Volume 21, Issue 1

We show that the universal central extensions of the little projective group of any Moufang polygon is precisely the Steinberg group obtained from its defining commutator relations, provided the defining structure is not too small. As an application, we get that also the universal central extensions of the little projective group of any 2-spherical Moufang twin building is precisely the Steinberg group obtained from its defining commutator relations.

We define the positive diameter of a finite group G with respect to a generating set A ⊆ G to be the smallest non-negative integer n such that every element of G can be written as a product of at most n elements of A . This invariant, which we denote by , can be interpreted as the diameter of the Cayley digraph induced by A on G . In this paper we study the positive diameters of a finite abelian group G with respect to its various generating sets A . More specifically, we determine the maximum possible value of and classify all generating sets for which this maximum value is attained. Also, we determine the maximum possible cardinality of A subject to the condition that is “not too small”. Conceptually, the problems studied are closely related to our earlier work [Klopsch and Lev, J. Algebra 261: 145–171, 2003] and the results obtained shed a new light on the subject. Our original motivation came from connections with caps, sum-free sets, and quasi-perfect codes.

Recently in [Kim and Song, Intrinsic ultracontractivity of non-symmetric diffusion semigroups in bounded domains: 2006, Kim and Song, Intrinsic ultracontractivity of non-symmetric diffusion with measure-valued drifts and potentials: 2006], we extended the concept of intrinsic ultracontractivity to non-symmetric semigroups and proved that for a large class of non-symmetric diffusions Z with measure-valued drift and potential, the semigroup of Z D (the process obtained by killing Z upon exiting D ) in a bounded domain is intrinsic ultracontractive under very mild assumptions. In this paper, we study the intrinsic ultracontractivity for non-symmetric discontinuous Lévy processes. We prove that, for a large class of non-symmetric discontinuous Lévy processes X such that the Lebesgue measure is absolutely continuous with respect to the Lévy measure of X , the semigroup of X D in any bounded open set D is intrinsic ultracontractive. In particular, for the non-symmetric stable process X discussed in [Vondraček, Glas. Mat. Ser. 37: 211–233, 2002], the semigroup of X D is intrinsic ultracontractive for any bounded set D . Using the intrinsic ultracontractivity, we show that the parabolic boundary Harnack principle is true for those processes. Moreover, we get that the supremum of the expected conditional lifetimes in a bounded open set is finite. We also have results of the same nature when the Lévy measure is compactly supported.

We define a category of quasi-coherent sheaves of topological spaces on projective toric varieties and prove a splitting result for its algebraic K -theory, generalising earlier results for projective spaces. The splitting is expressed in terms of the number of interior lattice points of dilations of a polytope associated to the variety. The proof uses combinatorial and geometrical results on polytopal complexes. The same methods also give an elementary explicit calculation of the cohomology groups of a projective toric variety over any commutative ring.

Let R be a left coherent ring, R ω a Wakamatsu tilting module and S = End( R ω ) a right coherent ring. It is shown that the FP -injective dimensions of R ω and ω S are identical provided both of them are finite. It is also shown that in this case all finitely presented left R -modules (right S -modules) have special ┴ ω -precovers and special ( ┴ ω ) ┴ -preenvelopes.

We construct an algorithm which computes the catenary and tame degree of a numerical monoid. As an example we explicitly calculate the catenary and tame degree of numerical monoids generated by arithmetical sequences in terms of their first element, the number of elements in the sequence and the difference between two consecutive elements of the sequence.

We give an example of a commutative Prüfer domain R with field of fractions F and a quaternion division algebra D with centre F such that R cannot be extended to a Prüfer order in D in the sense of [Alajbegović and Dubrovin, J. Algebra 135: 165–176, 1990]. This shows, that a general extension theorem for Prüfer orders in central simple algebras does not exist and finally answers a question given in [Marubayashi, Miyamoto, Ueda, Non-commutative Valuation Rings and Semihereditary Orders. K-Monographs in Mathematics 3, Kluwer, 1997]. Moreover, in our example R is a Bézout domain which is the intersection of a countable number of (non-discrete) real valuation rings.

Let a quadratic form on 𝔼 d be represented by its coefficient vector in 𝔼 (1/2) d ( d +1) . Then, to the family of all positive semidefinite quadratic forms on 𝔼 d there corresponds a closed convex cone 𝒬 d in 𝔼 (1/2) d ( d +1) with apex at the origin. We describe its exposed faces and show that the families of its extreme and exposed faces coincide. Using these results, flag transitivity, neighborliness, singularity and duality properties of 𝒬 d are shown. The isometries of the cone 𝒬 d are characterized and we state a conjecture describing its linear symmetries. While the cone 𝒬 d is far from being polyhedral, the results obtained show that it shares many properties with highly symmetric, neighborly and self dual polyhedral convex cones.