Showing papers on "Hopf algebra published in 1979"

The Cofree Irreducible Hopf Algebra on an Algebra

Abstract: In this paper we construct the cofree irreducible Hopf algebra CH(U) on an algebra U over a field K and, with emphasis on the commutative case, analyze its structure. The functor determined by UH>CH(U) is a right adjoint to the functor from irreducible Hopf algebras to associative algebras determined by HF-*H +. The underlying coalgebra structure of CH(U) is the cofree pointed irreducible coalgebra C(U) of the vector space U (see [8, Chapter 12]), while the multiplication arises from the multiplication on U together with the universal mapping property of C(U). CH(U) becomes the shuffle algebra on U when U has the trivial multiplication. (The shuffle algebra is discussed in [8] and [4]. The cocommutative cofree irreducible Hopf algebra is developed in [10].) In the commutative case, CH(U) has particular interest, as the theory of the shuffle algebra developed by one of us (see [5]) can with some modifications be applied. Thus, even though the explicit multiplication is rather complex, we develop an effective computational technique for analyzing the structure. Further, and most importantly, as any irreducible Hopf algebra can be embedded in a cofree irreducible Hopf algebra, one is able to reduce questions about irreducible Hopf algebras to questions about cofree irreducible Hopf algebras. (An irreducible Hopf algebra H over a field K can be embedded into a shuffle algebra if and only if (1) H is commutative and (2) the characteristic of K is 0 or p>O and (H+)(P)=(0).) In Section 1 we introduce CH(U), prove its universal mapping property and some of its other extrinsic properties, and finally give its explicit multiplication (though we do not use it in formal proofs). In Section 2 we show how the ring basis of the shuffle algebra introduced in [5] can be modified to apply to

An algebraic theory of hypergroups

Abstract: The category of groups forms a full subcategory of the category of hypergroups. This larger category also contains other group theoretic objects, such as the conjugacy class hypergroup and character hypergroup of a finite group. Definitions of hypergroups and hypergroup morphisms are given and related to double algebras (which are simultaneously algebras and cogebras, but not Hopf algebras in general). Quotient and orbit hypergroups are defined. Coefficients are allowed in a more or less arbitrary field. These concepts provide a new language in which groups and their character tables can be fruitfully discussed.

Homology of symmetric algebras

Abstract: If M is a simplicial module over a field K of characteristic p , the homology of the symmetric algebra Sd of M depends only on H [M] and it is the universal envelopping coalgebra U [V] of a vector space V. An explicit description of this Hopf algebra with divided powers is given. The proof is done in such a way thst it is possible to see what are the contributions of a generator of H [M] to H [S M].If n  2p, Hn [S M]is computed as in characteristic zero. This is no more true in higher dimensions: for instance a genera-tor of H 3[M] yields a generator of H 2p+1[SM].

Local $H$-maps of classifying spaces

Abstract: ABSTRACr. Let BU denote the localization at an odd prime p of the classifying space for stable complex bundles, and let f: BU -* BU be an H-map with fiber F. In this paper the Hopf algebra H*(F, Z/p) is computed for any such f. For certain H-maps f of geometric interest the p-local cohomology of F is given by means of the Bockstein spectral sequence. A direct description of H*(F, Z(p)) is also given for an important special case. Applications to the classifying spaces of surgery will appear later.

Affine Group Schemes: Examples

Abstract: A homomorphism of affine group schemes is a natural map G → H for which each G(R) → H(R) is a homomorphism. We have already seen the example det: GL n → G m . The Yoneda lemma shows as expected that such maps correspond to Hopf algebra homomorphisms. But since any map between groups preserving multiplication also preserves units and inverses, we need to check only that Δ is preserved. An algebra homomorphism between Hopf algebras which preserves Δ must automatically preserve S and e.

On the rigidity of graded algebras

Abstract: If ? is a graded algebra (separated and complete) over a field of characteristic zero and 9 is rigid in the category of algebras, then 9 is rigid in the category of filtered algebras.

Graded G-structures

Abstract: In this communication I would like to give a progress report on a program to construct graded geometric structures on graded differential manifolds. Most of the results so far are on the formal algebraic level and many important problems of analysis remain. Kostant I has given a definition of a graded differential manifold of dimension (n,m) and this is discussed in these proceedings by M. Batchelor. I only mention here that it consists of a pair (M,A) where M is an ordinary C ~ manifold of dimension n and A is a sheaf of Z2-graded commutative algebras whose underlying graded vector space V = V 0 + V I has dim V 0 = n, dim V 1 = m. A graded Lie group (G,A) then is a pair where G is a Lie group and the restricted dual A ° (vanishes on some ideal of finite codimension) has the structure of a graded Hopf algebra and can be represented as A ° = R(G) ^ E(G) where R(G) is the group ring, E(G) is the universal enveloping algebra of the graded Lie algebra G = G O + G I with G O the Lie algebra of G, and ^ denotes semidirect product. In analogy with the usual situation, graded frames can be introduced and the graded frame bundle constructed. This consists of a principal fibre bundle L(M,G) with group G = GL(n)×GL(m) such that the graded tangent bundle T(M,A) defined by Kostant is an associated vector bundle for L(M,G). The graded frame bundle (not really a bundle) is then given by the graded manifold (L(M,G), A8 H) where H ° = R(G) ^ E(G) , G = GL(n)× GL(m), G = gl(n,m) = End V. Now if G'C G is a closed Lie subgroup of G with a subHopf algebra H' C H , then (L(M,G'), A8 H') is called a graded G-struc ture if L(M,G') is reduced subbundle of L(M,G). There is a natural right action R a of H' defined on (L(M,G'), ASH'). Associated with the graded tangent bundle of a G-structure, there is the tangent sheaf derA + derH (der A = derivations of A), and this is a free A@ H module. A complement H to derH is called a horizontal subspace, and as usual H defines a connection if R * H = H. As underlying graded vector spaa ces H is isomorphic to V. In analogy with the ordinary ease, we study graded G-structures by studying graded Spencer cohomology. Let GC gl(n,m) = V8 V* be a graded Lie algebra. We define G (1) i i I I the first prolongation of G by all T E Hom(V,G) which satisfy T(u)v =(-i) luILvi T(V)U where u,v are homogeneous elements in V of degree lul, Ivl respectively. The k th prolongation is defined inductively by G (k) = (G(k-l)) (I) • We construct Z 2 gravector spaces C k+l'l = G (k) O Al(V *) where AI(V *) = AI(V~)0 Sl(V~) is the graded ded exterior algebra over V*. The cochain map ~:C k+l'/ >?,/+i is defined ~ by