Earl J. Taft

Quantum deformation of flag schemes and Grassmann schemes. I. A q-deformation of the shape-algebra for GL(n)

An affine algebraic group G has associated with it two (closely related) Hopf algebras: namely, its ring @[Cl of representative functions, and the universal enveloping algebra U(g) of its Lie.algebra g. The concept of “q-deformation” of a Hopf algebra leads to the notion of a “quantum group” [D2, Sect. 23 or “q-deformation of G,” by which is meant a q-deformation of one of these two Hopf algebras. (A slight abuse of terminology is customary here: The Hopf algebra resulting from such a q-deformation, being in general neither commutative nor cocommutative, is not in fact still thus associated with a Lie group or algebra.) Interest in these “quantum groups” seems first to have arisen, because of their utility in obtaining solutions to the quantum Yang-Baxter equations; an exposition of these matters may be found in [FRT]; cf. also [D2]. This line of approach, via quantum groups, to the quantum Yang-Baxter equations, first introduced (so we have been informed) by Kulish and Reshetikhin [KR], and Sklyanin [S] (Faddeev’s name should also be mentioned) led to constructions related to the quantum SL(2). Independently, Woronowicz introduced quantum groups in a different way [W]. Yu I. Manin has placed these constructions in a satisfying conceptual

Witt and Virasoro algebras as Lie bialgebras

Abstract We give a countably infinite number of Lie coalgebra structures on the Witt algebra W = Der k [ x ] over a field k , and on the Virasoro algebras W 1 =Der k [ x , x −1 ] and V = W 1 ⊕ kc with central charge c . These come from certain solutions of the classical Yang-Baxter equation, and yield Lie bialgebra structures in each case. For k of characteristic 0, we show that these Lie coalgebra structures on W are mutually non-isomorphic, using an analysis of the locally finite part of W . We also discuss the Lie bialgebra duals of each of these constructions, which can be identified with linearly recursive sequences (one-sided or two-sided).

Classification of the Lie bialgebra structures on the Witt and Virasoro algebras

An automatic meter reading and control system for communicating with remote terminal points includes a central station which selectively communicates with a transponder controller unit at each terminal point via a plurality of distribution units, each serving several transponder controller units. The distribution controller units are responsive to various commands issued by the central station to selectively route the commands to specified transponder controller units to direct the transponder controller units to selectively carry out a load control operation, a meter reading operation or transfer of previously stored meter data from the transponder units to the central station in accordance with functions specified by the various commands.

On antipodes in pointed Hopf algebras

Abstract If S is the antipode of a Hopf algebra H , the order of S is defined to be the smallest positive integer n such that S n = I (in case such integers exist) or ∞ (if no such integers exist). Although in most familiar examples of Hopf algebras the antipode has order 1 or 2, examples are known of infinite dimensional Hopf algebras in which the antipode has infinite order or arbitrary even order [1, 4, 6] and also of finite dimensional Hopf algebras in which the antipode has arbitrary even order [3, 5]. Some sufficient conditions for the antipode to have order ⩽4 are known [2, 4], but the following questions remain open: Does the antipode of a finite dimensional Hopf algebra necessarily have finite order? If the antipode S of a Hopf algebra H has finite order is that order bounded by some function of dim H ? In this paper, by constructing a certain basis for an arbitrary pointed coalgebra and studying the action of the antipode on the elements of such a basis for a pointed Hopf algebra, we obtain affirmative answers to the second question in case H is pointed and to the first question in case H is pointed over a field of prime characteristic. We use freely the definitions, notation, and results of [4].

The algebraic structure of linearly recursive sequences under hadamard product

We describe the algebraic structure of linearly recursive sequences under the Hadamard (point-wise) product. We characterize the invertible elements and the zero divisors. Our methods use the Hopf-algebraic structure of this algebra and classical results on Hopf algebras. We show that our criterion for invertibility is effective if one knows a linearly recursive relation for a sequence and certain information about finitely-generated subgroups of the multiplicitive group of the field.

There exist finite-dimensional Hopf algebras with antipodes of arbitrary even order

Abstract Let F be an arbitrary field and n be an arbitrary positive integer. Then there is a finite-dimensional Hopf algebra over F with antipode of order 2n.

Forms of certain hopf algebras

Let Ψ be a field, G a finite group of automorphisms of Ψ, and Φ the fixed field of G. Let H be a Hopf algebra over Ψ. For g ∈ G we define a Hopf algebra Hg which has the same underlying vector space as H and modified operations and show that the tensor product (over Ψ) ⊗g ∈ G Hg has a Φ-form. As a consequence we see that if n>0 is an integer and Φ is a field of characteristic zero or p>0 with (n,p)=1, then there is a finite dimensional Hopf algebra over Φ with antipode of order 2n.

Some Quantum-like Hopf Algebras which Remain Noncommutative when q = 1

Starting with only three of the six relations defining the standard (Manin) GLq(2), we try to construct a quantum group. The antipode condition requires some new relations, but the process stops at a Hopf algebra with a Birkhoff–Witt basis of irreducible monomials. The quantum determinant is group-like but not central, even when q = 1. So, the two Hopf algebras constructed in this way are not isomorphic to the Manin GLq(2), all of whose group-like elements are central. Analogous constructions can be made starting with the Dipper–Donkin version of GLq(2), but these turn out to be included in the two classes of Hopf algebras described above.

Hopf algebras with nonsemisimple antipode

An example is given to show that the antipode of a finite dimensional Hlopf algebra over a field of prime characteristic p > 2 need not be semisimple. (For p = 2 examples were previously known.) The example is a pointed irreducible Hopf algebra H (with antipode S) of dimension p3 such that S S2. Radford [4] has recently shown that the antipode S of a finite dimensional Hopf algebra H over a field K has finite order. Consequently, if K is of characteristic zero then the antipode of H is semisimple. On the other hand, if K is of characteristic 2 then S is semisimple only if S = I. (For otherwise S has even order, say 2k, and so 0 = 52k _ 1= (5k _ 1)2.) In this note we show that S may fail to be semisimple for any characteristic p > 3. We do this by constructing a pointed irreducible Hopf algebra of dimension p3 over an arbitrary field of characteristic p > 3 in which the antipode has order 2p (and hence is not semisimple). A related problem is that of finding a bound for the order of S. In [7] the authors have shown that if H is pointed, if G(H) has exponent e and if Ho C Hi C ... C Hm = H is the coradical filtration then (S2, _ I), = 0. Thus if K has characteristic p and pn-1 1 remains an open question. Presented to the Society, January 17, 1974 under the title Some finite dimensional pointed Hopf algebras with nonsemisimple antipode; received by the editors October 25, 1973. AMS (MOS) subject classifications (1970). Primary 16A24. lThis research was partially supported by National Science Foundation grant GP-38518. 2This research was partially supported by National Science Foundation grant GP-33226. Copyright

Hadamard invertibility of linearly recursive sequences in several variables

Abstract A linearly recursive sequence in n variables is a tableau of scalars (ƒ i 1… i n ) for i1,i2,…, in ⩾ 0, such that for each 1 ⩽ i ⩽ n, all rows parallel to the ith axis satisfy a fixed linearly recursive relation hi(x) with constant coefficients. We show that such a tableau is Hadamard invertible (i.e., the tableau (1/ ƒ i 1 … i n ) is linearly recursive) if and only if all ƒ i 1 … i n ≠ 0 , and each row is eventually an interlacing of geometric sequences. The procedure is effective, i.e., given a linearly recursive sequence ƒ = (ƒ i 1 … i n ) , it can be tested for Hadamard invertibility by a finite algorithm. These results extend the case n = 1 of Larson and Taft.