multivalued group in nLab

Contents

Contents

Idea

The notion of nn-valued groups is an analogue of groups where the multiplication takes values in the nn-th “symmetric power” of the underlying set. 2-valued formal groups appeared first in the study of characteristic classes by Buchstaber & Novikov 1971.

Definition

Let nn be a positive integer. Denote by Sym n(G)Sym^n(G) the nn-th symmetric power of a set GG, or equivalently the set of multisets of nn not necessarily different elements [g 1,…,g n][g_1,\ldots,g_n] in GG.

An nn-valued group is given by a set GG together with a function (“multiplication”)

*:G×G⟶Sym n(G), \ast \;\colon\; G \times G \longrightarrow Sym^n(G) \,,

satisfying:

• (associativity) The following n 2n^2-element multisets are equal:

  [a*(b*c) 1,a*(b*c) 2,…,a*(b*c) n]\big[ a\ast (b\ast c)_1, a\ast (b\ast c)_2,\ldots, a\ast (b\ast c)_n\big] and [(a*b) 1*c,…,(a*b) n*c]\big[(a\ast b)_1\ast c,\ldots,(a\ast b)_n\ast c\big],

• (unitality) there is a neutral element e\mathrm{e} such that for all x∈Gx\in G, e*x=x*e=[x,x,…,x]\mathrm{e} \ast x = x \ast \mathrm{e} = [x,x,\ldots,x],

• (inverses) For every g∈Gg\in G there exists g −1∈Gg^{-1}\in G such that e∈g −1*g\mathrm{e} \in g^{-1}\ast g and e∈g*g −1\mathrm{e} \in g\ast g^{-1}.

Literature

Origin of the notion in multivalued formal groups appearing in complex oriented cohomology theory:

• В. М. Бухштабер, С. П. Новиков, Формальные группы, степенные системы и операторы Адамса, Мат. Сборник (Н.С.) 84 (1971) 81–118;

  English translation:

  Victor M. Buchstaber, Sergei P. Novikov, Formal groups, power systems and Adams operators, Math. USSR-Sb. 13 (1971) 80-116 [doi:10.1070/SM1971v013n01ABEH001030]

• Victor M. Buchstaber, Two-valued formal groups. Some applications to cobordisms, Uspehi Mat. Nauk 26 3 (1971), (159) 195–196 [MR 0461533]

• Victor Buchstaber, §5 in: Cobordisms in problems of algebraic topology, J Math Sci 7 (1977) 629–653 [doi:10.1007/BF01084983]

The notion has in fact been sporadically studied much earlier, e.g. in the context of hypergroups.

• J. Delsarte, Hypergroupes et opérateurs de permutation et de transmutation, La théorie des équations aux dérivées partielles. Nancy, 9-15 avril 1956, pp. 29–45 Colloq. Internat. CNRS, LXXI [International Colloquia of the CNRS] Centre National de la Recherche Scientifique, Paris, 1956 MR0116151

Some newer articles

• D. Coulette, F. Déglise, J Fasel, J. Hornbostel, Formal ternary laws and Buchstaber’s 2-groups, Manuscripta Math. (2023) doi

• В. М. Бухштабер, Функциональные уравнения ассоциированные с теоремами сложения для эллиптических функций и двузначные алгебраические группы, Успехи Мат. Наук 45:3 (1990) 185–186 (transl.: V. M. Bukhstaber, Functional equations that are associated with addition theorems for elliptic functions and two-valued algebraic groups, Russian Math. Surveys 45:3 (1990) 213–215.)

In 2010. V. Dragović has observed that the integrability of Kovalevskaia top is related to associativity of certain nn-valued group. A related later developments on integrable systems are touched upon in:

• Victor M. Buchstaber, Vladimir Dragović, Two-valued groups, Kummer varieties, and integrable billiards, Arnold Math J. 4, 27–57 (2018) [doi:10.1007/s40598-018-0085-2]