Abstract:
The general classiffication problem for stable associative multiplications in complex cobordism theory is considered. It is shown that this problem reduces to the theory of a Hopf algebra SS (the Landweber–Novikov algebra) acting on the dual Hopf algebra S∗S∗ with distinguished "topologically integral" part ΛΛ that corresponds to the complex cobordism algebra of a point. We describe the formal group and its logarithm in terms of the algebra representations of SS. The notion of one-dimensional representations of a Hopf algebra is introduced, and examples of such representations motivated by well-known topological and algebraic results are given. Divided-difference operators on an integral domain are introduced and studied, and important examples of such operators arising from analysis, representation theory, and non-commutative algebra are described. We pay special attention to operators of division by a non-invertible element of a ring. Constructions of new associative multiplications (not necessarily commutative) are given by using divided-difference operators. As an application, we describe classes of new associative products in complex cobordism theory.
Citation:
B. I. Botvinnik, V. M. Buchstaber, S. P. Novikov, S. A. Yuzvinskii, “Algebraic aspects of the theory of multiplications in complex cobordism theory”, Russian Math. Surveys, 55:4 (2000), 613–633
\Bibitem{BotBucNov00}
\by B.~I.~Botvinnik, V.~M.~Buchstaber, S.~P.~Novikov, S.~A.~Yuzvinskii
\paper Algebraic aspects of the theory of multiplications in complex cobordism theory
\jour Russian Math. Surveys
\yr 2000
\vol 55
\issue 4
\pages 613--633
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Linking options:
https://www.mathnet.ru/eng/rm312
https://doi.org/10.1070/rm2000v055n04ABEH000312
https://www.mathnet.ru/eng/rm/v55/i4/p5
This publication is cited in the following 10 articles:
T. E. Panov, G. S. Chernykh, “$SU$-linear operations in complex cobordism and the $c_1$-spherical bordism theory”, Izv. Math., 87:4 (2023), 768–797
I. Yu. Limonchenko, T. E. Panov, G. S. Chernykh, “$SU$-bordism: structure results and geometric representatives”, Russian Math. Surveys, 74:3 (2019), 461–524
V. M. Buchstaber, “Complex cobordism and formal groups”, Russian Math. Surveys, 67:5 (2012), 891–950
V. M. Buchstaber, N. Yu. Erokhovets, “Polytopes, Fibonacci numbers, Hopf algebras, and quasi-symmetric functions”, Russian Math. Surveys, 66:2 (2011), 271–367
Victor M. Buchstaber, Kolmogorov's Heritage in Mathematics, 2007, 139
Savin A., Sternin B., “Pseudo differential subspaces and their applications in elliptic theory”, C(star)-Algebras and Elliptic Theory, Trends in Mathematics, 2006, 247–289
Braden H.W., Feldman K.E., “Functional equations and the generalised elliptic genus”, J. Nonlinear Math. Phys., 12, suppl. 1 (2005), 74–85
A. A. Bolibrukh, A. P. Veselov, A. B. Zhizhchenko, I. M. Krichever, A. A. Mal'tsev, S. P. Novikov, T. E. Panov, Yu. M. Smirnov, “Viktor Matveevich Buchstaber (on his 60th birthday)”, Russian Math. Surveys, 58:3 (2003), 627–635
Feldman, KE, “Chern numbers of Chern submanifolds”, Quarterly Journal of Mathematics, 53 (2002), 421
V. M. Maksimov, “Representation of the algebra of linear maps by a skew product of algebras”, Russian Math. Surveys, 56:1 (2001), 168–169
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