Classification of the ring of Witt vectors and the necklace ring associated with the formal group law \(X+Y - qXY\)
Necklace polynomials and the necklace algebra, \(N_r(A)\), for \(A\) a commutative ring with \(1\), where defined by \textit{N. Metropolis} and \textit{G.-C. Rota} [Adv. Math. 50, 95--125 (1983; Zbl 0545.05009)] who showed that \(N_r(A)\) is closely related to the ring of Witt vectors \(W(A)\). This was generalized by \textit{C. Lenart} [J. Algebra 199, 703--722 (1998; Zbl 0915.05011)] to a functor associating to a formal group law \(F(x,y)\) over a ring \(B\) and an \(B\)-algebra \(A\) a necklace algebra \(N^F_r (A)\). The paper under review studies this situation in the special case of a multiplicative formal group law of the form \(F(x, y) = F_q(x,y) = x+y -q x y\) with \(q \in {\mathbb Z}\), a case also considered in section 5 of Lenart's paper. The author gives classification theorems for \(W^{F_q}(A)\) and \(N_r^{F_q}(A)\) in terms of the prime divisors of \(q\), gives an isomorphism from \(N^{F_q}_r (A)\) to \(\Lambda_{F_q}(A)\), the Grothendieck ring of formal power series of constant term 1, and gives a bijective (i.e., combinatorial) proof of the analog of the cyclotomic identity for \(F_q(x,y)\).
scientific article; zbMATH DE number 5144395
scientific article