Lie Bialgebras of Generalized WITT Type,II

In an article by Michaelis, a class of infinite-dimensional Lie bialgebras containing the Virasoro algebra was presented. This type of Lie bialgebras was classified by Ng and Taft. In a recent article by Song and Su, Lie bialgebra structures on graded Lie algebras of generalized Witt type with finite dimensional homogeneous components were considered. In this article we consider Lie bialgebra structures on the graded Lie algebras of generalized Witt type with infinite dimensional homogeneous components. By proving that the first cohomology group H¹(𝒲, 𝒲 ? 𝒲) is trivial for any graded Lie algebras 𝒲 of generalized Witt type with infinite dimensional homogeneous components, we obtain that all such Lie bialgebras are triangular coboundary.     

Higher Order Peak Algebras

Using the theory of noncommutative symmetric functions, we introduce the higher order peak algebras (Sym(N))_N≥1, a sequence of graded Hopf algebras which contain the descent algebra and the usual peak algebra as initial cases (N=1 and N=2). We compute their Hilbert series, introduce and study several combinatorial bases, and establish various algebraic identities related to the multisection of formal power series with noncommutative coefficients. Received November 19, 2004     

Homotopy invariance of coherent Witt groups

 We prove the following general homotopy invariance theorem for coherent Witt groups: Let be a flat morphism of separated Gorenstein schemes of finite Krull dimension with affine fibers,i.e. π⁻¹(y) is an affine space over the residue field k(y) for all yY, then the induced homomorphism of coherent Witt groups is an isomorphism. As an application we calculate the (classical) Witt group of the affine hyperbolic sphere over a regular local ring. Received: 22 August 2001; in final form: 22 June 2002 / Published online: 1 April 2003     

Tensor products and discriminants of unital quadratic forms over commutative rings

A tensor product for unital quadratic forms is introduced which extends the product of separable quadratic algebras and is naturally associative and commutative. It admits a multiplicative functor vdis, the vector discriminant, with values in symmetric bilinear forms. We also compute the usual (signed) discriminant of the tensor product in terms of the discriminants of the factors. The orthogonal group scheme of a nonsingular unital quadratic formQ of even rank is isomorphic toZ ₂×SO(Q ⁰) whereQ ⁰ is the restriction of –Q to the space of trace zero elements. We use cohomology to interpret the action of separable quadratic algebras on unital quadratic forms, and to determine which forms of odd rank can be realized asQ ⁰.     

Coût des relations d’équivalence et des groupes

We study a new dynamical invariant for dicrete groups: the cost. It is a real number in {1−1/n}∪[1,∞], bounded by the number of generators of the group, and it is well behaved with respect to finite index subgroups. Namely, the quantities 1 minus the cost are related by multiplying by the index. The cost of every infinite amenable group equals 1. We compute it in some other situations, including free products, free products with amalgamation and HNN-extensions over amenable groups and for direct product situations. For instance, the cost of the free group on n generators equals n. We prove that each possible finite value of the cost is achieved by a finitely generated group. It is dynamical because it relies on measure preserving free actions on probability Borel spaces. In most cases, groups have fixed price, which implies that two freely acting groups which define the same orbit partition must have the same cost. It enables us to distinguish the orbit partitions of probability-preserving free actions of free groups of different ranks. At the end of the paper, we give a mercuriale, i.e. a list of costs of different groups. The cost is in fact an invariant of ergodic measure-preserving equivalence relations and is defined using graphings. A treeing is a measurable way to provide every equivalence class (=orbit) with the structure of a simplicial tree, this an example of graphing. Not every relation admits a treeing: we prove that every free action of a cost 1 non-amenable group is not treeable, but we prove that subrelations of treeable relations are treeable. We give examples of relations which cannot be produced by an action of any finitely generated group. The cost of a relation which can be decomposed as a direct product is shown to be 1. We define the notion for a relation to be a free product or an HNN-extension and compute the cost for the resulting relation from the costs of the building blocks. The cost is also an invariant of the pairs von Neumann algebra/Cartan subalgebra. Oblatum 27-I-1999 & 4-IV-1999 / Published online: 22 September 1999     

Leibniz algebras constructed by Witt algebras

ABSTRACT

We describe infinite-dimensional Leibniz algebras whose associated Lie algebra is the Witt algebra and we prove the triviality of low-dimensional Leibniz cohomology groups of the Witt algebra with the coefficients in itself.     

The Hopf Property for Subgroups of Hyperbolic Groups

A group is said to be Hopfian if every surjective endomorphism of the group is injective. We show that finitely generated subgroups of torsion-free hyperbolic groups are Hopfian. Our proof generalizes a theorem of Sela (Topology 35 (2) 1999, 301–321).     

The Witt ring kernel for a fourth degree field extension and related problems

In the first part of this paper we compute the Witt ring kernel for an arbitrary field extension of degree 4 and characteristic different from 2 in terms of the coefficients of a polynomial determining the extension. In the case where the lower field is not formally real we prove that the intersection of any power n of its fundamental ideal and the Witt ring kernel is generated by n-fold Pfister forms.In the second part as an application of the main result we give a criterion for the tensor product of quaternion and biquaternion algebras to have zero divisors. Also we solve the similar problem for three quaternion algebras.In the last part we obtain certain exact Witt group sequences concerning dihedral Galois field extensions. These results heavily depend on some similar cohomological results of Positselski, as well as on the Milnor conjecture, and the Bloch-Kato conjecture for exponent 2, which was proven by Voevodsky.     

A Gersten-Witt spectral sequence for regular schemes

A spectral sequence is constructed whose non-zero E₁-terms are the Witt groups of the residue fields of a regular scheme X, arranged in Gersten-Witt complexes, and whose limit is the four global Witt groups of X. This has several immediate consequences concerning purity for Witt groups of low-dimensional schemes. We also obtain an easy proof of the Gersten Conjecture in dimension smaller than 5. The Witt groups of punctured spectra of regular local rings are also computed.     

Tambara Functors on Profinite Groups and Generalized Burnside Functors

The Tambara functor was defined by Tambara in the name of TNR-functor, to treat certain ring-valued Mackey functors on a finite group. Recently Brun revealed the importance of Tambara functors in the Witt–Burnside construction. In this article, we define the Tambara functor on the Mackey system of Bley and Boltje. Yoshida's generalized Burnside ring functor is the first example. Consequently, we can consider a Tambara functor on any profinite group. In relation with the Witt–Burnside construction, we can give a Tambara-functor structure on Elliott's functor V _M , which generalizes the completed Burnside ring functor of Dress and Siebeneicher.     

p-Groupes abéliens de type (p,…,p) et disques ouverts p-adiques

Let k be an algebraically closed field of characteristic p>0, W(k) its ring of Witt vectors and R a complete discrete valuation ring dominating W(k). Consider finite groups G≃ (ℤ/pℤ) ⁿ , p≥ 2, n≥1. In a former paper we showed that a given realization of such a G as a group of k-automorphisms of k[[z]] must satisfy some conditions in order to have a lifting as a group of R-automorphisms of R[[Z]]. In this note, we give for every G (all p≥ 2, n>1) a realization as an automorphism group of k[[z]] which ca be lifted as a group of R-automorphisms of R[[Z]] for suitable R. Received: 22 December 1998     

Triangular Witt Groups Part I: The 12-Term Localization Exact Sequence

To a short exact sequence of triangulated categories with duality, we associate a long exact sequence of Witt groups. For this, we introduce higher Witt groups in a very algebraic and explicit way. Since those Witt groups are 4-periodic, this long exact sequence reduces to a cyclic 12-term one. Of course, in addition to higher Witt groups, we need to construct connecting homomorphisms, hereafter called residue homomorphisms.     

Steenrod Operation in Certain Cobordism Theories

We generalize results of Smirnow [Math. Zam. 65(2) (1999), 270–279] to the cases of MSO and MSU theories. Also we establish relation between the Steenrod operations in MG-cobordism theory (G=O, U, Sp, SO, SU) in our approach and the Steenrod–tom Dieck operations.     

Classification of the Congruent Embeddings of a Tetrahedron into a Triangular Prism

Let P(t) denote an infinitely long right triangular prism whose base is an equilateral triangle of edge length t. Let F(t){{\mathcal F}(t)} be the family of those subsets of P(t) that are congruent to a regular tetrahedron of unit edge. We present complete classification of the members of F(t){{\mathcal F}(t)} modulo rigid motions within the prism P(t), for every t > 0.     

Spectra of elements in the group ring of SU(2)

We present a new method for establishing the “gap” property for finitely generated subgroups of SU(2), providing an elementary solution of Ruziewicz problem on S² as well as giving many new examples of finitely generated subgroups of SU(2) with an explicit gap. The distribution of the eigenvalues of the elements of the group ring R[SU(2)] in the N-th irreducible representation of SU(2) is also studied. Numerical experiments indicate that for a generic (in measure) element of R[SU(2)], the “unfolded” consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for N even) and the GSE spacing law (for N odd) as N→∞; we establish several results in this direction. For certain special “arithmetic” (or Ramanujan) elements of R[SU(2)] the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction. Received June 1, 1998 / final version received September 8, 1998     

Segment Intersection Searching Problems in General Settings

   Abstract. We consider segment intersection searching amidst (possibly intersecting) algebraic arcs in the plane. We show how to preprocess n arcs in time O(n ^2+ɛ ) into a data structure of size O(n ^2+ɛ ) , for any ɛ >0 , such that the k arcs intersecting a query segment can be counted in time O( log n) or reported in time O( log n+k) . This problem was extensively studied in restricted settings (e.g., amidst segments, circles, or circular arcs), but no solution with comparable performance was previously presented for the general case of possibly intersecting algebraic arcs. Our data structure for the general case matches or improves (sometimes by an order of magnitude) the size of the best previously presented solutions for the special cases. As an immediate application of this result, we obtain an efficient data structure for the triangular windowing problem, which is a generalization of triangular range searching. As another application, the first substantially subquadratic algorithm for a red—blue intersection counting problem is derived. We also describe simple data structures for segment intersection searching among disjoint arcs, and for ray shooting among possibly intersecting arcs.     

On the commutativity degree of compact groups

In any finite group G, the commutativity degree of G (denoted by d(G)) is the probability that two randomly chosen elements of G commute. More generally, for every n ≥ 2 the nth commutativity degree (denoted by d _n (G)) is the probability that a randomly chosen ordered (n + 1)-tuple of the group elements is mutually commuting. The aim of this paper is to generalize the definition of d(G) and d _n (G) to every compact group G (infinite and even uncountable). We shall state some results concerning compact groups and we will extend some results in Erfanian et al. (Comm. Algebra 35 (2007), 4183–4197) and Lescot (J. Algebra 177 (1995), 847–869).     

Witt Cohomology, Mayer–Vietoris, Homotopy Invariance and the Gersten Conjecture

We establish a Mayer–Vietoris long exact sequence for Witt groups of regular schemes. We also establish homotopy invariance for Witt groups of regular schemes. For this, we introduce Witt groups with supports using triangulated categories. Subsequently, we use these results to prove the Gersten–Witt conjecture for semi-local regular rings of geometric type over infinite fields of characteristic different from two.     

On existentially closed and generic nilpotent groups

Letn≧2 be an integer. We prove the following results that are known in casen=2: The upper and the lower central series of an existentially closed nilpotent group of classn coincide. A finitely generic nilpotent group of classn is periodic and the center of a finitely generic torsion-free nilpotent group of classn is isomorphic toQ ⁺, whereas infinitely generic nilpotent groups do not enjoy these properties. We determine the structure of the torsion subgroup of existentially closed nilpotent groups of class 2. Finally we give an algebraic proof that there exist 2^κ non-isomorphic existentially closed nilpotent groups of classn in cardinalityK ≧N ₀. Some results of this paper were contained in [6].