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.     

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.     

The Steenrod Algebra and Other Copolynomial Hopf Algebras

We say that a Hopf algebra is copolynomial if its dual is polynomialas an algebra. We re-derive Milnor's result that the mod 2 Steenrodalgebra is copolynomial by means of a more general result thatis also applicable to a number of other related Hopf algebras.1991 Mathematics Subject Classification 55S10, 16W30.     

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.     

Explicit Construction and Uniqueness for Universal Operator Algebras of Directed Graphs

Given a directed graph, there exist a universal operator algebraand universal C*-algebra associated to the directed graph. Inthis paper we give intrinsic constructions for these objects.We also provide an explicit construction for the maximal C*-algebraof an operator algebra. We discuss uniqueness of the universalalgebras for finite graphs, showing that for finite graphs thegraph is an isomorphism invariant for the universal operatoralgebra of a directed graph. We show that the underlying undirectedgraph is a Banach algebra isomorphism invariant for the universalC*-algebra of a directed graph.     

Every projective Schur algebra is Brauer equivalent to a radical abelian algebra

We prove that any projective Schur algebra over a field K isequivalent in Br(K) to a radical abelian algebra. This was conjecturedin 1995 by Sonn and the first author of this paper. As a consequence,we obtain a characterization of the projective Schur group bymeans of Galois cohomology. The conjecture was known for algebrasover fields of positive characteristic. In characteristic zerothe conjecture was known for algebras over fields with a Henselianvaluation over a local or global field of characteristic zero.     

Property A, partial translation structures, and uniform embeddings in groups

We define the concept of a partial translation structure ona metric space X and show that there is a natural C*-algebraC*() associated with it, which is a subalgebra of the uniformRoe algebra C*_u(X). We introduce a coarse invariant of the metricwhich provides an obstruction to embedding the space in a group.When the space is sufficiently group-like, as determined byour invariant, properties of the Roe algebra can be deducedfrom those of C*(). We also give a proof of the fact that theuniform Roe algebra of a metric space is a coarse invariantup to Morita equivalence.     

The symplectic ideal and a double centraliser theorem

We interpret a result of Oehms as a statement about the symplecticideal. We use this result to prove a double centraliser theoremfor the symplectic group acting on , where V is the natural module for the symplectic group.This result was obtained in characteristic zero by Weyl. Furthermore,we use this to extend to arbitrary connected reductive groupsG with simply connected derived group the earlier result ofthe author that the algebra K[G] of infinitesimal invariantsin the algebra of regular functions on G is a unique factorisationdomain.     

Poincare Series of Multi-Filtered Algebras and Partitivity

It is proved that if an algebra R over a field can be endowedwith a pointed and finite-dimensional Nⁿ-filtration such thatthe associated Nⁿ-graded algebra T is semi-commutative, thenR is left and right finitely partitive. In order to do this,a multi-variable Poincaré series for every finitely generatedgraded T-module is considered and it is shown that this Poincaréseries is a rational function. The methods apply to some iteratedOre extensions such as quantum matrices and quantum Weyl algebrasas well as to the quantized enveloping algebra of sl(+1).     

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.     

Cluster-tilted algebras as trivial extensions

Given a finite-dimensional algebra C (over an algebraicallyclosed field) of global dimension at most two, we define itsrelation-extension algebra to be the trivial extension C Ext(DC,C) of C by the C–C-bimoduleExt(DC,C). We give a constructionfor the quiver of the relation-extension algebra in case thequiver of C has no oriented cycles. Our main result says thatan algebra is cluster-tilted if and only if there exists a tilted algebra C such that is isomorphic to the relation-extensionof C.     

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.     

Rigid representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

In this paper, we show that there is always an open adjointorbit in the nilpotent radical of a seaweed Lie algebra in gl_n(k),thus answering positively in this gl_n(k) case to a questionraised independently by Michel Duflo and Dmitri Panyushev. Theproof gives an explicit construction, using -filtered modulesof quasi-hereditary algebras arising from quotients of the doubleof quivers of type A. An example of a seaweed Lie algebra ina simple Lie algebra of type E₈ not admitting an open orbitin its nilpotent radical is given.     

Permanent Weak Amenability of Group Algebras of Free Groups

We show that all derivations from the group algebra l¹(G) ofa free group into its nth dual, where n is a positive even integer,are inner. Combined with the known result for odd values ofn, this shows that these algebras are permanently weakly amenable.1991 Mathematics Subject Classification 46M20, 43A20.     

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.     

Boundedness of Higher order Hankel Forms, Factorization in Potential Spaces and Derivations

We prove a bounded decomposition for higher order Hankel formsand characterize the first order Hochschild cohomology groupsof the disk algebra with coefficients in the space of boundedHankel forms of some fixed order. Although these groups arenon-trivial, we prove that every bounded derivation is innerand necessarily implemented by a Hankel form of order one higher.In terms of operators, this result extends the similarity resultof Aleksandrov and Peller. Both of the main structural theoremshere rely on estimates involving multilinear maps on the n-foldproduct of the disk algebra and we obtain several higher orderanalogues of the factorization results due to Aleksandrov andPeller. 2000 Mathematics Subject Classification: 47B35, 46E15,46E25.     

CARTAN DETERMINANTS OF CELLULAR ALGEBRAS

Let A be a finite-dimensional cellular algebra over a field.It is shown that the Cartan determinant of A is a positive integer;furthermore, the Cartan determinant of value 1 is equivalentto the condition that A is quasi-hereditary. In this article,we establish a formula for expressing the Cartan determinantsof cellular algebras. By applying this formula, we obtain anecessary and sufficient condition for a cellular algebra withCartan determinant 2 to be standardly stratified. Moreover,we provide a sufficient condition for an arbitrary cellularalgebra to be standardly stratified and an inductive constructionof standardly stratified cellular algebras.     

Kazhdan-Lusztig Cells and the Murphy Basis

Let H be the Iwahori–Hecke algebra associated with S_n,the symmetric group on n symbols. This algebra has two importantbases: the Kazhdan–Lusztig basis and the Murphy basis.We establish a precise connection between the two bases, allowingus to give, for the first time, purely algebraic proofs fora number of fundamental properties of the Kazhdan–Lusztigbasis and Lusztig's results on the a-function. 2000 MathematicsSubject Classification 20C08.     

On the Group of Symplectic Matrices Over a Free Associative Algebra

Symplectic groups are well known as the groups of isometriesof a vector space with a non-singular bilinear alternating form.These notions can be extended by replacing the vector spaceby a module over a ring R, but if R is non-commutative, it willalso have to have an involution. We shall here be concernedwith symplectic groups over free associative algebras (witha suitably defined involution). It is known that the generallinear group GL_n over the free algebra is generated by the setof all elementary and diagonal matrices (see [1, Proposition2.8.2, p. 124]). Our object here is to prove that the symplecticgroup over the free algebra is generated by the set of all elementarysymplectic matrices. For the lowest order this result was obtainedin [4]; the general case is rather more involved. It makes useof the notion of transduction (see [1, 2.4, p. 105]). When thereis only a single variable over a field, the free algebra reducesto the polynomial ring and the weak algorithm becomes the familiardivision algorithm. In that case the result has been provedin [3, Anhang 5].