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.     

Richardson Elements for Parabolic Subgroups of Classical Groups in Positive Characteristic

Let G be a simple algebraic group of classical type over an algebraically closed field k. Let P be a parabolic subgroup of G and let be the Lie algebra of P with Levi decomposition , where is the Lie algebra of the unipotent radical of P and ł is a Levi complement. Thanks to a fundamental theorem of Richardson (Bull. London Math. Soc. 6:21–24, 1974), P acts on with an open dense orbit; this orbit is called the Richardson orbit and its elements are called Richardson elements. Recently Baur (J. Algebra 297(1):168–185, 2006), the first author gave constructions of Richardson elements in the case for many parabolic subgroups P of G. In this note, we observe that these constructions remain valid for any algebraically closed field k of characteristic not equal to 2 and we give constructions of Richardson elements for the remaining parabolic subgroups. Presented by Peter Littelmann.     

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.     

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.     

Representation theory of $\mathcal{W}$-algebras

We study the representation theory of the -algebra associated with a simple Lie algebra at level k. We show that the “-” reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k∈ℂ. Moreover, we show that the character of each irreducible highest weight representation of is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra of . As a consequence we complete (for the “-” reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of -algebras. Mathematics Subject Classification (1991)  17B68, 81R10     

Periodic automorphisms of takiff algebras, contractions, and θ-groups

Let be an algebraic Lie algebra and a (generalised) Takiff algebra. Any finite-order automorphism θ of induces an automorphism of of the same order, denoted . We study invariant-theoretic properties of representations of the fixed point subalgebra of on other eigenspaces of in . We use the observation that, for special values of m, the fixed point subalgebra, , turns out to be a contraction of a certain Lie algebra associated with and θ. To my teacher Supported in part by R.F.B.R. grant 06-01-72550.     

Orderings and *-Orderings on Cocommutative Hopf Algebras

Our aim is to construct new examples of totally ordered and ∗-ordered noncommutative integral domains. We will discuss the following classes of rings: enveloping algebras U(L), group rings G and smash products U(L) G. All of them are examples of Hopf algebras. Characterizations of orderability for enveloping algebras and group rings and of ∗-orderability for enveloping algebras have been found before and will be recalled in the article. Our main results are: for and L finite–dimensional, we characterize the orderability of U(L) G; for , we give a necessary and a sufficient condition for ∗-orderability of G (G orderable, respectively, G residually ‘torsion-free nilpotent’). Moreover, for and L finite-dimensional, we reduce the problem of characterizing the ∗-orderability of U(L) G to the problem of characterizing the ∗-orderability of G. The latter remains open. The research of the first author was supported by the Ministry of Education, Science and Sport of the Republic of Slovenia under grant P1-0222 (Algebraic methods in operator theory). The research of the second and third author was supported by the Natural Sciences and Engineering Research Council of Canada.     

Nilpotent Bicone and Characteristic Submodule of a Reductive Lie Algebra

Let be a finite-dimensional complex reductive Lie algebra and S() its symmetric algebra. The nilpotent bicone of is the subset of elements (x, y) of whose subspace generated by x and y is contained in the nilpotent cone. The nilpotent bicone is naturally endowed with a scheme structure, as nullvariety of the augmentation ideal of the subalgebra of generated by the 2-order polarizations of invariants of . The main result of this paper is that the nilpotent bicone is a complete intersection of dimension , where and are the dimensions of Borel subalgebras and the rank of , respectively. This affirmatively answers a conjecture of Kraft and Wallach concerning the nullcone [KrW2]. In addition, we introduce and study in this paper the characteristic submodule of . The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal of definition. The main difficulty encountered for this work is that the nilpotent bicone is not reduced. To deal with this problem, we introduce an auxiliary reduced variety, the principal bicone. The nilpotent bicone, as well as the principal bicone, are linked to jet schemes. We study their dimensions using arguments from motivic integration. Namely, we follow methods developed by Mustaţǎ in [Mu]. Finally, we give applications of our results to invariant theory.     

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.     

An Application of U(g)-bimodules to Representation Theory of Affine Lie Algebras

Let be the affine Lie algebra associated to the simple finite-dimensional Lie algebra . We consider the tensor product of the loop -module associated to the irreducible finite-dimensional -module V() and the irreducible highest weight -module L _k,. Then L _k, can be viewed as an irreducible module for the vertex operator algebra M _k,0. Let A(L _k,) be the corresponding -bimodule. We prove that if the -module is zero, then the -module is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras.     

Special Quasigraded Lie Algebras and Integrable Hamiltonian Systems

We construct a family of special quasigraded Lie algebras of functions of one complex variables with values in finite-dimensional Lie algebra , labeled by the special 2-cocycles F on . The main property of the constructed Lie algebras is that they admit Kostant-Adler-Symes scheme. Using them we obtain new integrable finite-dimensional Hamiltonian systems and new hierarchies of soliton equations.     

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.     

On the Existence of Trace for Elements of \mathbb{C}_p

Let T be a transcendental element of and the orbit of T. On we have a Haar measure . The goal of this paper is to characterize all the elements of for which the integral , called the trace of T, is well defined.Presented by A. Verschoren     

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.     

Nested set complexes of Dowling lattices and complexes of Dowling trees

Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice (Proc. Internat. Sympos., 1971, pp. 101–115) and of its subposet of the G-symmetric partitions which was recently introduced by Hultman (, 2006), together with the complex of G-symmetric phylogenetic trees . Hultman shows that the complexes and are homotopy equivalent and Cohen–Macaulay, and determines the rank of their top homology. An application of the theory of building sets and nested set complexes by Feichtner and Kozlov (Selecta Math. (N.S.) 10, 37–60, 2004) shows that in fact is subdivided by the order complex of . We introduce the complex of Dowling trees and prove that it is subdivided by the order complex of . Application of a theorem of Feichtner and Sturmfels (Port. Math. (N.S.) 62, 437–468, 2005) shows that, as a simplicial complex, is in fact isomorphic to the Bergman complex of the associated Dowling geometry. Topologically, we prove that is obtained from by successive coning over certain subcomplexes. It is well known that is shellable, and of the same dimension as . We explicitly and independently calculate how many homology spheres are added in passing from to . Comparison with work of Gottlieb and Wachs (Adv. Appl. Math. 24(4), 301–336, 2000) shows that is intimely related to the representation theory of the top homology of . Research partially supported by the Swiss National Science Foundation, project PP002-106403/1.     

Relations between different types of spectra and spectral characterizations

In the study of the asymptotic behaviour of solutions of differential-difference equations the -spectrum has been useful, where and implies Fourier transform , with given , φ∈L ^∞(ℝ,X), X a Banach space, (half)line. Here we study and related concepts, give relations between them, especially weak Laplace half-line spectrum of φ, and thus ⊂ classical Beurling spectrum = Carleman spectrum =  ; also  = Beurling spectrum of “φ modulo ” (Chill-Fasangova). If satisfies a Loomis type condition (L _U ), then countable and uniformly continuous ∈U are shown to imply ; here (L _U ) usually means , indefinite integral Pf of f in U imply Pf in (the Bohl-Bohr theorem for = almost periodic functions, U=bounded functions). This spectral characterization and other results are extended to unbounded functions via mean classes , ℳ ^m U ((2.1) below) and even to distributions, generalizing various recent results for uniformly continuous bounded φ. Furthermore for solutions of convolution systems S*φ=b with in some we show . With these above results, one gets generalizations of earlier results on the asymptotic behaviour of solutions of neutral integro-differential-difference systems. Also many examples and special cases are discussed.     

Triebel-Lizorkin Spaces of Para-Accretive Type and a Tb Theorem

In this article, we use a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type $\dot{F}^{\alpha,q}_{b,p}In this article, we use a discrete Calderón-type reproducing formula and Plancherel-P?lya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type , which reduces to the classical Triebel-Lizorkin spaces when the para-accretive function is constant. Moreover, we give a necessary and sufficient condition for the boundedness of paraproduct operators. From this, we show that a generalized singular integral operator T with M _b TM _b ∈WBP is bounded from to if and only if and T ^* b=0 for , where ε is the regularity exponent of the kernel of T. Chin-Cheng Lin supported by National Science Council, Republic of China under Grant #NSC 97-2115-M-008-021-MY3. Kunchuan Wang supported by National Science Council, Republic of China under Grant #NSC 97-2115-M-259-009 and NCU Center for Mathematics and Theoretic Physics.