Cohomology of Hopf algebras and Massey products

V. M. Buchstaber   ; F. Yu. Popelenskii

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 79 (2024) no. 4, pp. 567-648

Résumé

The theory of the trigraded Buchstaber spectral sequence $\operatorname{Bss}$ for graded Hopf algebras is developed. It is shown that the differentials of $\operatorname{Bss}$ define an increasing exhaustive filtration as a new structure in the cohomology of Hopf algebras. This structure is described explicitly for a number of known Hopf algebras. For the tensor algebra $T(s \operatorname{Ext}^{1,*}_{A}(\Bbbk,\Bbbk))$ of the suspension of the one-dimensional cohomology of a Hopf algebra $A$ over a field $\Bbbk$, the construction of partial multivalued operations $\operatorname{Bss}_p$, $p\geqslant 1$, is presented. This construction is used to describe the differentials in the spectral sequence $\operatorname{Bss}$ and the exhaustive filtration in $\operatorname{Ext}_{A}^{*,*}(\Bbbk,\Bbbk)$. It is shown that the structure introduced is an effective tool for solving several well-known problems: (1) realising cohomology classes of Hopf algebras by Massey products; (2) interpreting differentials in $\operatorname{Bss}$ as Massey operations; (3) effective construction of a certain class of Massey products in the form of differentials in $\operatorname{Bss}$. Bibliography: 74 titles.

Détail

Keywords: Hopf algebras, Landweber–Novikov algebra, Buchstaber spectral sequence, Eilenberg–Moore spectral sequence, $\operatorname{Bss}$-operations, cohomology of nilmanifolds.

Bibliographie

[1] S. P. Novikov, “Cohomology of the Steenrod algebra”, Dokl. Akad. Nauk SSSR, 128:5 (1959), 893–895 (Russian) | MR | Zbl

[2] J. P. May, “A general algebraic approach to Steenrod operations”, The Steenrod algebra and its applications, A conference to celebrate N. E. Steenrod's sixtieth birthday (Battelle Memorial Inst., Columbus, OH 1970), Lecture Notes in Math., 168, Springer-Verlag, Berlin–New York, 1970, 153–231 | MR | Zbl

[3] J. Milnor, “Construction of universal bundles. I”, Ann. of Math. (2), 63:2 (1956), 272–284 | DOI | MR | Zbl

[4] J. Milnor, “Construction of universal bundles. II”, Ann. of Math. (2), 63:3 (1956), 430–436 | DOI | MR | Zbl

[5] J. McCleary, A user's guide to spectral sequences, Cambridge Stud. Adv. Math., 58, 2nd ed., Cambridge Univ. Press, Cambridge, 2001, xvi+561 pp. | DOI | MR | Zbl

[6] J. P. May, “The cohomology of restricted Lie algebras and of Hopf algebras”, J. Algebra, 3:2 (1966), 123–146 | DOI | MR | Zbl

[7] V. G. Drinfel'd, “Quantum groups”, J. Soviet Math., 41:2 (1988), 898–915 | DOI | MR | Zbl

[8] C. Kassel, Quantum groups, Grad. Texts in Math., 155, Springer-Verlag, New York, 1995, xii+531 pp. | DOI | MR | Zbl

[9] A. Yu. Lazarev and M. D. Movshev, “Deformations of Hopf algebras”, Russian Math. Surveys, 46:1 (1991), 253–254 | DOI | MR | Zbl

[10] H. Krause, S. Witherspoon, and J. Zhang, “Mini-workshop: Cohomology of Hopf algebras and tensor categories”, Oberwolfach Rep., 16:1 (2019), 663–693 | DOI | MR | Zbl

[11] W. S. Massey, “Some higher order cohomology operations”, Symposium internacional de topología algebraica, UNAM and UNESCO, México, 1958, 145–154 | MR | Zbl

[12] H. Uehara and W. S. Massey, “The Jacobi identity for Whitehead products”, Algebraic geometry and topology, A symposium in honor of S. Lefschetz, Princeton Univ. Press, Princeton, NJ, 1957, 361–377 | MR | Zbl

[13] J. P. May, “Matric Massey products”, J. Algebra, 12:4 (1969), 533–568 | DOI | MR | Zbl

[14] I. K. Babenko and I. A. Taimanov, “Massey products in symplectic manifolds”, Sb. Math., 191:8 (2000), 1107–1146 | DOI | MR | Zbl

[15] L. Smith, “Homological algebra and the Eilenberg–Moore spectral sequence”, Trans. Amer. Math. Soc., 129:1 (1967), 58–93 | DOI | MR | Zbl

[16] L. Smith, Lectures on the Eilenberg–Moore spectral sequence, Lecture Notes in Math., 134, Springer-Verlag, Berlin-New York, 1970, vii+142 pp. | DOI | MR | Zbl

[17] V. K. A. M. Gugenheim and J. P. May, On the theory and applications of differential torsion products, Mem. Amer. Math. Soc., 142, Amer. Math. Soc., Providence, RI, 1974, ix+94 pp. | MR | Zbl

[18] L. J. Santharoubane, “Cohomology of Heisenberg Lie algebras”, Proc. Amer. Math. Soc., 87:1 (1983), 23–28 | DOI | MR | Zbl

[19] M. A. Alvarez, “The Betti numbers for Heisenberg Lie algebras”, J. Algebraic Combin., 52:4 (2020), 461–467 | DOI | MR | Zbl

[20] E. Sköldberg, “The homology of Heisenberg Lie algebras over fields of characteristic two”, Math. Proc. R. Ir. Acad., 105:2 (2005), 47–49 | DOI | MR | Zbl

[21] G. Cairns and S. Jambor, “The cohomology of the Heisenberg Lie algebras over fields of finite characteristic”, Proc. Amer. Math. Soc., 136:11 (2008), 3803–3807 | DOI | MR | Zbl

[22] R. Howe, “Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond”, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., 8, Bar-Ilan Univ., Ramat Gan, 1995, 1–182 | MR | Zbl

[23] V. V. Zharinov, “On the cohomologies of the Heisenberg algebras”, Proc. Steklov Inst. Math., 228 (2000), 52–66 | MR | Zbl

[24] Soo Teck Lee and J. A. Packer, “The cohomology of the integer Heisenberg groups”, J. Algebra, 184:1 (1996), 230–250 | DOI | MR | Zbl

[25] V. M. Buhštaber, “The Chern–Dold character in cobordisms. I”, Math. USSR-Sb., 12:4 (1970), 573–594 | DOI | MR | Zbl

[26] J. Milnor, “On the cobordism ring $\Omega^*$ and a complex analogue. I”, Amer. J. Math., 82:3 (1960), 505–521 | DOI | MR | Zbl

[27] S. P. Novikov, “Some problems in the topology of manifolds connected with the theory of Thom spaces”, Soviet Math. Dokl., 1 (1960), 717–720 | MR | Zbl

[28] S. P. Novikov, “Homotopy properties of Thom complexes”, Topological library, Part 1. Cobordisms and their applications, Ser. Knots Everything, 39, World Sci. Publ., Hackensack, NJ, 2007, 211–250 | DOI | MR | Zbl

[29] S. P. Novikov, “The methods of algebraic topology from the viewpoint of cobordism theory”, Math. USSR-Izv., 1:4 (1967), 827–913 | DOI | MR | Zbl

[30] H. Hopf, “Über die Topologie der Gruppen-Mannigfaltigkeiten und Ihre Verallgemeinerungen”, Ann. of Math. (2), 42 (1941), 22–52 | DOI | MR | Zbl

[31] S. P. Novikov, “Various doublings of Hopf algebras. Operator algebras on quantum groups, complex cobordisms”, Russian Math. Surveys, 47:5 (1992), 198–199 | DOI | MR | Zbl

[32] V. M. Buchstaber, “Spectral sequences connected with the Chern–Dold character in cobordisms”, in: “Proceedings of the Moscow Mathematical Society”, Uspekhi Mat. Nauk, 26:1(157) (1971), 214–215 (Russian)

[33] N. V. Panov, “Characteristic numbers in $U$-theory”, Math. USSR-Izv., 5:6 (1971), 1365–1385 | DOI | MR | Zbl

[34] V. M. Bukhshtaber (Buchstaber) and A. V. Shokurov, “The Landweber–Novikov algebra and formal vector fields on the line”, Funct. Anal. Appl., 12:3 (1978), 159–168 | DOI | MR | Zbl

[35] A. V. Shokurov, “Topological applications of the apparatus of formal diffeomorphisms of the line”, Russian Math. Surveys, 34:6 (1979), 213–219 | DOI | MR | Zbl

[36] A. V. Shokurov, “Cohomology rings of operations in the theory of unitary cobordisms and homotopy invariants of continuous maps”, Russian Math. Surveys, 33:3 (1978), 179–180 | DOI | MR | Zbl

[37] L. V. Goncharova, “The cohomologies of Lie algebras of formal vector fields on the line”, Funct. Anal. Appl., 7:2 (1973), 91–97 | DOI | MR | Zbl

[38] L. V. Goncharova, “Cohomologies of Lie algebras of formal vector fields on the straight line”, Funct. Anal. Appl., 7:3 (1973), 194–203 | DOI | MR | Zbl

[39] I. K. Babenko and I. A. Taimanov, On nonformal simply connected symplectic manifolds, Preprints of the Sfb 288 “Differential Geometry and Quantum Physics”, no. 358, 1998, 21 pp.

[40] V. M. Bukhshtaber (Buchstaber), “Groups of polynomial transformations of a line, non-formal symplectic manifolds, and the Landweber–Novikov algebra”, Russian Math. Surveys, 54:4 (1999), 837–838 | DOI | MR | Zbl

[41] K. Nomizu, “On the cohomology of compact homogeneous spaces of nilpotent Lie groups”, Ann. of Math. (2), 59:3 (1954), 531–538 | DOI | MR | Zbl

[42] J. D. Stasheff, “Homotopy associativity of $H$-spaces. II”, Trans. Amer. Math. Soc., 108:2 (1963), 293–312 | DOI | MR | Zbl

[43] J. P. May, “The cohomology of principal bundles, homogeneous spaces, and two-stage Postnikov systems”, Bull. Amer. Math. Soc., 74 (1968), 334–339 | DOI | MR | Zbl

[44] E. O'Neill, “On Massey products”, Pacific J. Math., 76:1 (1978), 123–127 | DOI | MR | Zbl

[45] V. V. Lychagin, “Contact geometry and non-linear second-order differential equations”, Russian Math. Surveys, 34:1 (1979), 149–180 | DOI | MR | Zbl

[46] A. Kushner, V. Lychagin, and V. Rubtsov, Contact geometry and nonlinear differential equations, Encyclopedia Math. Appl., 101, Cambridge Univ. Press, Cambridge, 2007, xxii+496 pp. | DOI | MR | Zbl

[47] P. E. Conner and E. E. Floyd, Differentiable periodic maps, Ergeb. Math. Grenzgeb. (N. F.), 33, Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin–Göttingen–Heidelberg, 1964, vii+148 pp. | DOI | MR | Zbl

[48] G. Laures, “On cobordism of manifolds with corners”, Trans. Amer. Math. Soc., 352:12 (2000), 5667–5688 | DOI | MR | Zbl

[49] I. M. Gel'fand, B. L. Feigin, and D. B. Fuchs, “Cohomology of infinite-dimensional Lie algebras and Laplace operators”, Funct. Anal. Appl., 12:4 (1978), 243–247 | DOI | MR | Zbl

[50] V. S. Retakh and B. L. Feigin, “On the cohomology of certain Lie algebras and superalgebras of vector fields”, Russian Math. Surveys, 37:2 (1982), 251–252 | DOI | MR | Zbl

[51] B. L. Feigin and D. B. Fuchs, “Verma modules over the Virasoro algebra”, Topology (Leningrad 1982), Lecture Notes in Math., 1060, Springer-Verlag, Berlin, 1984, 230–245 | DOI | MR | Zbl

[52] F. V. Weinstein, “Filtering bases and cohomology of nilpotent subalgebras of the Witt algebra and the algebra of loops in $sl_2$”, Funct. Anal. Appl., 44:1 (2010), 4–21 | DOI | MR | Zbl

[53] A. Rocha-Caridi and N. R. Wallach, “Characters of irreducible representations of the Lie algebra of vector fields on the circle”, Invent. Math., 72:1 (1983), 57–75 | DOI | MR | Zbl

[54] D. V. Millionshchikov, Cohomology of positively graded Lie algebras and applications, D.Sc. thesis (physical and mathematical sciences), Moscow State University, Moscow, 2019, 262 pp. (Russian)

[55] B. L. Feigin, D. B. Fuchs, and V. S. Retakh, “Massey operations in the cohomology of the infinite-dimensional Lie algebra $L_1$”, Topology and geometry – Rohlin seminar, Lecture Notes in Math., 1346, Springer-Verlag, Berlin, 1988, 13–31 | DOI | MR | Zbl

[56] I. V. Artel'nykh, “Massey products and the Bukhshtaber spectral sequence”, Russian Math. Surveys, 55:3 (2000), 559–561 | DOI | MR | Zbl

[57] D. V. Millionschikov, “Algebra of formal vector fields on the line and Buchstaber's conjecture”, Funct. Anal. Appl., 43:4 (2009), 264–278 | DOI | MR | Zbl

[58] B. M. Levitan, Theory of generalized shift operators, Nauka, Moscow, 1973, 312 pp. (Russian) | MR | Zbl

[59] V. M. Buhštaber (Buchstaber), “Two-valued formal groups. Algebraic theory and applications to cobordism. I”, Izv. Math., 9:5 (1975), 987–1006 | DOI | MR | Zbl

[60] V. Buchstaber, “Polynomial Eulerian characteristic of nilmanifolds”, Funct. Anal. Appl., 58:1 (2024), 16–32 | DOI | MR | Zbl

[61] I. K. Babenko and I. A. Taimanov, “On the existence of informal simply connected symplectic manifolds”, Russian Math. Surveys, 53:5 (1998), 1082–1083 | DOI | MR | Zbl

[62] I. K. Babenko and I. A. Taĭmanov, “On nonformal simply-connected symplectic manifolds”, Siberian Math. J., 41:2 (2000), 204–217 | DOI | MR | Zbl

[63] S. MacLane, Homology, Grundlehren Math. Wiss., 114, Academic Press, New York; Springer-Verlag, Berlin–Göttingen–Heidelberg, 1963, x+422 pp. | MR | Zbl

[64] J. W. Milnor and J. C. Moore, “On the structure of Hopf algebras”, Ann. of Math. (2), 81:2 (1965), 211–264 | DOI | MR | Zbl

[65] D. Riley and H. Usefi, “The isomorphism problem for universal enveloping algebras of Lie algebras”, Algebr. Represent. Theory, 10:6 (2007), 517–532 | DOI | MR | Zbl

[66] P. S. Landweber, “Cobordism operations and Hopf algebras”, Trans. Amer. Math. Soc., 129:1 (1967), 94–110 | DOI | MR | Zbl

[67] V. M. Buchstaber and A. P. Veselov, “Chern–Dold character in complex cobordisms and theta divisors”, Adv. Math., 449 (2024), 109720, 35 pp. ; (2020 (v1 – 2020, v4 – 2024)), 29 pp., arXiv: 2007.05782v3 | DOI | MR | Zbl

[68] D. Kraines, “Massey higher products”, Trans. Amer. Math. Soc., 124 (1966), 431–449 | DOI | MR | Zbl

[69] A. I. Mal'cev, “On a class of homogeneous spaces”, Amer. Math. Soc. Transl., 1951 (1951), 39, 33 pp. | MR | Zbl

[70] I. K. Babenko, “Algebra, geometry, and topology of the substitution group of formal power series”, Russian Math. Surveys, 68:1 (2013), 1–68 | DOI | MR | Zbl

[71] D. V. Millionshchikov, “Cohomology of nilmanifolds and Goncharova's theorem”, Russian Math. Surveys, 56:4 (2001), 758–759 | DOI | MR | Zbl

[72] Shiing-Shen Chern, Complex manifolds, Textos de Matemática, 5, Instituto de Física e Matemática, Universidade do Recife, Recife, Pernambuco, 1959, v+181 pp. | MR | Zbl

[73] J. E. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts in Math., 9, 2nd rev. print., Springer-Verlag, New York–Berlin, 1978, xii+171 pp. | DOI | MR | Zbl

[74] A. J. Berrick and A. A. Davydov, “Splitting of Gysin extensions”, Algebr. Geom. Topol., 1 (2001), 743–762 | DOI | MR | Zbl