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Uspekhi Mat. Nauk, 2000, Volume 55, Issue 4(334), Pages 5–24 (Mi umn312)  

This article is cited in 7 scientific papers (total in 8 papers)

Algebraic aspects of the theory of multiplications in complex cobordism theory

B. I. Botvinnika, V. M. Buchstaberb, S. P. Novikovc, S. A. Yuzvinskiia

a University of Oregon
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
c University of Maryland

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 $S$ (the Landweber–Novikov algebra) acting on the dual Hopf algebra $S^*$ with distinguished "topologically integral" part $\Lambda$ 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 $S$. 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.

DOI: https://doi.org/10.4213/rm312

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English version:
Russian Mathematical Surveys, 2000, 55:4, 613–633

Bibliographic databases:

UDC: 513.836
MSC: Primary 57R77; Secondary 16W30, 57T05, 16G99, 55N22
Received: 01.06.2000

Citation: B. I. Botvinnik, V. M. Buchstaber, S. P. Novikov, S. A. Yuzvinskii, “Algebraic aspects of the theory of multiplications in complex cobordism theory”, Uspekhi Mat. Nauk, 55:4(334) (2000), 5–24; Russian Math. Surveys, 55:4 (2000), 613–633

Citation in format AMSBIB
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\paper Algebraic aspects of the theory of multiplications in complex cobordism theory
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\vol 55
\issue 4(334)
\pages 5--24
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\pages 613--633
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    2. Feldman, KE, “Chern numbers of Chern submanifolds”, Quarterly Journal of Mathematics, 53 (2002), 421  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    3. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    4. Braden H.W., Feldman K.E., “Functional equations and the generalised elliptic genus”, J. Nonlinear Math. Phys., 12, suppl. 1 (2005), 74–85  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    5. 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  crossref  mathscinet  zmath  isi
    6. V. M. Buchstaber, N. Yu. Erokhovets, “Polytopes, Fibonacci numbers, Hopf algebras, and quasi-symmetric functions”, Russian Math. Surveys, 66:2 (2011), 271–367  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. V. M. Buchstaber, “Complex cobordism and formal groups”, Russian Math. Surveys, 67:5 (2012), 891–950  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. I. Yu. Limonchenko, T. E. Panov, G. Chernykh, “$SU$-bordism: structure results and geometric representatives”, Russian Math. Surveys, 74:3 (2019), 461–524  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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