Showing papers on "Hopf algebra published in 1978"

Symmetry for finite dimensional Hopf algebras

Abstract: This note refines criteria given by R. G. Larson and M. E. Sweedler for a finite dimensional Hopf algebra to be a symmetric algebra, with applications to restricted universal enveloping algebras and to certain finite dimensional subalgebras of the hyperalgebra of a semisimple algebraic group in characteristic p. Let A be a finite dimensional associative algebra over a field K. Then A is called Frobenius if there exists a nondegenerate bilinear form f: A x A -* K which is associative in the sense that f (ab, c) = f (a, bc) for all a, b, c E A [3, Chapter IX]. A is called symmetric if there exists a symmetric form of this type [3, ?66]. For example, semisimple algebras and group algebras of finite groups are synumetric. We investigate here the extent to which finite dimensional Hopf algebras (with antipode) are symmetric; they are always Frobenius, thanks to the main theorem of [8]. 1. Hopf algebras. In this section H denotes a finite dimensional Hopf algebra over an arbitrary field K, with antipode s and augmentation e: H -* K. According to the main theorem of [8], existence of the antipode implies (and is implied by) the existence of a (nonsingular) left integral A E H, which is unique up to scalar multiples. By definition, A satisfies: hA = e(h)A, for all h E H. Equally well, H has a right integral A', unique up to scalar multiples. If A' is proportional to A, H is called unimodular. With a left integral A is associated a nondegenerate bilinear associative form b on H [8, ?7]. As a result, H is a Frobenius algebra. From the second corollary of Proposition 8 in [8], applied to the dual Hopf algebra (whose antipode has the same order as s), we obtain immediately: THEOREM 1. With notation as above, b is symmetric if and only if H is unimodular and s2= 1. In particular, if the latter conditions hold, then H is symmetric. We can apply this to the algebras un (n = 1, 2, .. .) defined in [6, Appendix U], [7]. These are finite dimensional Hopf subalgebras of the hyperalgebra UK of a simply connected, semisimple algebraic group G over an algebraically Reccived by the editors February 28, 1977. AMS (MOS) subject classifications (1970). Primary 16A24, 16A36, 17B50; Secondary 17B45.

Quotients of hopf algebras

Abstract: (1978). Quotients of hopf algebras. Communications in Algebra: Vol. 6, No. 17, pp. 1789-1800.

A Novice's guide to the adams-novikov spectral sequence

Abstract: Ever since its introduction by J. F. Adams [8] in 1958, the spectral sequence that bears his name has been a source of fascination to homotopy theorists. By glancing at a table of its structure in low dimensions (such have been published in [7], [i0] and [27]; one can also be found in ~2) one sees not only the values of but the structural relations among the corresponding stable homotopy groups of spheres. It cannot be denied that the determination of the latter is one of the central problems of algebraic topology. It is equally clear that the Adams spectral sequence and its variants provide us with a very powerful systematic approach to this question.

A topological note on the Adams spectral sequence based on Morava’s $K$-theory

Abstract: In this note, we compute the Hopf algebra K(n) (K{n)) and consider the Adams spectral sequence of K(n)J-) theory.

An identity on algebras over a Hopf algebra

Abstract: (for the proof see [1]). This formula has been proved useful in quite a few occasions (for example see [1], [2]). In this note we give a generalization of this formula for algebras over a Hopf algebra. Let A be a Hopf algebra over a commutative ring with unit. We assume that A is connected and that its comultiplication 4' is associative. Let X: A -* A be the canonical conjugation of A. Let M be a graded algebra over the Hopf algebra A. For the terminology and the basic results, we refer to [31. Under those assumptions, we have

The periodicity of cohomologies of Hopf algebras

Abstract: The results on periodicity of cohomologies of finite groups are extended to the case of Hopf algebras, the theory of the homologies of which has been developed by A V Yakovlev