Lie Triple Derivations on von Neumann Algebras

Let A be a von Neumann algebra with no central abelian projections. It is proved that if an additive map δ :A → A satisfies δ([[a, b], c]) = [[δ(a), b], c] + [[a, δ(b)], c] +[[a, b], δ(c)] for any a, b, c∈ A with ab = 0(resp. ab = P, where P is a fixed nontrivial projection in A), then there exist an additive derivation d from A into itself and an additive map f :A → ZA vanishing at every second commutator [[a, b], c] with ab = 0(resp.ab = P) such that δ(a) = d(a) + f(a) for any a∈ A.     

Local Superderivatio ns o n Lie Superalgebra $$\mathfrak{}$$ ( n)

Let \(\mathfrak{q}\)(n) be a simple strange Lie superalgebra over the complex field ?. In a paper by A.Ayupov, K.Kudaybergenov (2016), the authors studied the local derivations on semi-simple Lie algebras over ? and showed the difference between the properties of local derivations on semi-simple and nilpotent Lie algebras. We know that Lie superalgebras are a generalization of Lie algebras and the properties of some Lie superalgebras are similar to those of semi-simple Lie algebras, but \(\mathfrak{p}\)(n) is an exception. In this paper, we introduce the definition of the local superderivation on \(\mathfrak{q}\)(n), give the structures and properties of the local superderivations of \(\mathfrak{q}\)(n), and prove that every local superderivation on \(\mathfrak{q}\)(n), n > 3, is a superderivation.     

Generation of a finite group with Hall maximal subgroups by a pair of conjugate elements

For a finite group G, the set of all prime divisors of |G| is denoted by π(G). P. Shumyatsky introduced the following conjecture, which was included in the “Kourovka Notebook” as Question 17.125: a finite group G always contains a pair of conjugate elements a and b such that π(G) = π(〈a, b〉). Denote by \(\mathfrak{Y}\) the class of all finite groups G such that π(H) ≠ π(G) for every maximal subgroup H in G. Shumyatsky’s conjecture is equivalent to the following conjecture: every group from \(\mathfrak{Y}\) is generated by two conjugate elements. Let \(\mathfrak{V}\) be the class of all finite groups in which every maximal subgroup is a Hall subgroup. It is clear that \(\mathfrak{V} \subseteq \mathfrak{Y}\). We prove that every group from \(\mathfrak{V}\) is generated by two conjugate elements. Thus, Shumyatsky’s conjecture is partially supported. In addition, we study some properties of a smallest order counterexample to Shumyatsky’s conjecture.     

Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order

Let Q₀ be the classical generalized quadrangle of order q = 2ⁿ(n≥2) arising from a non-degenerate quadratic form in a 5-dimensional vector space defined over a finite field of order q. We consider the rank two geometry \(\mathcal {X}\) having as points all the elliptic ovoids of Q₀ and as lines the maximal pencils of elliptic ovoids of Q₀ pairwise tangent at the same point. We first prove that \(\mathcal {X}\) is isomorphic to a 2-fold quotient of the affine generalized quadrangle Q?Q₀, where Q is the classical (q,q²)-generalized quadrangle admitting Q₀ as a hyperplane. Further, we classify the cliques in the collinearity graph Γ of \(\mathcal {X}\). We prove that any maximal clique in Γ is either a line of \(\mathcal {X}\) or it consists of 6 or 4 points of \(\mathcal {X}\) not contained in any line of \(\mathcal {X}\), accordingly as n is odd or even. We count the number of cliques of each type and show that those cliques which are not contained in lines of \(\mathcal {X}\) arise as subgeometries of Q defined over \(\mathbb {F}_{2}\).     

Enveloping Algebras of the Nilpotent Malcev Algebra of Dimension Five

Pérez-Izquierdo and Shestakov recently extended the PBW theorem to Malcev algebras. It follows from their construction that for any Malcev algebra M over a field of characteristic ≠ 2, 3 there is a representation of the universal nonassociative enveloping algebra U(M) by linear operators on the polynomial algebra P(M). For the nilpotent non-Lie Malcev algebra \(\mathbb{M}\) of dimension 5, we use this representation to determine explicit structure constants for \(U(\mathbb{M})\); from this it follows that \(U(\mathbb{M})\) is not power-associative. We obtain a finite set of generators for the alternator ideal \(I(\mathbb{M}) \subset U(\mathbb{M})\) and derive structure constants for the universal alternative enveloping algebra \(A(\mathbb{M}) = U(\mathbb{M})/I(\mathbb{M})\), a new infinite dimensional alternative algebra. We verify that the map \(\iota\colon \mathbb{M} \to A(\mathbb{M})\) is injective, and so \(\mathbb{M}\) is special.     

Simple Functors of Admissible Linear Categories

Generalizing an idea used by Bouc, Thévenaz, Webb and others, we introduce the notion of an admissible R-linear category for a commutative unital ring R. Given an R-linear category \(\mathcal {L}\), we define an \(\mathcal {L}\)-functor to be a functor from \(\mathcal {L}\) to the category of R-modules. In the case where \(\mathcal {L}\) is admissible, we establish a bijective correspondence between the isomorphism classes of simple functors and the equivalence classes of pairs (G, V) where G is an object and V is a module of a certain quotient of the endomorphism algebra of G. Here, two pairs (F, U) and (G, V) are equivalent provided there exists an isomorphism F ← G effecting transport to U from V. We apply this to the category of finite abelian p-groups and to a class of subcategories of the biset category.     

Sets with even partition functions and 2-adic integers

For P ? \(\mathbb{F}_2 \)[z] with _P(0) = 1 and deg(_P) ≥ 1, let \(\mathcal{A}\) = \(\mathcal{A}\)(P) (cf. [4], [5], [13]) be the unique subset of ? such that Σ _n≥0 p(\(\mathcal{A}\), n)z ⁿ ≡ P(z) (mod 2), where p(\(\mathcal{A}\), n) is the number of partitions of n with parts in \(\mathcal{A}\). Let p be an odd prime and P ? \(\mathbb{F}_2 \)[z] be some irreducible polynomial of order p, i.e., p is the smallest positive integer such that P(z) divides 1 + z ^p in \(\mathbb{F}_2 \)[z]. In this paper, we prove that if m is an odd positive integer, the elements of \(\mathcal{A}\) = \(\mathcal{A}\)(P) of the form 2 ^k m are determined by the 2-adic expansion of some root of a polynomial with integer coefficients. This extends a result of F. Ben Saïd and J.-L. Nicolas [6] to all primes p.     

An asymptotic formula for the primitive of Hardy’s function

Let Z(t) be the classical Hardy function in the theory of Riemann’s zeta-function. An asymptotic formula with an error term O(T ^1/6log?T) is given for the integral of Z(t) over the interval [0,T], with special attention paid to the critical cases when the fractional part of \(\sqrt{T/2\pi }\) is close to \(\frac{1}{4}\) or \(\frac{3}{4}\).     

Commutative semigroups with cancellation law: a representation theorem

Any commutative, cancellative semigroup S with 0 equipped with a uniformity can be embedded in a topological group \(\widetilde{S}\). We introduce the notion of semigroup symmetry T which enables us to turn \(\widetilde{S}\) into an involutive group. In Theorem 2.8 we prove that if S is 2-torsion-free and T is 2-divisible then the decomposition of elements of \(\widetilde{S}\) into a sum of elements of the symmetric subgroup \(\widetilde{S}_{s}\) and the asymmetric subgroup \(\widetilde{S}_{a}\) is polar. In Theorem 3.7 we give conditions under which a topological group \(\widetilde{S}\) is a topological direct sum of its symmetric subgroup \(\widetilde{S}_{s}\) and its asymmetric subgroup \(\widetilde{S}_{a}\). Theorem 2.8 and Theorem 3.7 are designed to be useful tools in studying Minkowski–Rådström–Hörmander spaces (and related topological groups \(\widetilde{S}\)), which are natural extensions of semigroups of bounded closed convex subsets of real Hausdorff topological vector spaces.