Showing papers on "Hopf algebra published in 1986"

Crossed products and inner actions of Hopf algebras

Abstract: This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include \"inner\" group gradings of algebras. We prove that if 7r : H —► H is a Hopf algebra epimorphism which is split as a coalgebra map, then H is algebra isomorphic to A #„ H, a crossed product of H with the left Hopf kernel A of it. Given any crossed product A #CT H with H (weakly) inner on A, then A #CT if is isomorphic to a twisted product AT[H] with trivial action. Finally, if H is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of A implies that of A #„ H; in particular this is true if the (weak) action of H is inner. Introduction. The purpose of this paper is to begin to lay the foundations of a general theory of actions of Hopf algebras on noncommutative algebras. The importance of such a theory derives from three special cases in which the Hopf algebra is a group algebra, an enveloping algebra of a Lie algebra, or the dual of a group algebra. These cases show that the theory encompasses the study of actions of groups as automorphisms of algebras, the study of actions of Lie algebras as derivations of algebras, and the study of group graded rings, respectively. Each of these areas of study have been quite active lately (see [16, 9, 5]). A fundamental concept in the first two cases has been the notion of inner action. One of the major purposes of this paper is to study inner actions of Hopf algebras; as a new example, we will investigate what is meant by an inner grading. Another important concept in the first two cases is that of a semidirect product (smash product): for group actions we have skew group rings and for Lie algebra actions we have differential polynomial rings. This notion is also defined for Hopf algebra actions. More generally one can consider crossed products A ffa H of an algebra A with a Hopf algebra H, where the multiplication of the copy of H in A ffa H is twisted by a cocycle o, and their study is the other major purpose of this paper. It turns out that these two concepts (inner actions and crossed products) are closely interrelated. Now inner actions and crossed products of Hopf algebras were both studied by Sweedler [24] in the context of the cohomology theory of Hopf algebras. The present paper owes a great debt to Sweedler's work, especially in §§1 and 4. However, his set-up was restricted at crucial places to the situation where H is a cocommutative Received by the editors December 12, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A24; Secondary 16A72, 16A03, 46L40. The third author was partially supported by NSF Grant DMS 8500959 and by a Guggenheim Memorial Foundation Fellowship. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

The Yangians, Bethe Ansatz and combinatorics

Abstract: An axiomatic definition of a quantum monodromy matrix and the representations of its corresponding Hopf algebra are discussed. The connection between the quantum inverse transform method and the representation theory of a symmetric group is considered. A new approach to the completeness problem of Bethe vectors is also given.

Actions of Commutative Hopf Algebras

Abstract: We show that actions of finite-dimensional semisimple commutative Hopf algebras H on //-module algebras A are essentially group-gradings. Moreover we show that the centralizer of H in the smash product A # H equals A\" ® H. Using these we invoke results about group graded algebras and results about centralizers of separable subalgebras to give connections between the ideal structure of A, A and A # H. Examples of the above occur naturally when one considers: (1) finite abelian groups G of automorphisms of an algebra A with | G |~ e A; (2) G-graded algebras, for finite groups G; (3) finite-dimensional restricted Lie algebras L, with semisimple restricted enveloping algebra u(L), acting as derivations on an algebra A.

On the indecomposable elements of the bar construction

Abstract: An explicit formula for a canonical splitting s: Q4(9 ) ?4(& ) of the projection 4(l ) Q4(l ) of the bar construction on a commutative d.g. algebra 4' onto its indecomposables is given. We prove that s induces a d.g. algebra isomorphism A(Q4(9 )) R(o ) and that H(QtI(4`)) is isomorphic with QH(-4( )) If 6& is an augmented commutative d.g. algebra over a field I of characteristic zero, then the bar construction 4(9 ) on 4is a commutative d.g. Hopf algebra. Denote the augmentation ideal of 4(9 ) by If4( 61) (or I). The indecomposable elements of 4(6 ) are defined to be Q_(0 ) = I/I2. Since 2(9 ) = I D I?f(& ), there is a natural projection 4( 4') -Q4(& ). Our main result is the following. THEOREM. If & is a commutative d. g. algebra over the field I of characteristic zero, then there is a natural splitting s: Q2(9 ) -3 It4(9 ) of the natural projection rr: Ig(& ) -Q?6(9 ). The splitting s commutes with the differentials. Moreover, the map A(Q-4(&`)) --1(9 ) induced by s, from the free commutative d.g. algebra generated by Q2(6 ) into .4(& ), is a d.g. algebra isomorphism. In fact, the idempotent y = s o 7T is given by the formula n -y[all l an] = Y. E Y, (_l) m1 e( ) [a,(,)l ... la,(n)], m= 1 r1 +?r. = n aosh(r1,.* , rm) where sh(r1, .. ., rm) denotes the shuffles of (1, .. ., m } of type (r1, . . ., rm) and where E: -M { -1, 1 is the representation of the symmetric group obtained by assigning weight -1 + deg a1 to a1. COROLLARY. The natural map QH (?4(` )) -H (Q?1(9 )) is an isomorphism. 0 One version of the Poincare-Birkhoff-Witt (P.B.W.) theorem (cf. [6, appendix B]) states that if A is a commutative d.g. Hopf algebra, then there is a natural d.g. coalgebra isomorphism S(PA) A between A and the symmetric coalgebra on the primitives PA of A. Thus, the assertion that there is a d.g. algebra isomorphism Received by the editors March 20, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 55P62.

Hopf Algebra Forms of the Multiplicative Group and other Groups

Abstract: The multiplicative group functor, which associates with each k-algebra its group of units, is affine with Hopf algebra k[x,x−1]. The purpose of this paper is to determine explicitly all Hopf algebra forms of k[x,x−1] with only minor restrictions on k (2 not a zero-divisor and Pic(2)(k)=0). We also describe explicitly (by generators and relations) the Hopf algebra forms of kC3, kC4 and kC6, where Cn is the cyclic group of order n. Some of our results could be drawn from [1,III §5.3.3] where a similar result as ours is indicated (and left as an exercise). We prefer however a less technical approach, in particular we do not use the extended theory of algebraic groups and functor sheaves.

Tameness and local normal bases for objects of finite Hopf algebras

Abstract: Let R be a commutative ring, S an R-algebra, H a Hopf Ralgebra, both finitely generated and projective as R-modules, and suppose S is an H-object, so that H* = HomR(H, R) acts on S via a measuring. Let I be the space of left integrals of H*. We say S has normal basis if S H as H*modules, and S has local normal bases if Sp Hp as Hp-modules for all prime ideals p of R. When R is a perfect field, H is commutative and cocommutative, and certain obvious necessary conditions on S hold, then S has normal basis if and only if IS = R = SH*. If R is a domain with quotient field K, H is cocommutative, and L = S OR K has normal basis as (H* 0 K)-module, then S has local normal bases if and only if IS = R SH*. Suppose K is a number field with ring of integers R, L is a finite Galois extension of K with Galois group G, and S is the integral closure of R in L. Then G acts as a group of automorphisms of S. Relative Galois module theory seeks to understand S as an RG-module via this action. The most basic question is to inquire whether S is locally isomorphic to RG, that is, for all primes p of R, Sp has a local normal basis as a free Rp-module. This question was answered by Emmy Noether, who showed that S has a local normal basis at every prime p of R if and only if L/K is tamely ramified. Here, tamely ramified means that at each prime ideal p of R, the ramification index of any prime P of S lying over p is relatively prime to the residue field characteristic. It is well known that this latter condition is equivalent to the surjectivity of the trace map tr: S -* R, tr(s) = EZaCG (s). The purpose of this paper is to formulate and prove an extension of Noether's theorem to objects of finite cocommutative Hopf algebras. Assume now only that R is a commutative ring, H is a Hopf R-algebra which is a finitely generated projective R-module, and H* = HomR(H, R), the dual Hopf algebra. A commutative R-algebra S is an H-object if S is a right H-comodule via a map a: S -*S 0 H which is an R-algebra homomorphism. If S is an H-object, then a induces a measuring a*: H* ? S -* S in the sense of Sweedler [21]. Galois module theory in this setting is the study of S as an H*-module via the measuring a*. The "trivial" example is S = H itself, with a = A, the comultiplication on H. So we say that the H-object ? has normal basis if S H as H*-modules, and S has local normal bases if for all primes p of R, Sp Hp as H*-modules. Since RGf (RG)* as RG-modules for G any finite group, this notion of normal basis, when specialized to H* = RG, is equivalent to the classical notion. Received by the editors May 7, 1985 and, in revised form, January 7, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 13B05; Secondary 16A24. @)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

Cup products in sheaf cohomology

Abstract: Let k be an algebraically closed field, and let l be a prime number not equal to char(k). Let X be a locally fibrant simplicial sheaf on the big etale site for k, and let Y be a k scheme which is cohomologically proper. Then there is a Kunneth-type isomorphism which is induced by an external cup-product pairing. Reductive algebraic groups G over k are cohomologically proper, by a result of Friedlander and Parshall. The resulting Hopf algebra structure on may be used together with the Lang isomorphism to give a new proof of the theorem of Friedlander-Mislin which avoids characteristic 0 theory. A vanishing criterion is established for the Friedlander-Quillen conjecture.

X-inner automorphisms of crossed products and semi-invariants of Hopf algebras

Abstract: LetR*G be a crossed product of the groupG over the prime ringR and assume thatR*G is also prime. In this paper we study unitsq in the Martindale ring of quotientsQ0(R*G) which normalize bothR and the group of trivial units ofR*G. We obtain quite detailed information on their structure. We then study the group ofX-inner automorphisms ofR*G induced by such elements. We show in fact that this group is fairly close to the group of automorphisms ofR*G induced by certain trivial units inQ0(R)*G. As an application we specialize to the case whereR=U(L) is the enveloping algebra of a Lie algebraL. Here we study the semi-invariants forL andG which are contained inQ0(R*G) and we obtain results which extend known properties ofU(L). Finally, every cocommutative Hopf algebraH over an algebraically closed field of characteristic 0 is of the formH=U(L)*G. Thus we also obtain information on the semi-invariants forH contained inQ0(H).

Hopf algebras and Galois descent

Abstract: that: "some of the me-thods used in Galois theory andformulated in the Hopf algebrasto Chase and Sweedler to defineterms of Hopf algebras whichdon 't formulate a corresponding theory of descent" . The pur-pose of this notes is to develop this theory, using asGalois definition of object, the one given by Chase andSweedler in

An isomorphism for the Grothendieck ring of a Hopf algebra order

Abstract: If G is a finite abelian group, R is a principal ideal domain with field of quotients an algebraic number field K which splits G, and if A is a Hopf algebra order in KG, then the Grothendieck ring of the category of finitely generated A-modules is isomorphic to the Grothendieck ring of the category of finitely generated RG-modules. The Grothendieck group Ag9 of the category of finitely generated modules over a ring A is analogous to the free group generated by the characters of a finite group. If H is a Hopf algebra, then Hg9 iS a ring. We show if R is a principal ideal domain with field of quotients K which splits G, if G is a finite abelian group, and if A is a Hopf algebra order in KG, then RG9 Ag. Swan [6] has shown that if G is a finite group, and if R is a semilocal Dedekind domain, then RG.9 KG9. It follows that if R is a semilocal Dedekind domain with field of quotients K, if G is a finite abelian group split by K, and if A is a Hopf algebra order in KG, then A,9 KG9. Swan's result says that, factoring out relations induced by short exact sequences of representations, the representation theory of a group ring over a semilocal Dedekind domain is the same as the representation theory of the group algebra over the field of quotients. Our result says that, for abelian groups, the representation theory is the same for any Hopf algebra order in the group algebra as for the group ring. If C is an abelian category, the Grothendieck group of C is defined as follows: for each object M E C there is a generator [M]; for each short exact sequence 0 -OM' M -M" O0 in C there is a relation [Ml = [MI] + [M"]. For the remainder of this paper R will be a principal ideal domain with field of quotients an algebraic number field K. If A is an R-algebra which is a finitely generated projective R-module, we will denote by AC the category of finitely generated A-modules, and by AC the category of finitely generated A-modules which are torsion free as R-modules. We will denote the Grothendieck group of AC by Ag, and the Grothendieck group of AC by Ag. The embedding AC -4 AC induces an isomorphism Ag -Ag. (See [4] for details.) If H is a Hopf algebra over R, and M,N E HC, then M OR N E H?RHC. Pullback along the coproduct 6: H -H OR H gives an H-module structure on MORN. In this manner OR gives rise to a ring structure on Hg with multiplication given by [M][N] = [M ?R N]. Received by the editors May 21, 1984. 1980 Mathematics Subject Classification. Primary 16A24, 16A54.

A loop space whose homology has torsion of all orders

Abstract: has prim itiv e torsion elements of any orderis really its ke y propert y for the purpose s of this note. Interestingly, it isnot the first known example of a finitely presented graded Z algebrasatisfyin g this condition. In [5 , en d o f §3] a Hopf algebra denoted C isbuilt which also has primitive torsion elements of any prime order.Presumably it woul d not be to o hard to verif y that