Presenting affine <Emphasis Type="Italic">q</Emphasis>-Schur algebras

We obtain a presentation of certain affine q-Schur algebras in terms of generators and relations. The presentation is obtained by adding more relations to the usual presentation of the quantized enveloping algebra of type affine . Our results extend and rely on the corresponding result for the q-Schur algebra of the symmetric group, which were proved by the first author and Giaquinto.     

Modular <Emphasis Type="Italic">W</Emphasis> and <Emphasis Type="Italic">H</Emphasis> Type Lie Algebras

The modular Witt algebra W(p, n) and H(p, 2n) are defined on the polynomial rings Z_p[x₁,...,x_n] and Z_p[X₁,...,x_n, y₁,...,y_n] respectively. We generalize Z_p[x₁,...,xn] and Zp[x₁,...,xn, y₁,...,y_n], so we get the generalized W-type and H-type modular Lie algebras. We find all the derivations of W(p, 1).AMS Subject Classification: Primary 17B40, 17B56.     

Quantizations of the W-Algebra W（2, 2）

We quantize the W-algebra W(2, 2), whose Verma modules, Harish-Chandra modules, irreducible weight modules and Lie bialgebra structures have been investigated and determined in a series of papers recently.     

Free fermions, <Emphasis Type="Italic">W</Emphasis>-algebras,and isomonodromic deformations

We consider the theory of multicomponent free massless fermions in two dimensions and use it to construct representations of W-algebras at integer Virasoro central charges. We define the vertex operators in this theory in terms of solutions of the corresponding isomonodromy problem. We use this construction to obtain some new insights into tau functions of the multicomponent Toda-type hierarchies for the class of solutions given by the isomonodromy vertex operators and to obtain a useful representation for tau functions of isomonodromic deformations.     

(<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">d</Emphasis>)-bi-Koszul algebras

The paper focuses on the 1-generated positively graded algebras with non-pure resolutions and mainly discusses a new kind of algebras called(s,t,d)-bi-Koszul algebras as the generalization of bi-Koszul algebras. An(s,t,d)-bi-Koszul algebra can be obtained from two periodic algebras with pure resolutions. The generation of the Koszul dual of an(s,t,d)-bi-Koszul algebra is discussed. Based on it,the notion of strongly(s,t,d)-bi-Koszul algebras is raised and their homological properties are further discussed.     

<Emphasis Type="Italic">K</Emphasis>-theory for certain extension algebras of purely infinite simple <Emphasis Type="Italic">C</Emphasis>*-algebras

We consider extension algebras of unital purely infinite simple C*-algebras by purely infinite simple stable C*-algebras. K-theory of such extension algebras is described.     

Bidual of <Emphasis Type="Italic">r</Emphasis>-algebras

We prove that the order continuous bidual of an Archimedean r-algebra is a Dedekind complete r-algebra with respect to the Arens multiplications.     

Hardy algebras, <Emphasis Type="Italic">W</Emphasis>*-correspondences and interpolation theory

Given a von Neumann algebra M and a W*-correspondence E over M, we construct an algebra H(E) that we call the Hardy algebra of E. When M= =E, H(E) is the classical Hardy space H of bounded analytic functions on the unit disc. When M= and E= H(E) is the free semigroup algebra studied by Popescu, Davidson and Pitts and many others. We show that given any faithful normal representation of M on a Hilbert space H there is a natural correspondence E over the commutant (M), called the -dual of E, and that H(E) can be realized in terms of (B(H)-valued) functions on the open unit ball ((E)*) in the space of adjoints of elements in E. We prove analogues of the Nevanlinna-Pick theorem in this setting and discover other aspects of the value distribution theory for elements in H(E). We also analyze the boundary behavior of elements in H(E) and obtain generalizations of the Sz.-Nagy–Foia functional calculus and the functional calculus of Popescu for c.n.c. row contractions. The correspondence E has a dual that is naturally isomorphic to E and the commutants of certain, so-called induced representations of H(E) can be viewed as induced representations of H(E). For these induced representations a double commutant theorem is proved.Supported in part by grants from the National Science Foundation and from the U.S.-Israel Binational Science Foundation.Supported in part by the U.S.-Israel Binational Science Foundation and by the Fund for the Promotion of Research at the Technion.Revised version: 11 March 2004     

Rate of convergence to stationarity of the system <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">N</Emphasis>/<Emphasis Type="Italic">N</Emphasis>+<Emphasis Type="Italic">R</Emphasis>

We consider the M/M/N/N+R service system, characterized by N servers, R waiting positions, Poisson arrivals and exponential service times. We discuss representations and bounds for the rate of convergence to stationarity of the number of customers in the system, and study its behaviour as a function of R, N and the arrival rate λ, allowing λ to be a function of N.     

Commutativity of Almost <Emphasis Type="Italic">F</Emphasis>-algebras

Let A be an Archimedean vector lattice, let be its Dedekind completion and let B be a Dedekind complete vector lattice. If Ψ ₀:A × A→ B is a positive orthosymmetric bimorphism, then there exists a positive bimorphism extension Ψ of Ψ ₀ to × in B which is orthosymmetric. This leads to a new and short proof of the commutativity of the almost f-algebras multiplications.     

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp(n)/U(n). We prove that an eigenvalue λ _s (f ₂, f ₂, …, f _n ) of the Lie group Sp(n), corresponding to the representation with label (f ₁, f ₂, ..., f _n ), is an eigenvalue of the manifold Sp(n)/U(n), if and only if f ₁, f ₂, …, f _n are all even.     

Composition operators on <Emphasis Type="Italic">f</Emphasis>-algebras

By a composition operator between two f-algebras we mean a positive algebra homomorphism. This paper intends to give a systematic study of such operators. A particular attention is paid to their connection with separating regular operators as well as to their global behavior in the module of regular operators. The paper ends with some open problems.     

<Emphasis Type="Italic">K</Emphasis>-Radical classes and product radical classes of <Emphasis Type="Italic">MV</Emphasis>-algebras

For an MV-algebra let J ₀( ) be the system of all closed ideals of ; this system is partially ordered by the set-theoretical inclusion. A radical class X of MV-algebras will be called a K-radical class iff, whenever ∈ X and is an MV-algebra with J ₀( ) ≅ J ₀( ), then ∈ X. An analogous notation for lattice ordered groups was introduced and studied by Conrad. In the present paper we show that there is a one-to-one correspondence between K-radical classes of MV-algebras and K-radical classes of abelian lattice ordered groups. We also prove an analogous result for product radical classes of MV-algebras; product radical classes of lattice ordered groups were studied by Ton. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information, Grant I/2/2005.     

Structure theorems of <Emphasis Type="Italic">E</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)-Azumaya algebras

Let k be a field and E(n) be the 2 ⁿ⁺¹-dimensional pointed Hopf algebra over k constructed by Beattie, Dăscălescu and Grünenfelder [J. Algebra, 2000, 225: 743–770]. E(n) is a triangular Hopf algebra with a family of triangular structures R _M parameterized by symmetric matrices M in M _n (k). In this paper, we study the Azumaya algebras in the braided monoidal category $ E_{(n)} \mathcal{M}^{R_M } $ E_{(n)} \mathcal{M}^{R_M } and obtain the structure theorems for Azumaya algebras in the category $ E_{(n)} \mathcal{M}^{R_M } $ E_{(n)} \mathcal{M}^{R_M } , where M is any symmetric n×n matrix over k.     

Further Results for the Queueing System <Emphasis Type="Italic">M</Emphasis><Superscript><Emphasis Type="Italic">x</Emphasis></Superscript>/<Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">c</Emphasis>

For the multi-channel bulk-arrival queue, M ^x /M/c, Abol'nikov and Kabak independently obtained steady state results. In this paper the results of these authors are extended, corrected and simplified. A number of measures of efficiency are calculated for three cases where the arrival group size has: (i) a constant value, (ii) a geometric distribution, or (iii) a positive Poisson distribution. The paper also shows how to calculate fractiles for both the queue length and the waiting time distribution. Examples of extensive numerical results for certain measures of efficiency are presented in tabular and chart form.     

Cyclotomic Hecke Algebras of <Emphasis Type="Italic">G</Emphasis>(<Emphasis Type="Italic">r</Emphasis>, <Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)

In this note, we find a monomial basis of the cyclotomic Hecke algebra \({\mathcal{H}_{r,p,n}}\) of G(r,p,n) and show that the Ariki-Koike algebra \({\mathcal{H}_{r,n}}\) is a free module over \({\mathcal{H}_{r,p,n}}\), using the Gröbner-Shirshov basis theory. For each irreducible representation of \({\mathcal{H}_{r,p,n}}\), we give a polynomial basis consisting of linear combinations of the monomials corresponding to cozy tableaux of a given shape.     

The <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1+<Emphasis Type="Italic">G</Emphasis> queue revisited

We consider an M/G/1 queue with the following form of customer impatience: an arriving customer balks or reneges when its virtual waiting time, i.e., the amount of work seen upon arrival, is larger than a certain random patience time. We consider the number of customers in the system, the maximum workload during a busy period, and the length of a busy period. We also briefly treat the analogous model in which any customer enters the system and leaves at the end of his patience time or at the end of his virtual sojourn time, whichever occurs first.     

Rationality of the vertex operator algebra <Emphasis Type="Italic">V</Emphasis><Subscript><Emphasis Type="Italic">L</Emphasis></Subscript>+ for a positive definite even lattice <Emphasis Type="Italic">L</Emphasis>

The lattice vertex operator algebra V_L associated to a positive definite even lattice L has an automorphism of order 2 lifted from –1-isometry of L. The fixed point set V_L+ of V_L for the automorphism is naturally a vertex operator algebra. We prove that any ₀-graded weak V_L⁺-module is completely reducible.Supported by JSPS Research Fellowships for Young Scientists.     

Symbol <Emphasis Type="Italic">p</Emphasis>-algebras of prime degree and their <Emphasis Type="Italic">p</Emphasis>-central subspaces

We prove that the maximal dimension of a p-central subspace of the generic symbol p-algebra of prime degree p is \({p+1}\). We do it by proving the following number theoretic fact: let \({\{s_1,\dots,s_{p+1}\}}\) be \({p+1}\) distinct nonzero elements in the additive group \({G=(\mathbb{Z}/p \mathbb{Z}) \times (\mathbb{Z}/p \mathbb{Z})}\), then every nonzero element \({g \in G}\) can be expressed as \({d_1 s_1+\dots+d_{p+1} s_{p+1}}\) for some non-negative integers \({d_1,\dots,d_{p+1}}\) with \({d_1+\dots+d_{p+1}\leq p-1}\).