Пределы групповых алгебр для растущих симметрических групп и сплетений

Алгебра и анализ. 2026. Т. 38. № 1. С. 139–197.

Девяткова И. Е., Olshanski G.

Denote by S(\infty) the infinite symmetric group formed by finitary permutations of the set of natural numbers; this is a countable group. We define its virtual group algebra; it is a completion of the standard group algebra  С[S(\infty)]. The virtual group algebra is obtained from the finite-dimensional group algebras C[S(n)] as the result of a limit transition as n\to\infty, where the limit is taken in the so-called tame representations of the group S(\infty). (Note that our virtual group algebra is very different from the  C^*-envelope.) We describe the structure of the virtual group algebra, revealing a connection with the degenerate affine Hecke algebras introduced by Drinfeld and Lusztig. Then we extend the results to the wreath products  G\wr S(\infty) with arbitrary finite groups G.

Keywords: Infinite symmetric group, бесконечная симметрическая группа, tame representations, degenerate affine Hecke algebra, ручные представления, вырожденная аффинная алгебра Гекке