We study the injective hulls of faithful characteristic zero finite dimensional irreducible representations of uniform nilpotent pro-p groups, seen as modules over their corresponding Iwasawa algebras. Using this we prove that the kernels of these representations are classically localisable.

It is shown that ${\mathcal{Z}}$-stable projectionless C*-algebras have the property that every element is a limit of products of two nilpotents. This is then used to classify the approximate unitary equivalence classes of positive elements in such C*-algebras using traces.

The purpose of this paper is to provide new constructions of Hom-associative algebras using Hom-analogues of certain operators called twistors and pseudotwistors, by deforming a given Hom-associative multiplication into a new Hom-associative multiplication. As examples, we introduce Hom-analogues of the twisted tensor product and smash product. Furthermore, we show that the construction by the twisting principle introduced by Yau and the twisting of associative algebras using pseudotwistors admit a common generalization.

Let R be an arbitrary ring. We introduce and study a generalization of injective and flat complexes of modules, called weak injective and weak flat complexes of modules respectively. We show that a complex C is weak injective (resp. weak flat) if and only if C is exact and all cycles of C are weak injective (resp. weak flat) as R-modules. In addition, we discuss the weak injective and weak flat dimensions of complexes of modules. Finally, we show that the category of weak injective (resp. weak flat) complexes is closed under pure subcomplexes, pure epimorphic images and direct limits. As a result, we then determine the existence of weak injective (resp. weak flat) covers and preenvelopes of complexes.

We show that any evolution family with a strong nonuniform exponential dichotomy can always be transformed by a topological equivalence to a canonical form that contracts and/or expands the same in all directions. We emphasize that strong nonuniform exponential dichotomies are ubiquitous in the context of ergodic theory. The main novelty of our work is that we are able to control the asymptotic behaviour of the topological conjugacies at the origin and at infinity.

We classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Plücker polarization) and study the fixed loci.

We consider an artin algebra A and its crossed product algebra A α#σ G, where G is a finite group with its order invertible in A. Then, we prove that A is a tilted algebra if and only if so is A α#σ G.

We provide a combinatorial characterization of LG(3,6)(${\mathbb{K}}$) using an axiom set which is the natural continuation of the Mazzocca–Melone set we used to characterize Severi varieties over arbitrary fields (Schillewaert and Van Maldeghem, Severi varieties over arbitrary fields, Preprint). This fits within a large project aiming at constructing and characterizing the varieties related to the Freudenthal–Tits magic square.

The notions of central endolength and semigeneric tameness are introduced, and their behaviour under base field extension for finite-dimensional algebras over perfect fields are analysed. For k a perfect field, K an algebraic closure and Λ a finite-dimensional k-algebra, here there is a proof that Λ is semigenerically tame if and only if Λ ⊗kK is tame.

The purpose of this paper is to present a brief discussion of both the normed space of operator p-summable sequences in a Banach space and the normed space of sequentially p-limited operators. The focus is on proving that the vector space of all operator p-summable sequences in a Banach space is a Banach space itself and that the class of sequentially p-limited operators is a Banach operator ideal with respect to a suitable ideal norm- and to discuss some other properties and multiplication results of related classes of operators. These results are shown to fit into a general discussion of operator [Y,p]-summable sequences and relevant operator ideals.

Let R be a ring and M a monoid with twisting f:M × M → U(R) and action ω: M→ Aut(R). We introduce and study the concepts of CM-Armendariz and CM-quasi-Armendariz rings to generalise various Armendariz and quasi-Armendariz properties of rings by working on the context of the crossed product R*M over R. The following results are proved: (1) If M is a u.p.-monoid, then any M-rigid ring R is CM-Armendariz; (2) if I is a reduced ideal of an M-compatible ring R with M a strictly totally ordered monoid, then R/I being CM-Armendariz implies that R is CM-Armendariz; (3) if M is a u.p.-monoid and R is a semiprime ring, then R is CM-quasi-Armendariz. These results generalise and unify many known results on this subject.

We give upper and lower bounds for the number of rational points on Prym varieties over finite fields. Moreover, we determine the exact maximum and minimum number of rational points on Prym varieties of dimension 2.

We show that Artin–Schelter regularity of a $\mathbb{Z}$-graded algebra can be examined by its associated $\mathbb{Z}$r-graded algebra. We prove that there is exactly one class of four-dimensional Artin–Schelter regular algebras with two generators of degree one in the Jordan type. This class is strongly noetherian, Auslander regular, and Cohen–Macaulay. Their automorphisms and point modules are described.

In this note we study properties of partially ample line bundles on simplicial projective toric varieties. We prove that the cone of q-ample line bundles is a union of rational polyhedral cones, and calculate these cones in examples. We prove a restriction theorem for big q-ample line bundles, and deduce that q-ampleness of the anticanonical bundle is not invariant under flips. Finally we prove a Kodaira-type vanishing theorem for q-ample line bundles.

We investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The second is the set of generators of the Cox ring viewed as a subring of the coordinate ring of a product of a torus and another surface quotient singularity. In addition, we obtain an explicit description of the minimal resolution as a divisor in a toric variety.

We consider the Jack–Laurent symmetric functions for special values of parameters p0=n+k−1m, where k is not rational and m and n are natural numbers. In general, the coefficients of such functions may have poles at these values of p0. The action of the corresponding algebra of quantum Calogero–Moser integrals $\mathcal{D}$(k, p0) on the space of Laurent symmetric functions defines the decomposition into generalised eigenspaces. We construct a basis in each generalised eigenspace as certain linear combinations of the Jack–Laurent symmetric functions, which are regular at p0=n+k−1m, and describe the action of $\mathcal{D}$(k, p0) in these eigenspaces.

A companion basis for a quiver Γ mutation equivalent to a simply-laced Dynkin quiver is a subset of the associated root system which is a $\mathbb{Z}$-basis for the integral root lattice with the property that the non-zero inner products of pairs of its elements correspond to the edges in the underlying graph of Γ. It is known in type A (and conjectured for all simply-laced Dynkin cases) that any companion basis can be used to compute the dimension vectors of the finitely generated indecomposable modules over the associated cluster-tilted algebra. Here, we present a procedure for explicitly constructing a companion basis for any quiver of mutation type A or D.

The trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotients of contracted semigroup rings), where we describe all linear traces in terms of central maps on graph inverse semigroups and, under mild assumptions, those Leavitt path algebras that admit faithful traces.

Suppose E is an elliptic curve over $\Bbb Q$, and p>3 is a split multiplicative prime for E. Let q ≠ p be an auxiliary prime, and fix an integer m coprime to pq. We prove the generalised Mazur–Tate–Teitelbaum conjecture for E at the prime p, over number fields $K\subset \Bbb Q\big(\mu_{{q^{\infty}}},\;\!^{q^{\infty}\!\!\!\!}\sqrt{m}\big)$ such that p remains inert in $K\cap\Bbb Q(\mu_{{q^{\infty}}})^+$. The proof makes use of an improved p-adic L-function, which can be associated to the Rankin convolution of two Hilbert modular forms of unequal parallel weight.

Leti ${\Bbb F}$n be the free group of rank n and let $\bigoplus C^{*}({\Bbb F}_{n})$ denote the direct sum of full group C*-algebras $C^{*}({\Bbb F}_{n})$ of ${\Bbb F}_{n} (1\leq n<\infty$). We introduce a new comultiplication Δϕ on $\bigoplus C^{*}({\Bbb F}_{n})$ such that $(\bigoplus C^{*}({\Bbb F}_{n}),\,\Delta_{\varphi})$ is a non-cocommutative C*-bialgebra. With respect to Δϕ, the tensor product π⊗ϕπ′ of any two representations π and π′ of free groups is defined. The operation ×ϕ is associative and non-commutative. We compute its tensor product formulas of several representations.