Showing papers on "Hopf algebra published in 1987"

Some remarks on Koszul algebras and quantum groups

Abstract: La categorie des algebres quadratiques est munie d'une structure tensorielle. Ceci permet de construire des algebres de Hopf du type «(semi)-groupes quantiques»

Taming wild extensions with Hopf algebras

Abstract: Let K c L be a Galois extension of number fields with abelian Galois group G and rings of integers R c S, and let si be the order of 5 in KG. If si is a Hopf /¿-algebra with operations induced from KG, then S is locally isomorphic to si as j^module. Criteria are found for si to be a Hopf algebra when K = Q or when l./K is a Kummer extension of prime degree. In the latter case we also obtain a complete classification of orders over R in L which are tame or Galois //-extensions, // a Hopf order in KG, using a generalization of the discriminant. Galois module theory seeks to describe the ring of integers S of a Galois extension L D K of number fields with Galois group G as a G-module, either absolutely (i.e. over ZG) or relatively (i.e. over RG, R the ring of integers of K). In the relative case, the fundamental result is Noether's theorem: S is locally ÄG-isomorphic to RG, that is, S has a normal basis locally at each prime of R, if and only if L/K is tame, i.e. tamely ramified. However, nontame extensions L/K abound (e.g. K = Q, L = Q(\[m), m = 2 or 3 (mod4), or L = Q(f)> f a primitive /nth root of unity, m not square-free). In attempting to extend the tame results to nontame extensions, one approach, introduced by H. W. Leopoldt [17] and studied by H. Jacobinski [15], F. Bertrandias, J.-P. Bertrandias and M. J. Ferton [2, 3, 4], A. M. Berge [1], and recently, M. Taylor [24, 28], is to replace RG by a larger order over R in KG, in particular, the order s? of S in KG. s/= {a g KG\aSçz S), and consider 5 as an j^module. For K = Q this approach was successful: S = s/ as j^module when G = Gal(L/Q) is abelian. However, in [2 and 3] it is shown that S = si? as j^module may fail, even locally, if G is dihedral or if L/K is a Kummer extension of prime degree. This paper starts from the premise that it is of interest to know when jé is a Hopf A-algebra with operations induced from those on the Hopf A"-algebra KG (abusing language we henceforth call such an s/ a Hopf subalgebra of KG). There are several reasons for investigating such a premise. In general, as Bergman [29] has eloquently explained, for an algebra A to act on another algebra S and to respect the algebra structure of S, it is natural for A to be Received by the editors April 11, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 11R33. 13B05, 16A24: Secondary 11S15. ©1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page 111 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

The Conner-Floyd Map for Formal A-Modules

Abstract: A generalization of the Conner-Floyd map from complex cobordism to complex /C-theory is constructed for formal A-modules when A is the ring of algebraic integers in a number field or its p-adic completion. This map is employed to study the Adams-Novikov spectral sequence for formal A-modules and to confirm a conjecture of D. Ravenel. 0. Introduction. Let BP be the spectrum representing Brown-Peterson cohomology with respect to a prime p and let E be the Adams summand of complex if-theory with respect to this prime. The BP version of the Conner-Floyd map is a map of spectra BP —+ E which induces a natural equivalence BP, X ®Bp. E» ~ E.X. In particular this induces an isomorphism £* ®Bp. BP»BP®bp.£. ~ E.E and so provides a way of computing the Hopf algebra of stable co-operations for E from those for BP. Using this one can obtain a description of E*E similar to that for K*K contained in [AHS]. The study of BP and the computation of BP* BP are based on a study of formal group law, in particular the p-typical formal group law. In [Rl] Ravenel studied a generalization of this situation where the formal group law is replaced by a formal yl-module where A is the ring of integers in an algebraic number field K or its p-adic completion. The purpose of the present paper is to describe the corresponding generalization of the map (BP», BP* BP) —► (E*,E*E) induced by the Conner-Floyd map, and to compute the generalization of E*E. This is of interest because it provides some information about a conjecture (3.10) made in [Rl]. This conjecture concerned the value of a certain Ext group Exty,t(Va, Va) when K is an extension of the field Qp of p-adic numbers. Here (Va, VaT) is the Hopf algebroid corresponding to the A-typical formal A module. This group was conjectured to be, up to small factor, A/J^,^. Here J^iq_1\ is the ideal of A generated by the elements of the form aTM — 1 for units a of A congruent to 1 mod(7r) and (it) is the unique prime ideal in A. We will show, using the generalization of the Conner-Floyd map, that A/J*,^ occurs as E\' in the chromatic spectra sequence for formal ^-modules [Rl, Lemma 2.10] and that the small factor in the conjecture is contributed by the nontriviality of the differential d\ originating from this group. We will analyze this differential and show that it is nonzero for A Received by the editors June 27, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 55T25; Secondary 55N22, 14L05. ©1987 American Mathematical Society 0002-9947/87 SI.00 + $.25 per page 319 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

The homology of double loop spaces of complex stiefel manifolds

Abstract: The Hopf algebra structure of H*(Q2SU(n+l)/SU(m+l) : FJ and the action of the Steenrod algebra on it are determined.

The Berezinian in some monoidal categories

Abstract: It follows from the definition that a vector symmetry is a symmetric closed monoidal category [6]. According to [4], a constant unitary solution of a triangle equation R~EndV| determines a category ~ generated by tensor powers of V, their sums, and their subfactors. If the obtained symmetric category ~ is closed, then it is equivalent to the vector symmetry comod-H for some Hopf algebra H and a form 0.

Internal symmetry algebra of exactly integrable dynamical systems in the quantum domain

Abstract: For the example of the generalized Toda chain in two-dimensional space it is shown that in the quantum domain the semisimple algebras of the classical problem go over into the associative Hopf algebras described by Drinfel'd as quantum algebras. In terms of the quantum algebras, the Heisenberg operators of the interacting field can be expressed as functions of the in fields by means of the formulas of the classical theory, and the expressions obtained earlier for them acquire a simple algebraic meaning.