nLab multivalued group

Idea

The notion of $n$-valued groups is an analogue of groups where the multiplication takes values in the $n$-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 $n$ be a positive integer. Denote by $Sym^n(G)$ the $n$-th symmetric power of a set $G$, or equivalently the set of multisets of $n$ not necessarily different elements $[g_1,\ldots,g_n]$ in $G$.

An $n$-valued group is given by a set $G$ together with a function (“multiplication”)

$$\ast \;\colon\; G \times G \longrightarrow Sym^n(G) \,,$$

satisfying:

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

  $\big[ a\ast (b\ast c)_1, a\ast (b\ast c)_2,\ldots, a\ast (b\ast c)_n\big]$ and $\big[(a\ast b)_1\ast c,\ldots,(a\ast b)_n\ast c\big]$,

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

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

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 $n$-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]

category: algebra