Formal Group-Theoretic Generalizations of the Necklace Algebra, Including aq-Deformation
✍ Scribed by Cristian Lenart
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 292 KB
- Volume
- 199
- Category
- Article
- ISSN
- 0021-8693
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✦ Synopsis
95᎐125 studied the necklace polynomials, and were lead to define the necklace algebra as a combinato-Ž rial model for the classical ring of Witt ¨ectors which corresponds to the multi-. plicative formal group law X q Y y XY . In this paper, we define and study a generalized necklace algebra, which is associated with an arbitrary formal group law F over a torsion free ring A. The map from the ring of Witt vectors associated with F to the necklace algebra is constructed in terms of certain generalizations of the necklace polynomials. We present a combinatorial interpretation for these polynomials in terms of words on a given alphabet. The actions of the Verschiebung and Frobenius operators, as well as of the p-typification idempotent are described and interpreted combinatorially. A q-analogue and other generalizations of the cyclotomic identity are also presented. In general, the necklace algebra can only be defined over the rationalization A m .ޑ Nevertheless, we show that for the family Ž . of formal group laws over the integers F X,Y s X q Y y qXY, q g ,ޚ we can q define the corresponding necklace algebras over .ޚ We classify these algebras, and define isomorphic ring structures on the groups of Witt vectors and the groups of curves associated with the formal group laws F . The q-necklace polynomials, q which turn out to be numerical polynomials in two variables, can be interpreted combinatorially in terms of so-called q-words, and they satisfy an identity generalizing a classical one.