Schur Multiplier Research Articles

Multiplicative Lie algebras and Schur multiplier

In this paper, we attempt to study the structure of multiplicative Lie algebras, the theory of extensions, the second cohomology groups of multiplicative Lie algebras, and in turn the Schur multipliers. The Schur–Hopf formula is established for multiplicative Lie algebras. We also introduce the group of nontrivial relations satisfied by the Lie product in a multiplicative Lie algebra, and study it as a functor arising from the presentations of multiplicative Lie algebras. Some applications in K-theory are also discussed.

On the dimension of the c-nilpotent Schur Lie-multiplier of Leibniz algebras

The Schur -multiplier of Leibniz algebras is the Schur multiplier of Leibniz algebras defined relative to the Liezation functor. In this paper, we provide upperbounds for the dimensions of the c-nilpotent Schur -multiplier and c- stem cover of a finite-dimensional non-Lie Leibniz algebra. In the case of a -nilpotent non-Lie Leibniz algebra, we determine this upperbound in terms of the dimensions of its -center and -commutator. We also investigate the case of c-nilpotent Leibniz algebras of maximal class.

Central extensions of free periodic groups

It is proved that any countable abelian group can be embedded as a centre into a -generated group such that the quotient group is isomorphic to the free Burnside group of rank and of odd period . The proof is based on a certain modification of the method that was used by Adian in his monograph in 1975 for a positive solution of Kontorovich’s famous problem from the Kourovka Notebook on the existence of a finitely generated noncommutative analogue of the additive group of rational numbers with any number of generators (in contrast to the abelian case). More precisely, he proved that the desired analogues in which the intersection of any two non-trivial subgroups is infinite, can be constructed as a central extension of the free Burnside group , where , and is an odd number, using the infinite cyclic group as its centre. The paper also discusses other applications of the proposed generalization of Adian’s technique. In particular, the free groups of the variety defined by the identity and the Schur multipliers of the free Burnside groups for any odd are described. Bibliography: 14 titles.

On the dimension of the Schur multiplier of n-Lie algebras

ABSTRACT Let L be an n-Lie algebra over a field . In this paper, we introduce the exterior products of two n-Lie algebras and prove some properties of them. Also, we state some bounds for the dimension of the Schur multiplier of a given n-Lie algebra L as well as the dimension of , the non-abelian exterior (wedge) product of L.

Schur multipliers and mixed unitary maps

We consider the tensor product of the completely depolarising channel on d × d matrices with the map of Schur multiplication by a k × k correlation matrix and characterise, via matrix theory methods, when such a map is a mixed (random) unitary channel. When d = 1, this recovers a result of O’Meara and Pereira and for larger d, is equivalent to a result of Haagerup and Musat that was originally obtained via the theory of factorisation through von Neumann algebras. We obtain a bound on the distance between a given correlation matrix for which this tensor product is nearly mixed unitary and a correlation matrix for which such a map is exactly mixed unitary. This bound allows us to give an elementary proof of another result of Haagerup and Musat about the closure of such correlation matrices without appealing to the theory of von Neumann algebras.

Characterization of capable nilpotent n-Lie algebras of class two by their Schur multipliers

In this paper, we classify the generalized Heisenberg nilpotent n-Lie algebras of rank two with dimension at most $$2n+3$$ . Using them, we classify capable nilpotent n-Lie algebras with two-dimensional derived subalgebra. We also obtain the dimension of the Schur multiplier of nilpotent n-Lie algebras of class two with two-dimensional derived subalgebra.

Some Results on the c-Nilpotent Multiplier of a Pair of Lie Algebras

The Schur multiplier of a Lie algebra L, was introduced by Batten et al. (Commun Algebra 24:4319–4330, 1996). Salemkar and colleagues generalized the concept of the Schur multiplier to the c-nilpotent multiplier. Recently, the author introduced some exact sequences and gived some upper bounds for the dimension of the c-nilpotent multiplier of a pair of Lie algebras. In the present paper, we will extend these results. Moreover, we give some isomorphisms for the c-nilpotent multiplier of a pair of Lie algebras.

Characterization of finite $$\varvec{p}$$ p -groups by their Schur multiplier

Let G be a finite p-group of order $$p^n$$ and M(G) be its Schur multiplier. It is a well known result by Green that $$|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}$$ for some $$t(G) \ge 0$$ . In this article, we classify non-abelian p-groups G of order $$p^n$$ for $$t(G)=\log _p(|G|)+1$$ .

Herz–Schur multipliers of dynamical systems

We extend the notion of a Herz–Schur multiplier to the setting of non-commutative dynamical systems: given a C*-algebra A, a locally compact group G, and an action α of G on A, we define transformations on the reduced crossed product of A by α which, in the case A=C, reduce to the classical Herz–Schur multipliers. We introduce Schur A-multipliers, establish a characterisation that generalises the classical descriptions of Schur multipliers, and present a transference theorem in the new setting, identifying isometrically the Herz–Schur multipliers of the dynamical system (A,G,α) with the invariant part of the Schur A-multipliers. We discuss special classes of Herz–Schur multipliers, in particular, those associated to a locally compact abelian group G and its canonical action on the C*-algebra C⁎(Γ) of the dual group Γ.

On classification of groups having Schur multiplier of maximum order II

On classification of groups having Schur multiplier of maximum order II

Recent developments around partial actions

We give an overview of publications on partial actions and related concepts, paying main attention to some recent developments on diverse aspects of the theory, such as partial actions of semigroups, of Hopf algebras and groupoids, the globalization problem for partial actions, Morita theory of partial actions, twisted partial actions, partial projective representations and the Schur multiplier, cohomology theories related to partial actions, Galois theoretic results, ring theoretic properties and ideals of partial crossed products. Among the applications we consider in more detail the case of the Carlsen-Matsumoto $$C^*$$ -algebra related to an arbitrary subshift, but also mention many others. The total number of publications directly related to partial actions and partial representations is more than 130, so that it is impossible even to describe briefly the content of all of them within the constraints of the present survey. Thus, the majority of them are only cited with respects to specific topics, trying to give an idea about the involved matter. In order to complete the picture, we refer the reader to a recent book by Ruy Exel, to our previous surveys, as well as to those by other authors.

The structure, capability and the Schur multiplier of generalized Heisenberg Lie algebras

From Berkovich and Janko [3, Problem 1729] asked to obtain the Schur multiplier and the representation of a group G, when G is a special p-group minimally generated by d elements and |G′|=p12d(d−1). Here, we intend to give an answer to this question similarly for nilpotent Lie algebras. Furthermore, we give some results about the tensor square and the Schur multiplier of some nilpotent Lie algebras of class two.

The Schur multiplier of central product of groups

Let G be a central product of two groups H and K. We study second cohomology group of G, having coefficients in a divisible abelian group D with trivial G-action, in terms of the second cohomology groups of certain quotients of H and K. In particular, for D=C⁎, some of our results provide a refinement of results from Wiegold (1971) [10] and Eckmann et al. (1973) [2].

On dimension of Schur multiplier of nilpotent Lie algebras II

Let [Formula: see text] be a non-abelian nilpotent Lie algebra of dimension [Formula: see text] and put [Formula: see text], where [Formula: see text] denotes the Schur multiplier of [Formula: see text]. Niroomand and Russo in 2011 proved that [Formula: see text] and that [Formula: see text] if and only if [Formula: see text], in which [Formula: see text] is the Heisenberg algebra of dimension [Formula: see text] and [Formula: see text] is the abelian [Formula: see text]-dimensional Lie algebra. In the same year, they also classified all nilpotent Lie algebras [Formula: see text] satisfying [Formula: see text] or [Formula: see text]. In this paper, we obtain all nilpotent Lie algebras [Formula: see text] provided that [Formula: see text].

Finite p-groups having Schur multiplier of maximum order

Let G be a non-abelian p-group of order pn and M(G) denote the Schur multiplier of G. Niroomand proved that |M(G)|≤p12(n+k−2)(n−k−1)+1 for non-abelian p-groups G of order pn with derived subgroup of order pk. Recently Rai classified p-groups G of nilpotency class 2 for which |M(G)| attains this bound. In this article we show that there is no finite p-group G of nilpotency class c≥3 for p≠3 such that |M(G)| attains this bound. Hence |M(G)|≤p12(n+k−2)(n−k−1) for p-groups G of class c≥3 where p≠3. We also construct a p-group G for p=3 such that |M(G)| attains the Niroomand's bound.

Multilinear Schur multipliers and Schatten properties of operator Taylor remainders

We establish sharp conditions on scalar functions and perturbations that guarantee Schatten summability of nth order operator Taylor remainders. In the special case of dimension one, our estimates of these remainders deliver well known classical estimates of scalar Taylor remainders. We prove that if a scalar function f is in the set Cn and a perturbation is in the pth Schatten class Sp, p>n, then the respective nth order operator Taylor remainder is an element of Sp/n and has an estimate like the one in [16]. We construct examples of f∈Cn and perturbations in Sn such that the nth order Taylor remainder of the respective operator function is not in S1. Our construction relies, in particular, on novel dimension dependent estimates for Schatten norms of multilinear Schur multipliers from below that are of interest in their own right. Our results apply to both self-adjoint and unitary operators.

Stackings and the W-cycles Conjecture

AbstractWe prove Wise’s W-cycles conjecture. Consider a compact graph Γ' immersing into another graph Γ. For any immersed cycle Λ: S1 ⟶ Γ, we consider the map Λ' from the circular components 𝕊 of the pullback to Γ'. Unless Λ' is reducible, the degree of the covering map 𝕊 ⟶ S1 is bounded above by minus the Euler characteristic of Γ'. As a corollary, any finitely generated subgroup of a one-relator group has a finitely generated Schur multiplier.

Positive Herz–Schur multipliers and approximation properties of crossed products

For aC*-algebraAand a setXwe give a Stinespring-type characterisation of the completely positive SchurA-multipliers on κ(ℓ2(X)) ⊗A. We then relate them to completely positive Herz–Schur multipliers onC*-algebraic crossed products of the formA⋊α,rG, withGa discrete group, whose various versions were considered earlier by Anantharaman-Delaroche, Bédos and Conti, and Dong and Ruan. The latter maps are shown to implement approximation properties, such as nuclearity or the Haagerup property, forA⋊α,rG.

A note on the order of the Schur multiplier of p-groups

Let [Formula: see text] be a finite [Formula: see text]-group of order [Formula: see text] with [Formula: see text], and let [Formula: see text] denote its Schur multiplier. A classical result of Green states that [Formula: see text]. In 2009, Niroomand, improving Green’s and other bounds on [Formula: see text] for a non-abelian [Formula: see text]-group [Formula: see text], proved that [Formula: see text]. In this paper, we prove that a bound, obtained earlier by Ellis and Wiegold, is stronger than that of Niroomand. We derive from the bound of Ellis and Wiegold that [Formula: see text] for a non-abelian [Formula: see text]-group [Formula: see text]. We obtain an improvement to an old bound given by Vermani. Finally we prove, for a [Formula: see text]-group of coclass [Formula: see text], that [Formula: see text]. This improves a bound by Moravec.

Positive extensions of Schur multipliers

We introduce partially defined Schur multipliers and obtain necessary and sufficient conditions for the existence of extensions to fully defined positive Schur multipliers, in terms of operator systems canonically associated with their domains. We use these results to study the problem of extending a positive definite function defined on a symmetric subset of a locally compact group to a positive definite function defined on the whole group.