Deep Matrix Algebras of Finite Type

Deep matrix algebras based on a set over a ring are introduced and studied by McCrimmon when has infinite cardinality. Here, we construct a new -module that is faithful for of any cardinality. For a field of arbitrary characteristic and , is studied in depth. The algebra is shown to possess a unique proper non-zero ideal, isomorphic to . This leads to a new and interesting simple algebra, , the quotient of by its unique ideal. Both and the quotient algebra are shown to have centers isomorphic to . Via the new faithful representation, all automorphisms of are shown to be inner in the sense of Definition 18.Presented by D. Passman.     

Compactifications of Discrete Quantum Groups

Given a discrete quantum group we construct a Hopf -algebra which is a unital -subalgebra of the multiplier algebra of . The structure maps for are inherited from and thus the construction yields a compactification of which is analogous to the Bohr compactification of a locally compact group. This algebra has the expected universal property with respect to homomorphisms from multiplier Hopf algebras of compact type (and is therefore unique). This provides an easy proof of the fact that for a discrete quantum group with an infinite dimensional algebra the multiplier algebra is never a Hopf algebra.Partially supported by Komitet Badań Naukowych grants 2P03A04022 & 2P03A01324, the Foundation for Polish Science and Deutsche Forschungsgemeinschaft.     

Epi-topology and Epi-convergence for Archimedean Lattice-ordered Groups with Unit

is the category of archimedean -groups with distinguished weak order unit, with -group homomorphisms which preserve unit. This category includes all rings of continuous functions and all rings of measurable functions modulo null functions, with ring homomorphisms. The authors, and others, have studied previously the epimorphisms (right-cancellable morphisms) in . There is a rich theory. In this paper, we describe a topological approach to the analysis of these epimorphisms. On each – object, we define a topology and a convergence . These have the same closure operator, and this closure “captures epics” in the sense: a divisible subobject of is dense iff is epically embedded. The topology is , but only sometimes Hausdorff or an -group topology. The convergence is a Hausdorff -group convergence, but only sometimes topological. The associations of to , and to , are functorial. Dedicated to Bernhard Banaschewski for his 80th birthday.     

Rigids as Iterated Skew Commutators of Simples

Let be a finite-dimensional hereditary algebra of finite or tame representation type over a finite field, and let be a rigid -module. Then the element in the Ringel–Hall algebra is an iterated skew commutator of the isoclasses of simple -modules. This gives a new characterization of the rigidness of an indecomposable module over a tame hereditary algebra.The first author was partially supported by a grant from the NSF; and the second author was supported by the Doctorial Foundation of the Ministry of Education of People’s Republic of China, and the NSFC (Grant No. 10271113 and 10301033).     

Schur–Weyl Reciprocity between the Quantum Superalgebra and the Iwahori–Hecke Algebra

In this paper, we establish Schur–Weyl reciprocity between the quantum general super Lie algebra and the Iwahori–Hecke algebra . We introduce the sign -permutation representation of on the tensor space of dimensional -graded -vector space . This action commutes with that of derived from the vector representation on . Those two subalgebras of satisfy Schur–Weyl reciprocity. As special cases, we obtain the super case (), and the quantum case (). Hence this result includes both the super case and the quantum case, and unifies those two important cases.Presented by A. Verschoren.     

Stacks of Algebras and Their Homology

For any increasing function which takes only finitely many distinct values, a connected finite dimensional algebra is constructed, with the property that for all ; here is the -generated finitistic dimension of . The stacking technique developed for this construction of homological examples permits strong control over the higher syzygies of -modules in terms of the algebras serving as layers. The first author was supported in part by a fellowship stipend from the National Physical Science Consortium and the National Security Agency. The second author was partially supported by a grant from the National Science Foundation.     

Modular Group Algebras with Almost Maximal Lie Nilpotency Indices

Let be a field of positive characteristic and the group algebra of a group . It is known that, if is Lie nilpotent, then its upper (and lower) Lie nilpotency index is at most , where is the order of the commutator subgroup. The authors previously determined those groups for which this index is maximal and here they determine the groups for which it is `almost maximal', that is, it takes the next highest possible value, namely .Presented by V. Dl a b.Dedicated to Professor Vjacheslav Rudko on his 65th birthday.The research was supported by OTKA No. T 037202, No. T 038059 and Italian National Research Project “Group Theory and Application.”     

On Auslander–Reiten Components and Heller Lattices for Integral Group Rings

Let be a finite group, a complete discrete valuation ring of characteristic zero with residue class field of characteristic , and a block of the group ring . Suppose that is of infinite representation type and is sufficiently large to satisfy certain conditions. Let be the Auslander–Reiten quiver of and a connected component of . In this paper, we show that if contains some Heller lattices then the tree class of the stable part of is . Also, we show that has infinitely many components of type if a defect group of is neither cyclic nor a Klein four group.Presented by Jon Carlson.     

Generic Extensions of Prinjective Modules

Let be an algebraically closed field and let be a finite partially ordered set of finite prinjective type. We study generic extensions of prinjective modules over the incidence -algebra of . We prove that there exist generic extensions of prinjective -modules and describe properties of the monoid of generic extensions.Partially supported by Polish KBN Grant 5 P0 3A 015 21.     

Frames in spaces with finite rate of innovation

Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space modeling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applications, cellular radio and ultra wide-band communication. In particular, the space is generated by a family of well-localized molecules of similar size located on a relatively separated set using coefficients, and hence is locally finitely generated. Moreover that space includes finitely generated shift-invariant spaces, spaces of non-uniform splines, and the twisted shift-invariant space in Gabor (Wilson) system as its special cases. Use the well-localization property of the generator , we show that if the generator is a frame for the space and has polynomial (sub-exponential) decay, then its canonical dual (tight) frame has the same polynomial (sub-exponential) decay. We apply the above result about the canonical dual frame to the study of the Banach frame property of the generator for the space with , and of the polynomial (sub-exponential) decay property of the mask associated with a refinable function that has polynomial (sub-exponential) decay.      

Hilbert Space Representations of Cross Product Algebras II

In this paper, we study and classify Hilbert space representations of cross product -algebras of the quantized enveloping algebra with the coordinate algebras of the quantum motion group and of the complex plane, and of the quantized enveloping algebra with the coordinate algebras of the quantum group and of the quantum disc. Invariant positive functionals and the corresponding Heisenberg representations are explicitly described.Presented by S.L. Woronowicz.     

Koszul and Gorenstein Properties for Homogeneous Algebras

The Koszul property was generalized to homogeneous algebras of degree in [5], and related to -complexes. We show that if the -homogeneous algebra is generalized Koszul, AS-Gorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem to i.e., there is a Poincaré duality between Hochschild homology and cohomology of as for .     

Branching Rules, Kostka–Foulkes Polynomials and q-multiplicities in Tensor Product for the Root Systems B n , C n and D n

The Kostka–Foulkes polynomials related to a root system can be defined as alternating sums running over the Weyl group associated to . By restricting these sums over the elements of the symmetric group when is of type or , we obtain again a class of Kostka–Foulkes polynomials. When is of type or there exists a duality between these polynomials and some natural -multiplicities and in tensor products [11]. In this paper we first establish identities for the which implies in particular that they can be decomposed as sums of Kostka–Foulkes polynomials with nonnegative integer coefficients. Moreover these coefficients are branching coefficients This allows us to clarify the connection between the -multiplicities and the polynomials defined by Shimozono and Zabrocki. Finally we show that and coincide up to a power of with the one dimension sum introduced by Hatayama and co-workers when all the parts of are equal to , which partially proves some conjectures of Lecouvey and Shimozono and Zabrocki.Presented by P. Littelmann.     

Higher Level Affine Crystals and Young Walls

Using combinatorics of Young walls, we give a new realization of arbitrary level irreducible highest weight crystals for quantum affine algebras of type , , , , , and . The irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls. The notion of slices and splitting of blocks plays a crucial role in the construction of crystals.Presented by Peter Littelman.     

Hall Numbers and the Composition Algebra of the Kronecker Algebra

We present formulas for the structure constants (Hall numbers) of the Hall algebra associated to the Kronecker algebra. The formulas which in some cases involve the classical Hall polynomials enable us to determine every Hall number. Using again these formulas we construct new PBW-bases with simple structure constants for the composition algebra , making possible the definition of the generic composition algebra via Hall polynomials.Presented by C. Ringel.     

Intrinsic Ultracontractivity for Schrödinger Operators with Mixed Boundary Conditions

Let be a domain in , . Let be a divergence form uniformly elliptic operator with Dirichlet boundary conditions on and Neumann boundary conditions on , where is a closed subset of . We prove intrinsic ultracontractivity for the semigroup associated to the Schrödinger operator , where is a potential in the Kato class, provided that is locally Lipschitz and is given by the boundary of either a Hölder domain of order or a uniformly Hölder domain of order , . Our results extend to the mixed boundary case the results of Bañuelos, Bass and Burdzy, Bass and Hsu, and Davies and Simon.     

Solvable Groups Satisfying the Two-prime Hypothesis,Part I

In this paper, finite solvable groups satisfying the “-prime hypothesis” are considered. Specifically, a bound on the number of irreducible character degrees of such a group is obtained when . The general situation is also considered, and generalizations of the -prime hypothesis are analyzed. Presented by A. Verschoren.     

On the Affine Schur Algebra of Type A II

By studying certain centralizer subalgebras of the affine Schur algebra we show that is Noetherian and we determine its center. Assuming n ≥ r, we show that is Morita equivalent to , and the Schur functor is an equivalence under certain conditions. The author acknowledges support by National Natural Science Foundation of China No.10131010.     

Primitive Ideals and Automorphisms of Quantum Matrices

Let be a field and q be a nonzero element of that is not a root of unity. We give a criterion for 〈0〉 to be a primitive ideal of the algebra of quantum matrices. Next, we describe all height one primes of ; these two problems are actually interlinked since it turns out that 〈0〉 is a primitive ideal of whenever has only finitely many height one primes. Finally, we compute the automorphism group of in the case where m ≠ n. In order to do this, we first study the action of this group on the prime spectrum of . Then, by using the preferred basis of and PBW bases, we prove that the automorphism group of is isomorphic to the torus when m ≠ n and (m,n) ≠ (1, 3),(3, 1). This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme and by Leverhulme Research Interchange Grant F/00158/X.