Algebraic studies at Kazan University from V. V. Morozov to our days

A. N. Abyzov   ; Yu. A. Alpin   ; N. A. Koreshkov   ; M. F. Nasrutdinov   ; S. N. Tronin

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 154 (2012) no. 2, pp. 44-59

Résumé

This article reviews the researches on algebra at Kazan University in the period from 1947, when V. V. Morozov became the head of the Department of Algebra, up to the present.

Détail

Keywords: Lie algebras and groups, theory of rings and modules, operads, semimodules, theory of matrices, Hopf algebras.

Bibliographie

[1] Morozov V. V., “Algebra”, Usp. matem. nauk, 2:6 (1947), 3–8

[2] Arslanov M. M., “Strukturnaya teoriya stepenei nerazreshimosti”, NIIMM im. N. G. Chebotareva KGU 2003–2007, Sb. st., Izd-vo Kazan. un-ta, Kazan, 2008, 54–68

[3] Arslanov M. M., Kalimullin I. Sh., “Issledovaniya po teorii vychislimosti”, Na rubezhe vekov. NIIMM im. N. G. Chebotareva 1998–2002, Sb. st., Izd-vo Kazan. matem. o-va, Kazan, 2003, 50–67 | MR

[4] Ado I. D., “Predstavlenie algebr Li matritsami”, Usp. matem. nauk, 22:6 (1947), 159–173 | MR

[5] Morozov V. V., “O primitivnykh gruppakh”, Matem. sb., 5(47):2 (1939), 355–390 | MR | Zbl

[6] Morozov V. V., “O nilpotentnom elemente v poluprostoi algebre Li”, Dokl. AN SSSR, 36:3 (1942), 91–94 | MR

[7] Morozov V. V., “O tsentralizatore poluprostoi podalgebry v poluprostoi algebre Li”, Dokl. AN SSSR, 36:9 (1942), 275–277 | MR

[8] Dynkin E. B., “Klassifikatsiya prostykh grupp Li”, Matem. sb., 18(60):3 (1946), 347–352 | MR | Zbl

[9] Dynkin E. B., “Regulyarnye poluprostye podalgebry poluprostykh algebr Li”, Dokl. SSSR, 73:5 (1950), 877–880 | MR | Zbl

[10] Dynkin E. B., “Maksimalnye podgruppy poluprostykh grupp Li i klassifikatsiya primitivnykh grupp preobrazovanii”, Dokl. AN SSSR, 75:3 (1950), 33–336

[11] Karpelevich F. I., “O nepoluprostykh maksimalnykh podalgebrakh poluprostykh algebr Li”, Dokl. AN SSSR, 76:6 (1951), 755–778

[12] Morozov V. V., “Dokazatelstvo teoremy regulyarnosti”, Usp. matem. nauk, 11:5 (1956), 191–194 | MR | Zbl

[13] Zabotin Ya. I., “Poluprostye trazitivnye imprimitivnye gruppy chetyrekhmernogo prostranstva”, Izv. vuzov. Matem., 1958, no. 4, 67–79 | MR | Zbl

[14] Zabotin Ya. I., “O tranzitivnykh imprimitivnykh gruppakh s radikalom v chetyrekhmernom kompleksnom prostranstve”, Izv. vuzov. Matem., 1958, no. 6, 73–85 | MR | Zbl

[15] Eskin L. D., “Zamechanie ob operatorakh Laplasa na unimodulyarnoi gruppe”, Izv. vuzov. Matem., 1957, no. 2, 259–269 | MR

[16] Eskin L. D., “K teorii relyativistskikh sfericheskikh funktsii”, Nauch. dokl. vyssh. shkoly, 1959, no. 2, 95–97 | MR

[17] Eskin L. D., “O matrichnykh elementakh neprivodimykh predstavlenii gruppy Lorentsa”, Izv. vuzov. Matem., 1961, no. 6, 180–184 | MR | Zbl

[18] Eskin L. D., “Fita funktsiya na gruppe unitarnykh matrits”, Dokl. AN SSSR, 152:6 (1963), 1327–1328 | MR

[19] Eskin L. D., “Uravnenie teploprovodnosti v teorii kompaktnykh grupp”, Usp. matem. nauk, 19:2 (1964), 200–202

[20] Eskin L. D., “Uravnenie teploprovodnosti na gruppakh Li”, Pamyati N. G. Chebotareva. 1894–1947 gg., Sb. st., Izd-vo Kazan. un-ta, Kazan, 1964, 113–132 | MR

[21] Eskin L. D., “Uravnenie teploprovodnosti i preobrazovaniya Veiershtrassa na nekotorykh simmetricheskikh rimanovykh prostranstvakh”, Izv. vuzov. Matem., 1965, no. 5, 151–165 | MR | Zbl

[22] Eskin L. D., “K zadache V. Ya. Polubarinovoi-Kochinoi ob oporozhnenii basseina”, Izv. vuzov. Matem., 2004, no. 9, 73–84 | MR

[23] Eskin L. D., “Ob odnom obobschenii zadachi V. Ya. Polubarinovoi-Kochinoi ob oporozhnenii basseina”, Izv. vuzov. Matem., 2007, no. 2, 56–71 | MR | Zbl

[24] Stolov E. L., “Asimptotika odnogo integrala, soderzhaschego mnogochleny Gegenbauera”, Funkts. analiz i teoriya funktsii, Sb. st., 6, Izd-vo Kazan. un-ta, Kazan, 1969, 134–139

[25] Stolov E. L., “Ob odnom metode issledovaniya matrichnykh elementov predstavlenii grupp”, Izv. vuzov. Matem., 1969, no. 7, 79–86 | MR | Zbl

[26] Stolov E. L., “O matrichnykh elementakh predstavlenii $SO(n)$”, Izv. vuzov. Matem., 1970, no. 7, 86–91 | MR | Zbl

[27] Stolov E. L., “Dve asimptoticheskie formuly dlya spetsialnykh funktsii”, Izv. vuzov. Matem., 1972, no. 2, 72–77 | MR | Zbl

[28] Mubarakzyanov G. M., “O razreshimykh algebrakh Li”, Izv. vuzov. Matem., 1963, no. 1, 114–123 | MR | Zbl

[29] Mubarakzyanov G. M., “Klassifikatsiya veschestvennykh struktur razreshimykh algebr Li 5-go poryadka”, Izv. vuzov. Matem., 1963, no. 3, 99–106 | MR | Zbl

[30] Mubarakzyanov G. M., “Klassifikatsiya razreshimykh algebr Li 6-go poryadka s odnim nilpotentnym bazisnym elementom”, Izv. vuzov. Matem., 1963, no. 4, 104–116 | MR | Zbl

[31] Mubarakzyanov G. M., “Nekotorye teoremy o razreshimykh algebrakh Li”, Izv. vuzov. Matem., 1966, no. 6, 95–98 | Zbl

[32] Safiullina E. N., “Klassifikatsiya nilpotentnykh algebr Li 7-go poryadka”, Sb. aspir. rabot, Izd-vo Kazan. un-ta, Kazan, 1963, 103–105

[33] Morozov V. V., “Klassifikatsiya nilpotentnykh algebr Li shestogo poryadka”, Izv. vuzov. Matem., 1958, no. 4, 161–171 | MR | Zbl

[34] Suldin A. V., “O lineinykh predstavleniyakh algebr Li nad polem kharakteristiki $p>0$”, Dokl. AN SSSR, 1952, no. 4, 529–531 | MR | Zbl

[35] Dolotkazin A. Kh., “Algebry Li ranga odin s nenulevym vnutrennim proizvedeniem. II”, Izv. vuzov. Matem., 1966, no. 5, 70–77 | MR | Zbl

[36] Ermolaev Yu. B., “Vychislenie tsentralnogo elementa universalnoi obertyvayuschei algebry algebry Vitta”, Izv. vuzov. Matem., 1975, no. 5, 20–26 | MR | Zbl

[37] Ermolaev Yu. B., “Minimalnyi mnogochlen tsentralnogo elementa universalnoi obertyvayuschei algebry algebry Vitta”, Izv. vuzov. Matem., 1976, no. 10, 32–41 | MR | Zbl

[38] Ermolaev Yu. B., “O tsentralnykh elementakh universalnoi obertyvayuschei algebry algebry Tsassenkhauza”, Izv. vuzov. Matem., 1978, no. 6, 73–88 | MR | Zbl

[39] Ermolaev Yu. B., “O strukture tsentra universalnoi obertyvayuschei algebry algebry Tsassenkhauza”, Izv. vuzov. Matem., 1978, no. 12, 46–59 | MR | Zbl

[40] Kostrikin A. I., Shafarevich I. R., “Graduirovannye algebry Li konechnoi kharakteristiki”, Izv. AN SSSR. Ser. matem., 33:2 (1969), 251–322 | MR | Zbl

[41] Ermolaev Yu. B., “Monogennye graduirovannye algebry Li”, Izv. vuzov. Matem., 1981, no. 8, 70–74 | MR | Zbl

[42] Ermolaev Yu. B., “K voprosu o suschestvovanii prostykh algebr Li s graduirovkoi dostatochno bolshoi dliny”, Izv. vuzov. Matem., 1981, no. 10, 66–69 | MR | Zbl

[43] Ermolaev Yu. B., “Algebry Li s monogennoi graduirovkoi dostatochno bolshoi dliny”, Izv. vuzov. Matem., 1984, no. 2, 60–63 | MR | Zbl

[44] Tselousov M. Yu., “Differentsirovaniya algebr Li kartanovskogo tipa”, Izv. vuzov. Matem., 1970, no. 7, 126–134 | MR | Zbl

[45] Elsting G. O., “O sistemakh obrazuyuschikh v nekotorykh prostykh algebrakh Li”, Izv. vuzov. Matem., 1975, no. 4, 112–115 | MR | Zbl

[46] Elsting G. O., “Ob odnoi invariantnoi podalgebre v gamiltonovoi i kontaktnoi algebrakh Li”, Izv. vuzov. Matem., 1976, no. 1, 129–131 | MR | Zbl

[47] Kats V. G., “Opisanie filtrovannykh algebr Li, s kotorymi assotsiirovany graduirovannye algebry Li kartanovskogo tipa”, Izv. AN SSSR. Ser. matem., 38:4 (1974), 800–838 | MR | Zbl

[48] Skryabin S. M., Kanonicheskii vid gamiltonovykh i kontaktnykh form nad algebrami razdelennykh stepenei, Deponirovano v VINITI No 8594, 1986

[49] Skryabin S. M., “Klassifikatsiya gamiltonovykh form nad algebrami razdelennykh stepenei”, Matem. sb., 181:1 (1990), 114–133 | MR | Zbl

[50] Koreshkov N. A., “O neprivodimykh predstavleniyakh odnoi algebry Li”, Izv. vuzov. Matem., 1978, no. 9, 49–57 | MR | Zbl

[51] Koreshkov N. A., “O neprivodimykh predstavleniyakh $p$-algebry Li $W_2$”, Izv. vuzov. Matem., 1980, no. 4, 39–46 | MR | Zbl

[52] Koreshkov N. A., “Tsentralnye elementy v algebre $U(K_m)$”, Izv. vuzov. Matem., 1991, no. 5, 16–22 | MR | Zbl

[53] Skryabin S. M., “Structure of $H$-semiprime artinian algebras”, Algebr. Represent. Theory, 14:5 (2011), 803–822 | DOI | MR | Zbl

[54] Skryabin S. M., “Projectivity of Hopf algebras over subalgebras with semilocal central localization”, J. K-Theory, 2:1 (2008), 1–40 | DOI | MR | Zbl

[55] Skryabin S. M., “Projectivity and freeness over comodule algebras”, Trans. Amer. Math. Soc., 359 (2007), 2597–2623 | DOI | MR | Zbl

[56] Skryabin S. M., “Models of quasiprojective homogeneous spaces for Hopf algebras”, J. Reine Angew. Math., 643 (2010), 201–236 | MR | Zbl

[57] Eryashkin M. S., Skryabin S. M., “Naibolshaya podalgebra Khopfa v bialgebre”, Matem. zametki, 86:6 (2009), 942–946 | DOI | MR | Zbl

[58] Eryashkin M. S., “Invarianty deistviya poluprostoi konechnomernoi algebry Khopfa na algebrakh spetsialnogo vida”, Izv. vuzov. Matem., 2011, no. 8, 14–22 | MR | Zbl

[59] Bass H., “Finitistic dimension and a homological generalization of semi-primary rings”, Trans. Amer. Math. Soc., 95 (1960), 466–488 | DOI | MR | Zbl

[60] Skornyakov L. A., “Gomologicheskaya klassifikatsiya kolets”, Trudy IV Vsecoyuz. matem. s'ezda (M., 1961 g.), v. 2, Nauka, M., 1964, 22–32

[61] Endo S., “On semi-hereditary rings”, J. Math. Soc. Japan, 13:2 (1961), 109–119 | DOI | MR | Zbl

[62] Vasconcelos W. V., “On finitely generated flat modules”, Trans. Amer. Math. Soc., 138 (1969), 505–512 | DOI | MR | Zbl

[63] Sakhaev I. I., “O proektivnosti konechno porozhdennykh ploskikh modulei”, Sib. matem. zhurn., 6:3 (1965), 564–573 | Zbl

[64] Sakhaev I. I., “O proektivnosti konechno porozhdennykh ploskikh modulei nad polulokalnymi koltsami”, Matem. zametki, 37:2 (1985), 152–162 | MR | Zbl

[65] Jøndrup S., “On Finitely generated flat modules”, Math. Scand., 26 (1970), 233–240 | MR

[66] Facchini A., Herbera D., Sakhajev I., “Finitely generated flat modules and a characterisation of semiperfect rings”, Commun. Algebra, 31:9 (2003), 4195–4214 | DOI | MR | Zbl

[67] Facchini A., Herbera D., Sakhajev I., “Flat modules and lifting of finitely generated projective modules”, Pacific J. Math., 220:1 (2005), 49–67 | DOI | MR | Zbl

[68] Chirkov G. V., “Levo-polusovershennoe koltso, ne yavlyayuscheesya levo-$f$-polusovershennym”, Izv. vuzov. Matem., 1971, no. 6, 102–110 | MR | Zbl

[69] Nasrutdinov M. F., “O polulokalnykh gruppovykh algebrakh”, Matem. zametki, 78:3 (2005), 409–412 | DOI | MR | Zbl

[70] Nasrutdinov M. F., “Stabilnaya konechnost gruppovykh kolets”, Izv. vuzov. Matem., 2006, no. 11, 29–32 | MR

[71] Lazard D., “Liberté des gros modules projectifs”, J. Algebra, 31 (1974), 437–451 | DOI | MR | Zbl

[72] Sakhaev I. I., “O konechno porozhdennosti proektivnykh modulei”, Izv. vuzov. Matem., 1977, no. 9, 69–79 | MR | Zbl

[73] Gerasimov V. N., Sakhaev I. I., “Kontrprimer k dvum gipotezam o proektivnykh i ploskikh modulyakh”, Sib. matem. zhurn., 25:6 (1984), 31–53 | MR

[74] Lomp C., “On semilocal modules and rings”, Comm. Algebra, 27:4 (1999), 1921–1935 | DOI | MR | Zbl

[75] Puninski G., “Projective modules over the endomorphism ring of a biuniform module”, J. Pure Appl. Algebra, 188:1 (2004), 227–246 | DOI | MR | Zbl

[76] Prihoda P., “Projective modules are determined by their radical factors”, J. Pure Appl. Algebra, 210:3 (2007), 827–835 | DOI | MR | Zbl

[77] Hamza H., “$I_0$-rings and $I_0$-modules”, Math. J. Okayama Univ., 40 (1998), 91–97 | MR

[78] Abyzov A. N., “Slabo regulyarnye moduli nad normalnymi koltsami”, Sib. matem. zhurn., 49:4 (2008), 721–738 | MR | Zbl

[79] Abyzov A. N., Tuganbaev A. A., “Koltsa, nad kotorymi vse moduli yavlyayutsya $I_0$-modulyami”, Fundament. i prikl. matem., 14:2 (2008), 3–12 | MR | Zbl

[80] Abyzov A. N., “Obobschennye SV-moduli”, Sib. matem. zhurn., 50:3 (2009), 481–488 | MR | Zbl

[81] Ilin S. N., “Obratimost matrits nad uporyadochennymi algebraicheskimi sistemami”, Sib. matem. zhurn., 39:3 (1998), 551–559 | MR | Zbl

[82] Ilin S. N., “Obratimye matritsy nad (neassotsiativnymi) polukoltsami”, Universalnaya algebra i ee prilozheniya, Tr. uchastnikov mezhdunar. seminara, posvyasch. pamyati prof. L. A. Skornyakova, Volgograd, 2000, 81–89

[83] Ilin S. N., “Polukoltsa, nad kotorymi vse polumoduli in'ektivny (proektivny)”, Matem. vestn. pedvuzov i un-tov Volgo-Vyatskogo regiona (Kirov), 2006, no. 8, 50–53

[84] Ilin S. N., “O primenimosti k polukoltsam dvukh teorem teorii kolets i modulei”, Mat. zametki, 83:4 (2008), 536–544 | DOI | MR | Zbl

[85] Ilin S. N., “Pryamye summy in'ektivnykh polumodulei i pryamye proizvedeniya proektivnykh polumodulei nad polukoltsami”, Izv. vuzov. Matem., 2010, no. 10, 31–43 | MR | Zbl

[86] Il'in S. N., Katsov Y., “On $p$-Schreier varieties of semimodules”, Comm. Algebra, 39:4, 1491–1501 | DOI | MR | Zbl

[87] Morozov V. V., “O kommutativnykh matritsakh”, Uchen. zap. Kazan. un-ta, 112, no. 9, 1952, 17–20

[88] Alpin Yu. A., “Granitsy dlya perronova kornya neotritsatelnoi matritsy, uchityvayuschie svoistva eë grafa”, Matem. zametki, 58:4 (1995), 635–637 | MR | Zbl

[89] Alpin Yu. A., “Sblizhenie granits Frobeniusa dlya perronova kornya neotritsatelnoi matritsy”, Zhurn. vychisl. matem. i matem fiz., 37:2 (1997), 131–138 | MR | Zbl

[90] Alpin Yu. A., Kolotilina L. Yu., “Inequalities for the Perron root related to Levinger's theorem”, Linear Algebra Appl., 283:1–3 (1998), 99–113 | DOI | MR | Zbl

[91] Alpin Yu. A., Chien M.-T., Yeh L., “The numerical radius and bounds for zeros of a polynomial”, Proc. Amer. Math. Soc., 131 (2003), 725–730 | DOI | MR | Zbl

[92] Alpin Yu. A., Kolotilina L. Yu., Korneeva N. N., “Sovmestnye otsenki dlya perronovskikh kornei neotritsatelnykh matrits i ikh prilozheniya”, Zap. nauch. semin. POMI RAN, 334, 2006, 30–56 | MR | Zbl

[93] Alpin Yu. A., “Granitsy dlya sovmestnykh spektralnykh radiusov neotritsatelnykh matrits”, Matem. zametki, 87:1 (2010), 13–16 | DOI | MR | Zbl

[94] Alpin Yu. A., Elsner L. Ikramov Kh. D., “On condensed forms for partially commuting matrices”, Linear Algebra Appl., 306:1–3 (2000), 165–182 | DOI | MR | Zbl

[95] Alpin Yu. A., George A., Ikramov Kh. D., “Solving the two-dimensional CIS problem by a rational algorithm”, Linear Algebra Appl., 312:1–3 (2000), 115–123 | DOI | MR | Zbl

[96] Alpin Yu. A., Ikramov Kh. D., “Reducibility theorems for pairs of matrix as rational criteria”, Linear Algebra Appl., 313:1–3 (2000), 155–161 | DOI | MR | Zbl

[97] Alpin Yu. A., Koreshkov N. A., “Ob odnovremennoi triangulizuemosti matrits”, Matem. zametki, 68:5 (2000), 648–652 | DOI | MR | Zbl

[98] Alpin Yu. A., Ikramov Kh. D., “Ob unitarnom podobii matrichnykh semeistv”, Matem. zametki, 74:6 (2003), 815–826 | DOI | MR | Zbl

[99] Alpin Yu. A., Ikramov Kh. D., “Kriterii unitarnoi kongruentnosti matrits”, Dokl. RAN, 437:1 (2011), 7–8 | MR | Zbl

[100] Smirnov V. A., Operadnye i simplitsialnye metody v algebraicheskoi topologii, Faktorial Press, M., 2002, 272 pp.

[101] Markl M., Shnider S., Stasheff J., Operads in Algebra, Topology and Physics, Amer. Math. Soc., 2002, 350 pp. | MR | Zbl

[102] Leinster T., Higher Operads, Higher Categories, Cambridge Univ. Press, Cambridge, 2004, 448 pp. | MR | Zbl

[103] Loday J.-L., Valette B., Algebraic Operads, 2010, XVIII+512 pp. URL: http://math.unice.fr/~brunov/Operads.pdf

[104] Tronin S. N., O retraktsiyakh svobodnykh algebr i modulei, Dis. $\dots$ kand. fiz.-mat. nauk, Kishinev, 1989, 105 pp.

[105] Tronin S. N., “O kommutativnykh assotsiativnykh proektivnykh algebrakh ranga 2 nad sovershennym polem”, Matem. zametki, 41:6 (1987), 776–780 | MR | Zbl

[106] Tronin S. N., “Retrakty i retraktsii svobodnykh algebr”, Izv. vuzov. Matem., 1998, no. 1, 67–78 | MR | Zbl

[107] Tronin S. N., “Proizvedeniya v kategoriyakh chastnykh i universalnoe obraschenie gomomorfizmov”, Matem. sb., 188:10 (1997), 109–130 | DOI | MR | Zbl

[108] Tronin S. N., Kopp O. A., “Matrichnye lineinye operady”, Izv. vuzov. Matem., 2000, no. 6, 53–62 | MR | Zbl

[109] Kapranov M., Manin Yu., “Modules and Morita theorem for operads”, Amer. J. Math., 123:5 (2001), 811–838 | DOI | MR | Zbl

[110] Tronin S. N., “Abstraktnye klony i operady”, Sib. matem. zhurn., 43:4 (2002), 924–936 | MR | Zbl

[111] Tronin S. N., Operadnye i kategornye metody v teorii mnogoobrazii universalnykh algebr, Dis. $\dots$ d-ra fiz.-mat. nauk, Kazan, 2011, 350 pp.

[112] Tronin S. N., “Multikategorii i mnogoobraziya mnogosortnykh algebr”, Sib. matem. zhurn., 49:5 (2008), 1185–1201 | MR | Zbl

[113] Tronin S. N., “Operady i mnogoobraziya algebr, opredelyaemye polilineinymi tozhdestvami”, Sib. matem. zhurn., 47:3 (2006), 670–694 | MR | Zbl

[114] Tronin S. N., “Superalgebry i operady. I”, Sib. matem. zhurnal, 50:3 (2009), 631–646 | MR | Zbl

[115] Tronin S. N., “O superalgebrakh nad operadami”, Sib. matem. zhurnal, 50:6 (2009), 1401–1412 | MR | Zbl

[116] Tronin S. N., “Estestvennye multipreobrazovaniya multifunktorov”, Izv. vuzov. Matem., 2011, no. 11, 58–71 | MR | Zbl