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.     

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.     

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.     

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.     

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).     

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.     

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.     

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.     

Almost-triangular Hopf Algebras

In this paper, we consider a finite dimensional semisimple cosemisimple quasitriangular Hopf algebra with (we call this type of Hopf algebras almost-quasitriangular) over an algebraically closed field . We denote by the vector space generated by the left tensorand of . Then is a sub-Hopf algebra of . We proved that when is odd, has a triangular structure and can be obtained from a group algebra by twisting its usual comultiplication [14]; when is even, is an extension of an abelian group algebra and a triangular Hopf algebra, and may not be triangular. In general, an almost-triangular Hopf algebra can be viewed as a cocycle bicrossproduct.      

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.     

Wedge Products and Cotensor Coalgebras in Monoidal Categories

The construction of the cotensor coalgebra for an “abelian monoidal” category which is also cocomplete, complete and AB5, was performed in Ardizzoni et al. (Comm Algebra 35(1):25–70, 2007). It was also proved that this coalgebra satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration for a coalgebra E in is filled by considering a direct limit of a filtration consisting of wedge products of a subcoalgebra D of E. The main aim of this paper is to characterize hereditary coalgebras , where D is a coseparable coalgebra in , by means of a cotensor coalgebra: more precisely, we prove that, under suitable assumptions, is hereditary if and only if it is formally smooth if and only if it is the cotensor coalgebra if and only if it is a cotensor coalgebra , where N is a certain D-bicomodule in . Because of our choice, even when we apply our results in the category of vector spaces, new results are obtained. This paper was written while A. Ardizzoni was member of G.N.S.A.G.A. with partial financial support from Mi.U.R.     

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.     

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.     

Weak Quantum Enveloping Algebras of Borcherds Superalgebras

In present paper we define a new kind of weak quantized enveloping algebra of Borcherds superalgebras . It is a noncommutative and noncocommutative weak graded Hopf algebra. Using localizing with some Ore set, we obtain a different kind of quantized enveloping algebras of Borcherds superalgebras . It has a homomorphic image which is isomorphic to the usual quantum enveloping algebra of . Moreover, is isomorphic to a direct sum of and an other algebra as algebras. The author is sponsored by ZJNSF No. Y607136.     

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.”     

Covers and Directed Colimits

We find conditions on , a class of objects of a Grothendieck category, sufficient for the existence of -covers. The theory includes the existence of flat covers of modules.     

Sober Approach Spaces are Firmly Reflective for the Class of Epimorphic Embeddings

In [2] the subconstruct of sober approach spaces was introduced and it was shown to be a reflective subconstruct of the category of approach spaces. The main result of this paper states that moreover is firmly -reflective in for the class of epimorphic embeddings. ‘Firm -reflective’ is a notion introduced in [3] by G.C.L. Brümmer and E. Giuli and is inspired by the exemplary behaviour of the usual completion in the category of Hausdorff uniform spaces with uniformly continuous maps. It means that is -reflective in and that the reflector is such that belongs to if and only if is an isomorphism. Firm -reflectiveness implies uniqueness of completion in the sense that whenever is a map with and sober, the associated is an isomorphism. Our result generalizes the fact that in the category the subconstruct of sober topological spaces is firmly reflective for the class of b-dense embeddings in . Also firmness in some other subconstructs of will be easily obtained.A. Gerlo and C. Van Olmen are research assistants at the Fund of Scientific Research Vlaanderen (FWO). E. Vandersmissen is a research assistant supported by the FWO-grant G.0244.05.     

SU(d)-Biinvariant Random Walks on SL\,{\left( {d,\,\mathbb{C}} \right)} and their Euclidean Counterparts

We establish a deformation isomorphism between the algebras of -biinvariant compactly supported measures on and -conjugation invariant measures on the Euclidean space of all Hermitian -matrices with trace . This isomorphism concisely explains a close connection between the spectral problem for sums of Hermititan matrices on one hand and the singular spectral problem for products of matrices from on the other, which has recently been observed by Klyachko [13]. From this deformation we further obtain an explicit, probability preserving and isometric isomorphism between the Banach algebra of bounded -biinvariant measures on and a certain (non-invariant) subalgebra of the bounded signed measures on . We demonstrate how this probability preserving isomorphism leads to limit theorems for the singular spectrum of -biinvariant random walks on in a simple way. Our construction relies on deformations of hypergroup convolutions and will be carried out in the general setting of complex semisimple Lie groups.Margit Rösler was partially supported by the Netherlands Organisation for Scientific Research (NWO), project nr. B 61-544.     

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.