Cyclic group actions on polynomial rings

We consider a cyclic group of order pⁿ, acting on a module incharacteristic p, and show how to reduce the calculation ofthe symmetric algebra to that of the exterior algebra.     

Embedding Properties of Metabelian Lie Algebras and Metabelian Discrete Groups

We show that for every natural number m a finitely generatedmetabelian group G embeds in a quotient of a metabelian groupof type FP_m. Furthermore, if m 4, the group G can be embeddedin a metabelian group of type FP_m. For L a finitely generatedmetabelian Lie algebra over a field K and a natural number mwe show that, provided the characteristic p of K is 0 or p >m, then L can be embedded in a metabelian Lie algebra of typeFP_m. This result is the best possible as for 0 < p m everymetabelian Lie algebra over K of type FP_m is finite dimensionalas a vector space.     

Arens Regularity for Quotients Ap(E) of the Herz Algebra

If the subset T supports a synthesizable p-pseudofunction T,then the restriction A_p(E) to E of the Herz algebra on T isnot Arens regular. The proof is direct, by brute force, anddoes not show the existence of Day points for A_p(E). 2000 MathematicsSubject Classification 43A15, 43A10 (primary), 46L10 (secondary).     

A Remark About the Lie Algebra of Infinitesimal Conformal Transformations of the Euclidean Space

Infinitesimal conformal transformations of Rⁿ are always polynomialand finitely generated when n > 2. Here we prove that theLie algebra of infinitesimal conformal polynomial transformationsover Rⁿ, n 2, is maximal in the Lie algebra of polynomial vectorfields. When n is greater than 2 and p, q are such that p +q = n, this implies the maximality of an embedding of so(p +1, q + 1, R) into polynomial vector fields that was revisitedin recent works about equivariant quantizations. It also refinesa similar but weaker theorem by V. I. Ogievetsky. 1991 MathematicsSubject Classification 17B66, 53A30.     

Algebras of Symmetric Holomorphic Functions on lp

The authors study the algebra of uniformly continuous holomorphicsymmetric functions on the ball of l_p, investigating in particularthe spectrum of such algebras. To do so, they examine the algebraof symmetric polynomials on l_p-spaces, as well as finitely generatedsymmetric algebras of holomorphic functions. Such symmetricpolynomials determine the points in l_p up to a permutation.     

Quasitriangular Hopf Algebras of Dimension pq

Let p and q be odd prime numbers. It is shown that all quasitriangularHopf algebras of dimension pq over an algebraically closed fieldk of characteristic zero are semisimple and therefore isomorphicto a group algebra.     

An Algebra of Pseudodifferential Operators with Slowly Oscillating Symbols

Let VR denote the Banach algebra of absolutely continuous functionsof bounded total variation on R, and let B_p be the Banach algebraof bounded linear operators acting on the Lebesgue space L^pRfor 1 < p < . We study the Banach algebra A B_p generatedby the pseudodifferential operators of zero order with slowlyoscillating VR-valued symbols on R. Boundedness and compactnessconditions for pseudodifferential operators with symbols inL R, VR are obtained. A symbol calculus for the non-closed algebraof pseudodifferential operators with slowly oscillating VR-valuedsymbols is constructed on the basis of an appropriate approximationof symbols by infinitely differentiable ones and by use of thetechniques of oscillatory integrals. As a result, the quotientBanach algebra A = A K, where K is the ideal of compact operatorsin B_p, is commutative and involutive. An isomorphism betweenthe quotient Banach algebra A of pseudodifferential operatorsand the Banach algebra of their Fredholm symbols is established. A Fredholm criterionand an index formula for the pseudodifferential operators A A are obtained in terms of their Fredholm symbols. 2000 MathematicsSubject Classification 47G30, 47L15 (primary), 47A53, 47G10(secondary).     

The Decomposition of Lie Powers

Let G be a group, F a field of prime characteristic p and Va finite-dimensional FG-module. Let L(V) denote the free Liealgebra on V regarded as an FG-submodule of the free associativealgebra (or tensor algebra) T(V). For each positive integerr, let L^r (V) and T^r (V) be the rth homogeneous components ofL(V) and T(V), respectively. Here L^r (V) is called the rth Liepower of V. Our main result is that there are submodules B₁,B₂, ... of L(V) such that, for all r, B_r is a direct summandof T^r(V) and, whenever m 0 and k is not divisible by p, themodule is the direct sum of , . Thus every Lie power is a direct sum of Lie powers of p-powerdegree. The approach builds on an analysis of T^r (V) as a bimodulefor G and the Solomon descent algebra. 2000 Mathematics SubjectClassification 17B01 (primary), 20C07, 20C20 (secondary).     

A Geometric Invariant for Metabelian Pro-p Groups

In [2] Bieri and Strebel introduced a geometric invariant forfinitely generated abstract metabelian groups that determineswhich groups are finitely presented. For a valuable survey oftheir results, see [6]; we recall the definition briefly inSection 4. We shall introduce a similar invariant for pro-pgroups. Let F be the algebraic closure of F_p and U be the formal powerseries algebra F[T], with group of units U^x. Let Q be a finitelygenerated abelian pro-p group. We write Z_p[Q] for the completedgroup algebra of Q over Z_p. Let T(Q) be the abelian group Hom(Q,U^x) of continuous homomorphisms from Q to U^x. We write 1 forthe trivial homomorphism. Each vT(Q) extends to a unique continuousalgebra homomorphism from Z_p[Q]to U.     

Polynomial and Polygonal Connections between Idempotents in Finite-Dimensional Real Algebras

Let P (A) be the set of idempotents in a finite-dimensionalreal algebra A. Let p and q be idempotents that lie in the samecomponent of P (A). Then, among the continuous paths connectingp and q in P (A), there exist a polynomial path of degree atmost 3 and a polygonal path consisting of at most three segments.2000 Mathematics Subject Classification 46H10, 16P10.     

Chern Character for the Schwartz Algebra of p-Adic GL(n)

Let G denote the p-adic group GL(n), and let S(G) denote theSchwartz algebra of G. We construct a Chern character from theK-theory of the reduced C*-algebra of G to the periodic cyclichomology of S(G) which becomes an isomorphism after tensoringover Z with C.     

Noether Numbers for Subrepresentations of Cyclic Groups of Prime Order

Let W be a finite-dimensional Z/p-module over a field, k, ofcharacteristic p. The maximum degree of an indecomposable elementof the algebra of invariants, k[W]^Z/p, is called the Noethernumber of the representation, and is denoted by rß(W).A lower bound for rß(W) is derived, and it is shownthat if U is a Z/p submodule of W, then rß(U) rß(W).Aset of generators, in fact a SAGBI basis, is constructed fork[V₂ V₃]^Z/p, where V_n is the indecomposable Z/p-module of dimensionn. 2000 Mathematics Subject Classification 13A50, 20J06.     

Characterization of the Mod 3 Cohomology of the Compact, Connected, Simple, Exceptional Lie Groups of Rank 6

It is shown that the mod 3 cohomology of a 1-connected, homotopyassociative mod 3 H-space that is rationally equivalent to theLie group E₆ is isomorphic to that of E₆ as an algebra. Moreover,it is shown that the mod 3 cohomology of a nilpotent, homotopy-associativemod 3 H-space that is rationally equivalent to E₆, and whosefundamental group localized at 3 is non-trivial, is isomorphicto that of the Lie group Ad E₆ as a Hopf algebra over the mod3 Steenrod algebra. It is also shown that the mod 3 cohomologyof the universal cover of such an H-space is isomorphic to thatof E₆ as a Hopf algebra over the mod 3 Steenrod algebra. 2000Mathematics Subject Classification 57T05, 57T10, 57T25.     

The Amenability of Measure Algebras

In this paper we shall prove that the measure algebra M(G) ofa locally compact group G is amenable as a Banach algebra ifand only if G is discrete and amenable as a group. Our contributionis to resolve a conjecture by proving that M(G) is not amenablein the case where the group G is not discrete. Indeed, we shallprove a much stronger result: the measure algebra of a non-discrete,locally compact group has a non-zero, continuous point derivationat a certain character on the algebra.     

K0 and the dimension filtration for p-torsion Iwasawa modules

Let G be a compact p-adic analytic group. We study K-theoreticquestions related to the representation theory of the completedgroup algebra kG of G with coefficients in a finite field kof characteristic p. We show that if M is a finitely generatedkG-module with canonical dimension smaller than the dimensionof the centralizer, as a p-adic analytic group, of any p-regularelement of G, then the Euler characteristic of M is trivial.Writing _i for the abelian category consisting of all finitelygenerated kG-modules of dimension at most i, we provide an upperbound for the rank of the natural map from the Grothendieckgroup of _i to that of _d, where d denotes the dimension of G.We show that this upper bound is attained in some special cases,but is not attained in general.     

A Version of the Poincare-Birkhoff-Witt Theorem

In this note I shall prove that if L is a finite-dimensionalLie algebra over a field F of characteristic zero which is generatedas an algebra by a set of elements {e₁, e₂,...,e_k}, then theuniversal enveloping algebra U(L) of L is linearly generatedby monomials spanned by the elements {e_i} of an a priori boundedwidth. As an application, a criterion of Kostant for a leftideal of U(L) to be of finite codimension is proved by purelyalgebraic means.     

On the Equations zm = F(x, y) and Axp + Byq = Czr

We investigate integer solutions of the superelliptic equation where F is a homogeneous polynomialwith integer coefficients, and of the generalized Fermat equation where A, B and C are non-zero integers.Call an integer solution (x, y, z) to such an equation properif gcd(x, y, z) = 1. Using Faltings' Theorem, we shall givecriteria for these equations to have only finitely many propersolutions. We examine (1) using a descent technique of Kummer, which allowsus to obtain, from any infinite set of proper solutions to (1),infinitely many rational points on a curve of (usually) highgenus, thus contradicting Faltings' Theorem (for example, thisworks if F(t, 1) = 0 has three simple roots and m 4). We study (2) via a descent method which uses unramified coveringsof P₁ \ {0, 1, } of signature (p, q, r), and show that (2) hasonly finitely many proper solutions if l/p + l/q + 1/r <1. In cases where these coverings arise from modular curves,our descent leads naturally to the approach of Hellegouarchand Frey to Fermat's Last Theorem. We explain how their ideamay be exploited for other examples of (2). We then collect together a variety of results for (2) when 1/p+ 1/q + 1/r 1. In particular, we consider ‘local-global’principles for proper solutions, and consider solutions in functionfields.     

A Matrix Setting for the q-Schur Algebra

The Schur algebra S(n, r) has a basis (described in [6, 2.3])consisting of certain elements _i,j, where i, jI(n, r), the setof all ordered r-tuples of elements from the set n={1, 2, ...,n}. The multiplication of two such basis elements is given bya formula known as Schur's product rule. In recent years, aq-analogue S_q(n, r) of the Schur algebra has been investigatedby a number of authors, particularly Dipper and James [3, 4].The main result of the present paper, Theorem 3.6, shows howto embed the q-Schur algebra in the rth tensor power T^r(M_n)of the nxn matrix ring. This embedding allows products in theq-Schur algebra to be computed in a straightforward manner,and gives a method for generalising results on S(n, r) to S_q(n,r). In particular we shall make use of this embedding in subsequentwork to prove a straightening formula in S_q(n, r) which generalisesthe straightening formula for codeterminants due to Woodcock[12]. We shall be working mainly with three types of algebra: thequantized enveloping algebra U(gl_n) corresponding to the Liealgebra gl_n, the q-Schur algebra S_q(n, r), and the Hecke algebra,H(A_r–1). It is often convenient, in the case of the q-Schuralgebra and the Hecke algebra, to introduce a square root ofthe usual parameter q which will be denoted by v, as in [5].This corresponds to the parameter v in U(gl_n). We shall denotethis ‘extended’ version of the q-Schur algebra byS_v(n, r), and we shall usually refer to it as the v-Schur algebra.All three algebras are associative and have multiplicative identities,and the base field will be the field of rational functions,Q(v), unless otherwise stated. The symbols n and r shall bereserved for the integers given in the definitions of thesethree algebras.     

The (Q, q)-Schur Algebra

In this paper we use the Hecke algebra of type B to define anew algebra S which is an analogue of the q-Schur algebra. Weshow that S has ‘generic’ basis which is independentof the choice of ring and the parameters q and Q. We then constructWeyl modules for S and obtain, as factor modules, a family ofirreducible S-modules defined over any field. 1991 MathematicsSubject Classification: 16G99, 20C20, 20G05.