P.I. envelopes of classical simple Lie superalgebras

Let be a classical simple Lie superalgebra. We describe the prime ideals in the enveloping algebra such that satisfies a polynomial identity. If the factor algebra is not artinian, then it is an order in a matrix algebra over .     

Arens-Michael enveloping algebras and analytic smash products

Let be a finite-dimensional complex Lie algebra, and let be its universal enveloping algebra. We prove that if , the Arens-Michael envelope of is stably flat over (i.e., if the canonical homomorphism is a localization in the sense of Taylor (1972), then is solvable. To this end, given a cocommutative Hopf algebra and an -module algebra , we explicitly describe the Arens-Michael envelope of the smash product as an ``analytic smash product' of their completions w.r.t. certain families of seminorms.

     

A note on branching theorems

Let be a complex, simply connected semisimple analytic group with a closed connected reductive subgroup. Suppose is an irreducible holomorphic -module and an irreducible holomorphic -module. We prove that Hom possesses the structure of an irreducible -module whenever is . Moreover, for all and if and only if is commutative.

     

Open subgroups and the centre problem for the Fourier algebra

Let be the Fourier algebra of a locally compact group and the -algebra of uniformly continuous linear functionals on . We study how the centre problem for the algebra (resp. ) is related to the centre problem for the algebras (resp. ) of -compact open subgroups of . We extend some results of Lau-Losert on the centres of and .

     

Linear functionals on the Cuntz algebra

For a pure state on , which is an extension of a pure state on with the property that if is a corresponding representation, then , induces a unital shift of of the Powers index . We describe states on by using sequences of unit vectors in . We study the linear functionals on the Cuntz algebra whose restrictions are the product pure state on . We find conditions on the sequence of unit vectors for which the corresponding linear functionals on become states under these conditions.

     

Lie algebras of cohomological codimension one

We show that if is a finite dimensional real Lie algebra, then has cohomological dimension if and only if is a unimodular extension of the two-dimensional non-Abelian Lie algebra .

     

Extensions of Hopf Algebras and Lie Bialgebras

Let , be finite-dimensional Lie algebras over a field of characteristic zero. Regard and , the dual Lie coalgebra of , as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair of Lie bialgebras is given, which has structure maps . Then it induces a matched pair of Hopf algebras, where is the universal envelope of and is the Hopf dual of . We show that the group of cleft Hopf algebra extensions associated with is naturally isomorphic to the group of Lie bialgebra extensions associated with . An exact sequence involving either of these groups is obtained, which is a variation of the exact sequence due to G.I. Kac. If , there follows a bijection between the set of all cleft Hopf algebra extensions of by and the set of all Lie bialgebra extensions of by .

     

On zeros of Eisenstein series for genus zero Fuchsian groups

Let SL be a genus zero Fuchsian group of the first kind with as a cusp, and let be the holomorphic Eisenstein series of weight on that is nonvanishing at and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on and on a choice of a fundamental domain , we prove that all but possibly of the nontrivial zeros of lie on a certain subset of . Here is a constant that does not depend on the weight, is the upper half-plane, and is the canonical hauptmodul for

     

Free summands of conormal modules and central elements in homotopy Lie algebras of local rings

If is a surjective local homomorphism with kernel , such that and the conormal module has a free summand of rank , then the degree central subspace of the homotopy Lie algebra of has dimension greater than or equal to . This is a corollary of the Main Theorem of this note. The techniques involved provide new proofs of some well known results concerning the conormal module.

     

On a conjecture about MRA Riesz wavelet bases

Let be a compactly supported refinable function in such that the shifts of are stable and for a -periodic trigonometric polynomial . A wavelet function can be derived from by . If is an orthogonal refinable function, then it is well known that generates an orthonormal wavelet basis in . Recently, it has been shown in the literature that if is a -spline or pseudo-spline refinable function, then always generates a Riesz wavelet basis in . It was an open problem whether can always generate a Riesz wavelet basis in for any compactly supported refinable function in with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function does not generate a Riesz wavelet basis in . Our proof is based on some necessary and sufficient conditions on the -periodic functions and in such that the wavelet function , defined by , generates a Riesz wavelet basis in .

     

Open subgroups of

Let be a locally compact group and let denote the -algebra generated by left translation operators on . Let and be the spaces of almost periodic and weakly almost periodic functionals on the Fourier algebra , respectively. It is shown that if contains an open abelian subgroup, then (1) if and only if is norm dense in ; (2) is a -algebra if is norm dense in , where denotes the set of elements in with compact support. In particular, for any amenable locally compact group which contains an open abelian subgroup, has the dual Bohr approximation property and is a -algebra.

     

Invariant distributions supported on the nilpotent cone of a semisimple Lie algebra

Let be a semisimple complex Lie algebra with adjoint group and be the algebra of differential operators with polynomial coefficients on . If is a real form of , we give the decomposition of the semisimple -module of invariant distributions on supported on the nilpotent cone.

     

Lifts of - and -morphisms to -morphisms

Let be the Hochschild complex of cochains on and let be the space of multivector fields on . In this paper we prove that given any -structure (i.e. Gerstenhaber algebra up to homotopy structure) on , and any -morphism (i.e. morphism of a commutative, associative algebra up to homotopy) between and , there exists a -morphism between and that restricts to . We also show that any -morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a -morphism, using Tamarkin's method for any -structure on . We also show that any two of such -morphisms are homotopic.

     

Almost-disjoint coding and strongly saturated ideals

We show that Martin's Axiom plus implies that there is no -saturated -ideal on .

     

A note concerning the index of the shift

Let be a finite, positive Borel measure with support in such that - the closure of the polynomials in - is irreducible and each point in is a bounded point evaluation for . We show that if 0$">and there is a nontrivial subarc of such that

-\infty,\end{displaymath}">

then for each nontrivial closed invariant subspace for the shift on .

     

Types in reductive -adic groups: The Hecke algebra of a cover

In this paper, is a non-Archimedean local field and is the group of -points of a connected reductive algebraic group defined over . Also, is an irreducible representation of a compact open subgroup of , the pair being a type in . The pair is assumed to be a cover of a type in a Levi subgroup of . We give conditions, generalizing those of earlier work, under which the Hecke algebra is the tensor product of a canonical image of and a sub-algebra , for a compact open subgroup of containing .

     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

     

The Banach algebra induced by a double centralizer

Given a Banach algebra , R. Larsen defined, in his book ``An introduction to the theory of multipliers", a Banach algebra by means of a multiplier on , and essentially used it in the case of a commutative semisimple Banach algebra to prove a result on multiplications which preserve regular maximal ideals. Here, we consider the analogue Banach algebra induced by a bounded double centralizer of a Banach algebra . Then, our main concern is devoted to the relationships between , , and the algebras of bounded double centralizers and of and , respectively. By removing the assumption of semisimplicity, we generalize some results proven by Larsen.

     

On polynomial products in nilpotent and solvable Lie groups

We are dealing with Lie groups which are diffeomorphic to , for some . After identifying with , the multiplication on can be seen as a map . We show that if is a polynomial map in one of the two (sets of) variables or , then is solvable. Moreover, if one knows that is polynomial in one of the variables, the group is nilpotent if and only if is polynomial in both its variables.

     

Phantom depth and flat base change

We prove that if is a flat local homomorphism, is Cohen-Macaulay and -injective, and and share a weak test element, then a tight closure analogue of the (standard) formula for depth and regular sequences across flat base change holds. As a corollary, it follows that phantom depth commutes with completion for excellent local rings. We give examples to show that the analogue does not hold for surjective base change.