Pluriadjoint bundles of polarized surfaces

LetX be a complex connected projective smooth algebraic surface and letL be an ample line bundle onX. The maps associated with the pluriadjoint bundles (K _X L) ¹,t2, are studied by combining an ampleness result forK _X L with a very recent result by Reider. It turns out that apart from some exceptions and up to reductions, 1) (K _X L)³ is very ample; 2) (K _X L) ² is ample and spanned by global sections and is very ample unless eitherg (L)=2 (arithmetic genus ofL) orX contains an elliptic curveE withE ²=0,E·L=1;3) when (K _X L) ² is not very ample, the associated map has degree 4, equality implying thatg (L)=2 and .     

Embeddings of Banach Spaces Into Banach Lattices and the Gordon–Lewis Property

In this paper we first show that if X is a Banach space and is a left invariant crossnorm on lX, then there is a Banach lattice L and an isometric embedding J of X into L, so that I J becomes an isometry of lX onto l_m J(X). Here I denotes the identity operator on l and l_m J(X) the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to prove the main results which characterize the Gordon–Lewis property GL and related structures in terms of embeddings into Banach lattices.     

Differential geometry of tensor product immersions II

In the first part of this series, we prove that the tensor product immersionf ₁ f _2k of2k isometric spherical immersions of a Riemannian manifoldM in Euclidean space is of-type with k and classify tensor product immersionsf ₁ f _2k which are ofk-type. In this article we investigate the tensor product immersionsf ₁ f _2k which are of (k+1)-type. Several classification theorems are obtained.     

Tame and wild subspace problems

Assume thatB is a finite-dimensional algebra over an algebraically closed fieldk, B _d =Spec k[(B _d ] is the affine algebraic scheme whoseR-points are theB _k k[B_d]-module structures onR ^d, and M_d is a canonical B_k k[B_d]-module supported by k[B_d]^d. Further, say that an affine subscheme of B_d isclass true if the functor F_gn X M_{d k[B]} X induces an injection between the sets of isomorphism classes of indecomposable finite-dimensional modules over k[] andB. If B_d contains a class-true plane for somed, then the schemes B_e contain class-true subschemes of arbitrary dimensions. Otherwise, each B_d contains a finite number of classtrue puncture straight linesL(d, i) such that for eachn, almost each indecomposableB-module of dimensionn is isomorphic to someF _{L(d, i)} (X); furthermore,F _{L(d, i)} (X) is not isomorphic toF _{L(l, j)} (Y) if(d, i) (l, j) andX 0. The proof uses a reduction to subspace problems, for which an inductive algorithm permits us to prove corresponding statements.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 313–352, March, 1993.     

The Factorization Problem and the Smash Biproduct of Algebras and Coalgebras

We consider the factorization problem for bialgebras. Let L and H be algebras and coalgebras (but not necessarily bialgebras) and consider two maps R : H L L H and W : L H H L. We introduce a product K = L _{W R} H and we give necessary and sufficient conditions for K to be a bialgebra. Our construction generalizes products introduced by Majid and Radford. Also, some of the pointed Hopf algebras that were recently constructed by Beattie, Dsclescu and Grünenfelder appear as special cases.     

Lifting-Probleme für Vektorfunktionen und ⊗-Sequenzen

In this article the topologically exact sequences of locally convex spaces are characterized for which for every locally convex space F the map id : FE F Q is a homomorphism, or equivalently, the map id L : FK F E is a topological injection. This is motivated by the problem of lifting Q-valued functions with certain given properties to E-valued functions with the same or slightly weaker properties, which may also be considered as the investigation of parameter dependences of solutions of linear (differential) equations. Applications to partial differential equations and to Fredholm functions are given.     

Some homological properties of commutative semitrivial ring extensions

LetR be a commutative ring with 1 andM anR-module. If:M _R MR is anR-module homomorphism satisfying(mm)=(mm) and(mm)m=m(mm), the additive abelian groupRM becomes a commutative ring, if multiplication is defined by (r,m)(r,m)=(rr+(mm),rm+rm). This ring is called the semitrivial extension ofR byM and and it is denoted byR M. This generalizes the notion of a trivial extension and leads to a more interesting variety of examples. The purpose of this paper is to studyR M; in particular, we are interested in some homological properties ofR M as that of being Cohen-Macaulay, Gorenstein or regular. A sample result: Let (R,m) be a local Noetherian ring,M a finitely generatedR-module and Im() m. ThenR M is Gorenstein if and only if eitherRM is Gorenstein orR is Gorenstein,M is a maximal Cohen-Macaulay module andMM ^*, where the isomorphism is given by the adjoint of.     

Homotopy Classification of Categorical Torsors

The long-known results of Schreier on group extensions are here raised to a categorical level by giving a factor set theory for torsors under a categorical group (G,) over a small category . We show a natural bijection between the set of equivalence classes of such torsors and [B({}),B(G,)], the set of homotopy classes of continuous maps between the corresponding classifying spaces. These results are applied to algebraically interpret the set of homotopy classes of maps from a CW-complex X to a path-connected CW-complex Y with _i (Y)=0 for all i1,2.     

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

Normality preserving multiplication operators

We show that a multiplication operator (T)=ATB is normality preserving if and only if it is hyponormality preserving, if and only if it is either of the formA=fg,B=h f, orA=D,B=D* for someC andD* D=I. Also we show that is (semi-) Fredholmness prserving if and only ifA andB are (semi-) Fredholm operators.Supported by the Science Foundation of Zhejiang Province and NSF.     

Note on Fano 3-folds

LetE be an ample vector bundle of rankr on a compact complex manifoldX of dimension 3 with detE=–K _x, andi(X) the index ofX. Then it is proved in this note thati(X)r unless (X,E)(¹ × ²,p*O(1) q*), wherep,q are the projections and is isomorphic toO(2) O(1) or the tangent bundleT of ². This result gives a counterexample to the conjecture formed by T. Peternell.     

Perfect crystals andq-deformed Fock spaces

In [S], [KMS] the semi-infinite wedge construction of level 1U _q (A _n ⁽¹⁾ ) Fock spaces and their decomposition into the tensor product of an irreducibleU _q (A _n ⁽¹⁾ )-module and a bosonic Fock space were given. Here a general scheme for the wedge construction ofq-deformed Fock spaces using the theory of perfect crystals is presented.LetU _q (g) be a quantum affine algebra. LetV be a finite-dimensionalU _q (g)-module with a perfect crystal base of levell. LetV _aff V [z,z ^–1] be the affinization ofV, with crystal base (L _aff,B _aff). The wedge spaceV _aff V _aff is defined as the quotient ofV _aff V _aff by the subspace generated by the action ofU _q (g) [z ^a z ^b +z ^b z ^a ]_a,b onv v (v an extremal vector). The wedge space ^r V _aff (r ) is defined similarly. Normally ordered wedges are defined by using the energy functionH :B _aff B _aff . Under certain assumptions, it is proved that normally ordered wedges form a base of ^r V _aff.Aq-deformed Fock space is defined as the inductive limit of ^r V _aff asr , taken along the semi-infinite wedge associated to a ground state sequence. It is proved that normally ordered wedges form a base of the Fock space and that the Fock space has the structure of an integrableU _q (g)-module. An action of the bosons, which commute with theU _q (g)-action, is given on the Fock space. It induces the decomposition of theq-deformed Fock space into the tensor product of an irreducibleU _q (g)-module and a bosonic Fock space.As examples, Fock spaces for typesA _2n ⁽²⁾ ,B _n ⁽¹⁾ ,A _2n ^–1/(2) ,D _n ⁽¹⁾ andD _n ^+1/(2) at level 1 andA ₁ ⁽¹⁾ at levelk are constructed. The commutation relations of the bosons in each of these cases are calculated, using two point functions of vertex operators.     

Subfactors associated to compact Kac algebras

We construct inclusions of the form (B ₀P) ^G (B ₁P) ^G , whereG is a compact quantum group of Kac type acting on an inclusion of finite dimensional C^*-algebrasB ₀B ₁ and on aII ₁ factorP. Under suitable assumptions on the actions ofG, this is a subfactor, whose Jones tower and standard invariant can be computed by using techniques of A. Wassermann. The subfactors associated to subgroups of compact groups, to projective representations of compact groups, to finite quantum groups, to finitely generated discrete groups, to vertex models and to spin models are of this form.     

Das gemischte Volumen als Distribution

It is shown that there exists a distribution T on ⁿ ( unit sphere of Euclidean n-space) such that the mixed volume V(K₁,...,K_n) equals for all convex bodies K_i, where is the support function of k_i and denotes the tensor product. As a consequence, mixed volumes are approximated uniformly by n-fold integrals of the corresponding support functions.     

Über die Riemannsche Einbettungsgeometrie der Veronesemannigfaltigkeiten

Let E be a n-dimensional euclidean vector space. The subset V _k ⁿ ={x ... x | x E} of ^kE is called a Veronesemanifold. The scalar product of E induces a euclidean structure on ^kE. Passing to the corresponding projective space , one may consider as a riemannian submanifold of the space form . In this paper we study properties of the pair of riemannian manifolds.     

Abstract Bott periodicity inKK-theory

We show that the universalC*-algebras KqA and K²A are homotopy equivalent and define abstract analogues of the Bott elements inKK-theory.     

Quantum Poisson processes and dilations of dynamical semigroups

Summary The notion of a quantum Poisson process over a quantum measure space is introduced. This process is used to construct new quantum Markov processes on the matrix algebraM _n with stationary faithful state . If (, ) is the quantum measure space in question ( a von Neumann algebra and a faithful normal weight), then the semigroupe ^tL of transition operators on (M _n , ) has generator whereu is an arbitrary unitary element of the centraliser of (M _n ,).Supported by the Netherlands Organization for Scientific Research NWO     

A note on sub-multiplicative norms

Summary IfX is a finite-dimensional linear space andL(X) the linear space of linear operators onX thenL(X) may be represented asXX ^*. IfE={e ₁, ...,e _n } is a basis forX and e _j y _j ^* is a typical element ofXX ^*, then norms can be introduced onL(X) in the form y _j ^* e _j . Given that the norm onX isE-absolute we derive a necessary and sufficient condition for the norm onL(X) to be submultiplicative.     

K-invariants in the universal enveloping algebra of the desitter group

Let G be a non-compact connected semisimple Lie group with finite center and let G^K denote the centralizer of a maximal compact subgroup K of G inG, the universal enveloping algebra over of the Lie algebra of G. In [4] Lepowsky defines an injective anti-homo morphism P:G ^KK ^MA, where M is the centralizer in K of a Cartan subalgebraa of the symmetric pair (G,K),K andA are the universal enveloping algebras over corresponding to K anda, respectively, andK ^M is the centralizer of M inK. The subalgebra P(G ^K) ofK ^MA has considerable significance in the infinite dimensional representation theory of G. In this paper we explicitly compute P(G ^K) when G=S0_o(4,1), and show how this result leads to the determination of all irreducible representations of G and its universal covering group Spin(4,1).Partially supported by CONICET (Argentina) grants.     

Maslov Index and a Central Extension of the Symplectic Group

In this paper we show that over any field K of characteristic different from 2, the Maslov index gives rise to a 2-cocycle on the stable symplectic group with values in the Witt group. We also show that this cocycle admits a natural reduction to I ²(K) and that the induced natural homomorphism from K ₂ Sp(K)I ²(K) is indeed the homomorphism given by the symplectic symbol {x, y} mapping to the Pfister form 1, -x 1, –y.