.25603 · doi:10.2307/2372999
[34] K. KÜNNEMAN, On the Chow motive of an abelian scheme, in: Motives, Proc. Symposia pure Math., 55 (I), AMS (1994), pp. 189-205. Zbl0823.14032 MR1265530 · Zbl 0823.14032
[35] P. Y. LEDUC, catégories semi-simples et catégories primitives, Canad. J. Math., 20 (1968), pp. 612-628. Zbl0159.02104 MR228568 · Zbl 0159.02104 · doi:10.4153/CJM-1968-060-7
[36] S. MAC LANE, Categories for the working mathematician, 2ème éd., Springer GTM, 5 (1998). Zbl0906.18001 MR1712872 · Zbl 0906.18001
[37] G. A. MARGULIS, Discrete subgroups of semisimple Lie groups, Springer, Berlin, 1991. Zbl0732.22008 MR1090825 · Zbl 0732.22008
[38] B. MITCHELL, Rings with several objects, Adv. Math., 8 (1972), pp. 1-161. Zbl0232.18009 MR294454 · Zbl 0232.18009 · doi:10.1016/0001-8708(72)90002-3
[39] V. V. MOROZOV, Sur un élément nilpotent dans une algèbre de Lie semi-simple (en russe), Dokl. Akad. Nauk SSSR, 36 (1942), pp. 83-86. Zbl0063.04103 MR7750 · Zbl 0063.04103
[40] V. V. MOROZOV, Sur le centralisateur d’une sous-algèbre semi-simple d’une algèbre de Lie semi-simple (en russe), Dokl. Akad. Nauk SSSR, 36 (1942), pp. 259-261. Zbl0063.04104 · Zbl 0063.04104
[41] J. P. MURRE, On a conjectural filtration on the Chow groups of an algebraic variety, parts I and II, Indag. Math., 4 (1993), pp. 177-201. Zbl0805.14001 MR1225267 · Zbl 0805.14001 · doi:10.1016/0019-3577(93)90038-Z
[42] M. NAGATA, On the nilpotency of nil-algebras, J. Math. Soc. Japan, 4 (1952), pp. 296-301. Zbl0049.02402 MR53088 · Zbl 0049.02402 · doi:10.2969/jmsj/00430296
[43] M. NATHANSON, Classification problems in K-categories, Fund. Math., 105 3 (1979/80), pp. 187-197. Zbl0457.18007 MR580581 · Zbl 0457.18007 · eudml:211072
[44] P. O’SULLIVAN, lettres aux auteurs, 29 avril et 12 mai 2002.
[45] D. I. PANYUSHEV, Nilpotent pairs in semisimple Lie algebras and their characteristics, Internat. Math. Res. Notices, 2000, 1-21. Zbl0954.17007 MR1741606 · Zbl 0954.17007 · doi:10.1155/S1073792800000015
[46] A. PIARD, Indecomposable representations of a semi-direct product sl(2) l 3A and semi-simple groups containing sl(2) l3A, in: Sympos. Math. XXXI (Roma, 1988), pp. 185-195, Acad. Press, 1990. Zbl0718.22010 MR1059502 · Zbl 0718.22010
[47] PLATON, Phédon, § LIII.
[48] N. POPESCU, Abelian categories with applications to rings and modules, Acad. Press, 1973. Zbl0271.18006 MR340375 · Zbl 0271.18006
[49] M. PREST, Model theory and modules, L.M.S. Lecture note series 130, Cambridge Univ. Press 1988. Zbl0634.03025 MR933092 · Zbl 0634.03025
[50] C. M. RINGEL, Recent advances in the representation theory of finite dimensional algebras, in: Representation theory of finite groups and finite-dimensional algebras, Birkhäuser Progress in Math., 95 (1991). Zbl0757.16006 MR1112160 · Zbl 0757.16006
[51] L. ROWEN, Ring theory, vol. 1, Acad. Press, 1988. Zbl0651.16002 · Zbl 0651.16002
[52] W. RUMP, Doubling a path algebra, or: how to extend indecomposable modules to simple modules, in: Representation theory of groups, algebras and orders (Costanţa, 1995), An. Ştiinţ. Univ. Ovidius Constanţa Ser Mat., 4 2 (1996), pp. 174-185. Zbl0876.16006 MR1428466 · Zbl 0876.16006
[53] N. SAAVEDRA RIVANO, Catégories tannakiennes, Lect. Notes in Math. 265, Springer, 1972. Zbl0241.14008 MR338002 · Zbl 0241.14008
[54] J.-P. SERRE, Gèbres, L’Ens. Math., 39 (1993), pp. 33-85. Zbl0810.16039 · Zbl 0810.16039
[55] J.-P. SERRE, Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques, in: Motives, Proc. Symposia pure Math., 55 (I), AMS (1994), pp. 377-400. Zbl0812.14002 MR1265537 · Zbl 0812.14002
[56] D. SIMSON - A. SKOWROŃSKI, The Jacobson radical power series of module categories and the representation type, Bol. Soc. Mat. Mexicana, 5 2 (1999), pp. 223-236. Zbl0960.16012 MR1738424 · Zbl 0960.16012
[57] R. STREET, Ideals, radicals and structure of additive categories, Appl. Cat. Structures, 3 (1995), pp. 139-149. Zbl0826.18003 MR1329188 · Zbl 0826.18003 · doi:10.1007/BF00877633
[58] R. THOMASON, The classification of triangulated categories, Compos. Math., 105 (1997), pp. 1-27. Zbl0873.18003 MR1436741 · Zbl 0873.18003 · doi:10.1023/A:1017932514274
[59] V. VOEVODSKY, A nilpotence theorem for cycles algebraically equivalent to zero, International Mathematics Research Notices, 4 (1995), pp. 1-12. Zbl0861.14006 MR1326064 · Zbl 0861.14006 · doi:10.1155/S1073792895000158
[60] P. VOGEL, Invariants de Witten-Reshetikin-Turaev et théories quantiques des champs, in: Panoramas et synthèses, 7 (1999), pp. 117-143. MR1691795
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