Crystal Bases for Quantum Generalized Kac-Moody Algebras

In this paper, we develop the crystal basis theory for quantumgeneralized Kac–Moody algebras. For a quantum generalizedKac–Moody algebra U_q(g), we first introduce the categoryO_int of U_q(g)-modules and prove its semisimplicity. Next, wedefine the notion of crystal bases for U_q(g)-modules in thecategory O_int and for the subalgebra . We then prove the tensor product rule and the existence theoremfor crystal bases. Finally, we construct the global bases forU_q(g)-modules in the category O_int and for the subalgebra . 2000 Mathematics Subject Classification17B37, 17B67.     

On the Structure of Modular Categories

For a braided tensor category C and a subcategory K there isa notion of a centralizer C_C K, which is a full tensor subcategoryof C. A pre-modular tensor category is known to be modular inthe sense of Turaev if and only if the center Z₂C C_CC (not tobe confused with the center Z₁ of a tensor category, relatedto the quantum double) is trivial, that is, consists only ofmultiples of the tensor unit, and dimC 0. Here , the X_i being the simple objects. We prove several structural properties of modular categories.Our main technical tool is the following double centralizertheorem. Let C be a modular category and K a full tensor subcategoryclosed with respect to direct sums, subobjects and duals. ThenC_CC_CK = K and dim K·dim C_CK = dim C. We give several applications. (1) If C is modular and K is a full modular subcategory,then L=C_CK is also modular and C is equivalent as a ribbon categoryto the direct product: . Thus every modular category factorizes (non-uniquely, in general)into prime modular categories. We study the prime factorizationsof the categories D(G)-Mod, where G is a finite abelian group. (2) If C is a modular *-category and K is a full tensorsubcategory then dim C dim K · dim Z₂K. We give exampleswhere the bound is attained and conjecture that every pre-modularK can be embedded fully into a modular category C with dim C=dimK·dim Z₂K. (3) For every finite group G there is a braided tensor*-category C such that Z₂CRep,G and the modular closure/modularization is non-trivial. 2000 MathematicsSubject Classification 18D10.     

Hopf-Cyclic Homology and Relative Cyclic Homology of Hopf-Galois Extensions

Let H be a Hopf algebra and let M_s (H) be the category of allleft H-modules and right H-comodules satisfying appropriatecompatibility relations. An object in M_s (H) will be calleda stable anti-Yetter–Drinfeld module (over H) or a SAYDmodule, for short. To each M M_s (H) we associate, in a functorialway, a cyclic object Z_* (H, M). We show that our constructioncan be used to compute the cyclic homology of the underlyingalgebra structure of H and the relative cyclic homology of H-Galoisextensions. Let K be a Hopf subalgebra of H. For an arbitrary M M_s (K)we define a right H-comodule structure on so that becomes an object in M_s (H). Under some assumptions on K and M we computethe cyclic homology of . As a direct application of this result, we describe the relativecyclic homology of strongly graded algebras. In particular,we calculate the cyclic homology of group algebras and quantumtori. Finally, when H is the enveloping algebra of a Lie algebra g,we construct a spectral sequence that converges to the cyclichomology of H with coefficients in a given SAYD module M. Wealso show that the cyclic homology of almost symmetric algebrasis isomorphic to the cyclic homology of H with coefficientsin a certain SAYD module. 2000 Mathematics Subject Classification16E40 (primary), 16W30 (secondary).     

The Generic Local Spectrum of Any Operator is The Full Spectrum

Let T=(T₁, ..., T_n) be a commuting n-tuple of bounded linearoperators acting on some complex Banach space X. We show thatif T has the single-valued extension property, then the localspectrum _T(x) coincides with the spectrum (T), for all vectorsx X, except on a set of the first Baire category. 1991 MathematicsSubject Classification 47A11, 47A13.     

Derived Functors of Inverse Limits Revisited

We prove, correct and extend several results of an earlier paperof ours (using and recalling several of our later papers) aboutthe derived functors of projective limit in abelian categories.In particular we prove that if C is an abelian category satisfyingthe Grothendieck axioms AB3 and AB4* and having a set of generatorsthen the first derived functor of projective limit vanisheson so-called Mittag-Leffler sequences in C. The recent examplesgiven by Deligne and Neeman show that the condition that thecategory has a set of generators is necessary. The conditionAB4* is also necessary, and indeed we give for each integerm 1 an example of a Grothendieck category C_m and a Mittag-Lefflersequence in C_m for which the derived functors of its projectivelimit vanish in all positive degrees except m. This leads toa systematic study of derived functors of infinite productsin Grothendieck categories. Several explicit examples of theapplications of these functors are also studied.     

The Singular Homology of the Hawaiian Earring

The singular homology groups of compact CW-complexes are finitelygenerated, but the groups of compact metric spaces in generalare very easy to generate infinitely and our understanding ofthese groups is far from complete even for the following compactsubset of the plane, called the Hawaiian earring: Griffiths [11] gave a presentation of the fundamental groupof H and the proof was completed by Morgan and Morrison [15].The same group is presented as the free -product of integers Z in [4, Appendix]. Hence the firstintegral singular homology group H₁(H) is the abelianizationof the group . These results have been generalized to non-metrizable counterparts H_I of H(see Section 3). In Section 2 we prove that H₁(X) is torsion-free and H_i(X) =0 for each one-dimensional normal space X and for each i 2.The result for i 2 is a slight generalization of [2, Theorem5]. In Section 3 we provide an explicit presentation of H₁(H)and also H₁(H_I) by using results of [4]. Throughout this paper, a continuum means a compact connectedmetric space and all maps are assumed to be continuous. Allhomology groups have the integers Z as the coefficients. Thebouquet with n circles is denoted by B_n. The base point (0, 0) of B_n is denoted by o forsimplicity.     

Algebras and Modules in Monoidal Model Categories

In recent years the theory of structured ring spectra (formerlyknown as A- and E-ring spectra) has been simplified by the discoveryof categories of spectra with strictly associative and commutativesmash products. Now a ring spectrum can simply be defined asa monoid with respect to the smash product in one of these newcategories of spectra. In this paper we provide a general methodfor constructing model category structures for categories ofring, algebra, and module spectra. This provides the necessaryinput for obtaining model categories of symmetric ring spectra,functors with smash product, Gamma-rings, and diagram ring spectra.Algebraic examples to which our methods apply include the stablemodule category over the group algebra of a finite group andunbounded chain complexes over a differential graded algebra.1991 Mathematics Subject Classification: primary 55U35; secondary18D10.     

K0 and the dimension filtration for p-torsion Iwasawa modules

Let G be a compact p-adic analytic group. We study K-theoreticquestions related to the representation theory of the completedgroup algebra kG of G with coefficients in a finite field kof characteristic p. We show that if M is a finitely generatedkG-module with canonical dimension smaller than the dimensionof the centralizer, as a p-adic analytic group, of any p-regularelement of G, then the Euler characteristic of M is trivial.Writing _i for the abelian category consisting of all finitelygenerated kG-modules of dimension at most i, we provide an upperbound for the rank of the natural map from the Grothendieckgroup of _i to that of _d, where d denotes the dimension of G.We show that this upper bound is attained in some special cases,but is not attained in general.     

Correction: the Embedding of Radical Rings in Simple Radical Rings

As G. M. Bergman has pointed out, in the proof of the lemmaon p. 187, we cannot conclude that $$\stackrel{\¯}{S}$$is universal in the sense stated. However, the proof can becompleted as follows: Any element of $$\stackrel{\¯}{S}$$can be obtained as the first component of the solution u ofa system (A–I)u+a = 0, (1) where A S_n, a ⁿS and A–I has an inverse over L. SinceS is generated by R and k{s}, A can (by the last part of Lemma3.2 of [1]) be taken to be linear in these arguments, say A= A₀ + sA1, where A₀ R_n, A₀ R_n, A₁ K_n. Multiplying by (I–sA₁)^–1,we reduce this equation to the form (S_vB_v–I)u+a=0, (2) with the same solution u as before, where B_v R_n, s_v k{s}¹and a ⁿS. Now consider the retraction S k{s} (3) obtained by mapping R 0. If we denote its effect by x x^*,then (2) goes over into an equation –I.v + a^* 0, (4) which clearly has a unique solution v in k{s}; therefore theretraction (3) can be extended to a homomorphism $$\stackrel{\¯}{S}$$ k{s}, again denoted by x x^*, provided we can show that u₁^*does not depend on the equation (1) used to define it. Thisamounts to showing that if an equation (1), or equivalently(2), has the solution u₁ = 0, then after retraction we get v₁= 0 in (4), i.e. a₁^* = 0. We shall use induction on n; if u₁= 0 in (2), then by leaving out the first row and column ofthe matrix on the left of (2), we have an equation for u₂,...,u_n and by the induction hypothesis, their values after retractionare uniquely determined. Now from (2) we have where B = (b_ij^v). Applying ^* and observing that b_ij^vR, we seethat a₁ ^* = 0, as we wished to show. The proof still appliesfor n = 1, so we have a well-defined mapping $$\stackrel{\¯}{S}$$ k{s}, which is a homomorphism. Now the proof of the lemma canbe completed as before.     

On the Discreteness and Convergence in n-Dimensional Mobius Groups

Throughout this paper, we adopt the same notations as in [1,6, 8] such as the Möbius group M(Rⁿ), the Clifford algebraC_n–1, the Clifford matrix group SL(2, _n), the Cliffordnorm of ||A||=(|a|²+|b|²+|c|²+|d|²) (1) and the Clifford metric of SL(2, _n) or of the Möbius groupM(Rⁿ) d(A₁,A₂)=||A₁–A₂||(|a₁–a₂|²+|b₁–b₂|²+|c₁–c₂|²+|d₁–d₂|²)(2) where |·| is the norm of a Clifford number and represents f_i M(), i = 1,2, and so on. In addition, we adopt some notions in [6, 12]:the elementary group, the uniformly bounded torsion, and soon. For example, the definition of the uniformly bounded torsionis as follows.     

Link Cobordism and the Intersection of Slice Discs

We work in the smooth category. An (oriented) (ordered) m-component n-(dimensional) link isa smooth oriented submanifold L = {K₁, ..., K_m} of Sⁿ⁺² whichis the ordered disjoint union of m manifolds, each PL-homeomorphicto the standard n-sphere. If m = 1, then L is called a knot. We say that m-component n-dimensional links L₀ and L₁ are (link-)concordantor (link-)cobordant if there is a smooth oriented submanifoldC = {C₁, ..., C_m} of Sⁿ⁺² x [0, 1] which meets the boundarytransversely in C, is PL-homeomorphic to L₀ x [0, 1], and meetsSⁿ⁺² x {l} in L_l (l = 0, 1). If m = 1, then we say that n-knotsL₀ and L_l are (knot-)concordant or (knot-)cobordant. Then wecall C a concordance-cylinder of the two n-knots L₀ and L_l. If an n-link L is concordant to the trivial link, then we callL a slice link. If an n-link L = {K₁, ..., K_m} Sⁿ⁺² = Bⁿ⁺³ Bⁿ⁺³ is slice,then there is a disjoint union of (n + 1)-discs in Bⁿ⁺³ such that is called a set of slice discs for L. If m = 1, then is called a slice disc for the knotL. 1991 Mathematics Subject Classification 57M25, 57Q45.     

Indecomposable flat cotorsion modules

An additive functor from the category of flat right R-modulesto the category of abelian groups is continuous if it is isomorphicto a functor of the form–_R M, where M is a left R-module.It is shown that for any simple subfunctor A of– M thereis a unique indecomposable flat cotorsion module U_R for whichA(U)0. It is also proved that every subfunctor of a continuousfunctor contains a simple subfunctor. This implies that everyflat right R-module may be purely embedded into a product ofindecomposable flat cotorsion modules. If CE(R) is the cotorsion envelope of R_R and S= End;_R CE(R),then a local ring monomorphism is constructed from R/J(R) toS/J(S). This local morphism of rings is used to associate asemiperfect ring to any semilocal ring. It also proved thatif R is a semilocal ring and M a simple left R-module, thenthe functor–_R M on the category of flat right R-modulesis uniform, and therefore contains a unique simple subfunctor.     

Complex Dynamics of Convergence Acceleration

Generalized Steffensen methods are nonderivative algorithmsfor the computation of fixed points of a function f. They replacethe functional iteration Z_m+1=f(Z_m) with Z_m+1=F_n(Z_m, where F_nis explicitly provided for every n 1 as a quotient of two Hankeldeterminants. In this paper we derive rules pertaining to thelocal behaviour of these methods. Specifically, and subjectto analyticity, given that is a bounded fixed point of f, thenit is also a fixed point of F_n. Moreover, unless f'() vanishesor is a root of unity, becomes a superattractive fixed pointof F_n of degree n; if f'() is a root of unity of minimal degreeq2, then is (as a fixed point of F_n) superattractive of degreemin {q-1, n}; if f'()=1, then is attractive for F_n; and, finally,if is superattractive of degree s (as a fixed point of f),then it becomes superattractive of degree (s + 1)^n–1(ns+ s + 1)–1. Attractivity rules change at infinity (providedthat f()=). Broadly speaking, infinity becomes less attractivefor F_n, Since one is interested in convergence to finite fixedpoints, this further enhances the appeal of generalized Steffensenmethods.     

UNEXPECTED SUBSPACES OF TENSOR PRODUCTS

We describe complemented copies of ₂ both in C(K₁) C(K₂) when at least one of the compact spaces K_iis not scattered and in L₁(µ₁)L₁(µ₂) when at least one of the measures is not atomic.The corresponding local construction gives uniformly complementedcopies of the in c₀ c₀. We continue the study of c₀ c₀ showing that it contains a complementedcopy of Stegall's space and proving that (c₀ c₀)' is isomorphicto , together with other results. In the last section we use Hardy spaces to find an isomorphiccopy of L_p in the space of compact operators from L_q to L_r,where 1 < p, q, r < and 1/r = 1/p + 1/q.     

Ganea's Conjecture on Lusternik-Schnirelmann Category

A series of complexes Q_p indexed by all primes p is constructedwith catQ_p=2 and catQ_pxSⁿ=2 for either n2 or n=1 and p=2. Thisdisproves Ganea's conjecture on Lusternik–Schnirelmann(LS) category. 1991 Mathematics Subject Classification 55M30.     

Coherent Products in a Finite Group along a Linear Ordering

Let C = (C, ) be a linear ordering, E a subset of {(x, y):x< y in C} whose transitive closure is the linear orderingC, and let :E G be a map from E to a finite group G = (G, •).We showed with M. Pouzet that, when C is countable, there isF E whose transitive closure is still C, and such that (p) = (x_o, x₁)•(x₁, x₂)•....•(x_{n– 1}, x_n) G depends only upon the extremities x₀, x_n ofp, where p = (x_o, x₁...,x_n) (with 1 n < ) is a finite sequencefor which (x_i, x_{i + 1}) F for all i < n. Here, we show thatthis property does not hold if C is the real line, but is stilltrue if C does not embed an ₁-dense linear ordering, or evena 2-dense linear ordering when Martin's Axiom holds (it followsin particular that it is independent of ZFC for linear orderingsof size ). On the other hand, we prove that this property isalways valid if E = {(x,y):x < y in C}, regardless of anyother condition on C.     

Strong Weighted Mean Summability and Kuttner's Theorem

In this paper we study sequence spaces that arise from the conceptof strong weighted mean summability. Let q = (q_n) be a sequenceof positive terms and set Q_n = ⁿ_k=1q_k. Then the weighted meanmatrix M_q = (a_nk) is defined by if kn, a_nk=0 if k>n. It is well known that M_q defines a regular summability methodif and only if Q_n. Passing to strong summability, we let 0<p<.Then , are the spaces of all sequences that are strongly M_q-summablewith index p to 0, strongly M_q-summable with index p and stronglyM_q-bounded with index p, respectively. The most important specialcase is obtained by taking M_q = C₁, the Cesàro matrix,which leads to the familiar sequence spaces w₀(p), w(p) and w(p), respectively, see [4, 21]. We remark that strong summabilitywas first studied by Hardy and Littlewood [8] in 1913 when theyapplied strong Cesàro summability of index 1 and 2 toFourier series; orthogonal series have remained the main areaof application for strong summability. See [32, 6] for furtherreferences. When we abstract from the needs of summability theory certainfeatures of the above sequence spaces become irrelevant; forinstance, the q_k simply constitute a diagonal transform. Hence,from a sequence space theoretic point of view we are led tostudy the spaces     

Isoperimetric Inequalities for Volumes Associated with Decomposable Forms

We give sharp estimates for volumes in Rⁿ defined by decomposableforms. In particular, we show that if F(X₁..., X_n) = (_i1X₁ + ... + _inX_n) is a decomposableform with _ij C, degree d > n, and discriminant D_F 0, andif V_F is the volume of the region {xRⁿ:|F(x)| 1}, then |D_F|^(d–n)!/d!V_F C_n, where C_n is the value of |D_F|^(d–n)!/d! V_F whenF(X₁..., X_n) = X₁... X_n(X₁ +... + X_n); moreover, we show thatthe sequence {C_n} is asymptotic to (2/)e^1–(2n)ⁿ. Theseresults generalize work of the first author on binary formsand will likely find application in the enumeration of solutionsof decomposable form inequalities.     

On the Discrepancy for Cartesian Products

Let B₂ denote the family of all circular discs in the plane.It is proved that the discrepancy for the family {B₁ x B₂ :B₁, B₂ B₂} in R⁴ is O(n^1/4+) for an arbitrarily small constant > 0, that is, it is essentially the same as that for thefamily B₂ itself. The result is established for the combinatorialdiscrepancy, and consequently it holds for the discrepancy withrespect to the Lebesgue measure as well. This answers a questionof Beck and Chen. More generally, we prove an upper bound forthe discrepancy for a family {^k_i=1A_i:A_iA_i, i = 1, 2, ..., k},where each A_i is a family in R^d_i, each of whose sets is describedby a bounded number of polynomial inequalities of bounded degree.The resulting discrepancy bound is determined by the ‘worst’of the families A_i, and it depends on the existence of certaindecompositions into constant-complexity cells for arrangementsof surfaces bounding the sets of A_i. The proof uses Beck's partialcoloring method and decomposition techniques developed for therange-searching problem in computational geometry.     

Filtration-Closed Auslander-Reiten Components for Wild Hereditary Algebras

Let H=kQ be a finite-dimensional connected wild hereditary pathalgebra, over some field k. Denote by H-reg the category offinite-dimensional regular H-modules, that is, the categoryof modules M with for all integers m, where _H denotes the Auslander–Reiten translation.Call a filtration of a regular H-module M a regular filtration if all subquotients M_i/M_i+1are regular. Call a regular filtration (*) a regular compositionseries if it is strictly decreasing and has no proper refinement.A regular component C in the Auslander–Reiten quiver (H) of H-mod is called filtration closed if, for each M addC, the additive closure of C, and each regular filtration (*)of M, all the subquotients M_i/M_i+1 are also in add C. We showthat most wild hereditary algebras have filtration-closed Auslander–Reitencomponents. Moreover, we deduce from this that there are alsoalmost serial components, that is regular components C, suchthat any indecomposable XC has a unique regular compositionseries. This composition series coincides with the Auslander–Reitenfiltration of X, given by the maximal chain of irreducible monosending at X. 1991 Mathematics Subject Classification: 16G70,16G20, 16G60, 16E30.