Some inequalities for the dimension of the Schur multiplier of a pair of (nilpotent) Lie algebras

The purpose of this paper is to obtain some inequalities for the dimension of the Schur multiplier of a pair of finite dimensional Lie algebras and their factor Lie algebras. Moreover, we present some inequalities for the Schur multiplier of a pair of finite dimensional nilpotent Lie algebras.     

On c-nilpotent multiplier and c-covers of a pair of Lie algebras

In this paper, we characterize the structure of the c-nilpotent multiplier of a pair of Lie algebras concerning the c-covering pairs and discuss some results on the existence of c-covers of a pair of Lie algebras. Moreover, we show that every nilpotent pair of Lie algebras of class at most k with non-trivial c-nilpotent multiplier does not admit any c-covering pair, if c>k. Also, we obtain the structure of c-covers of a c-capable pair of Lie algebras.     

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 Schur multiplier of nilpotent lie algebras

We give a bound on the dimension of the Schur multiplier of a finite dimensional nilpotent Lie algebra which sharpens the earlier known bounds.     

On dimension of the Schur multiplier of nilpotent Lie algebras

Let L be an n-dimensional non-abelian nilpotent Lie algebra and $ s(L) = \frac{1} {2}(n - 1)(n - 2) + 1 - \dim M(L) $ s(L) = \frac{1} {2}(n - 1)(n - 2) + 1 - \dim M(L) where M(L) is the Schur multiplier of L. In [Niroomand P., Russo F., A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra (in press)] it has been shown that s(L) ≥ 0 and the structure of all nilpotent Lie algebras has been determined when s(L) = 0. In the present paper, we will characterize all finite dimensional nilpotent Lie algebras with s(L) = 1; 2.     

Some Inequalities for the Order of the Schur Multiplier of a Pair of Groups

Let (H, N) be a pair of finite p-groups, in which N is a normal subgroup of H. In the present article, some upper bounds for the order of the commutator subgroup [N, H] are given. It is also constructed an upper bound for the order of the Schur multiplier of the pair of finite p-groups.     

Schur type functions associated with polynomial sequences of binomial type

We introduce a class of Schur type functions associated with polynomial sequences of binomial type. This can be regarded as a generalization of the ordinary Schur functions and the factorial Schur functions. This generalization satisfies some interesting expansion formulas, in which there is a curious duality. Moreover, this class includes examples which are useful to describe the eigenvalues of Capelli type central elements of the universal enveloping algebras of classical Lie algebras.      

On Components of the Partial Schur Multiplier

In this paper we study properties of the partial Schur multiplier. As an application a characterization of components of the partial Schur multiplier for some groups is given. We also describe the classical Schur multiplier of elementary abelian 2-groups by using partial factor sets.     

Semidirect products of weak multiplier Hopf algebras: Smash products and smash coproducts

In this paper, we will develop the smash product of weak multiplier Hopf algebras unifying the cases of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras. We will show that the smash product R#A has a regular weak multiplier Hopf algebra structure if R and A are regular weak multiplier Hopf algebras. We shall investigate integrals on R#A. We also consider the result in the ?-situation and new examples. Dually, we consider the smash coproduct of weak multiplier Hopf algebras under an appropriate form and integrals on the smash coproduct and we obtain results in the ?-situation.     

The Schur Multiplier of a Pair of Groups

In this article we develop the theory of a Schur multiplier for pairs of groups. The idea of such a multiplier is implicit in the work of J.-L. Loday (1978) and others on algebraicK -theory, and in the work of Eckmann et al. (1972) and others on group homology. In contrast to their work, we focus on the general group-theoretic properties of the multiplier. These properties are systematically derived from: 1) the functoriality of the multiplier; 2) an exact homology sequence; 3) and a transfer homomorphism.     

Integrable factors in compact Schur multipliers

It is shown that a Schur multiplier is compact if and only if it is the Schur product of two multipliers, one of which is a Hankel-Schur multiplier generated by an integrable function. This is illuminated by factoring exotic, singular measures and is brought into relation with Paley set-based multipliers.

     

Some properties of c-covers of a pair of Lie algebras

In [H. Safa and H. Arabyani, On c-nilpotent multiplier and c-covers of a pair of Lie algebras, Commun. Algebra 45(10) (2017), 4429–4434], we characterized the structure of the c-nilpotent multiplier of a pair of Lie algebras in terms of its c-covering pairs and discussed some results on the existence of c-covers of a pair of Lie algebras. In the present paper, it is shown under some conditions that a relative c-central extension of a pair of Lie algebras is a homomorphic image of a c-covering pair. Moreover, we prove that a c-cover of a pair of finite dimensional Lie algebras, under some assumptions, has a unique domain up to isomorphism and also that every perfect pair of Lie algebras admits at least one c-cover. Finally, we discuss a result concerning the isoclinism of c-covering pairs.     

Pairing and Quantum Double of Multiplier Hopf Algebras

We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.     

L-R smash products for multiplier Hopf algebras

The theory of L-R smash product is extended to multiplier Hopf algebras and a sufficient condition for L-R smash product to be regular multiplier Hopf algebras is given. In particular, the result of the paper implies Delvaux＇s main theorem in the case of smash products.     

SVEP for multipliers on a faithful commutative Banach algebra

We give some sufficient conditions that each multiplier on a faithful commutative Banach algebra has SVEP. On the other hand, we show that there exist a faithful commutative Banach algebra and a multiplier on it without SVEP. Such examples of multipliers can actually be found within the class of multiplication operators on unital commutative Banach algebras. This answers in negative a question that is stated as Open problem 6.2.1 by Laursen and Neumann, 2000.

     

On corepresentations of multiplier Hopf algebras

We present a general construction producing a unitary corepresentation of a multiplier Hopf algebra in itself and study RR-corepresentations of twisted tensor coproduct multiplier Hopf algebra. Then we investigate some properties of functors, integrals and morphisms related to the categories of the crossed modules and covariant modules over multiplier Hopf algebras. By doing so, we can apply our theory to the case of group-cograded multiplier Hopf algebras, in particular, the case of Hopf group-coalgebras.     

Simplicity of quadratic Lie conformal algebras

In this paper, simplicity of quadratic Lie conformal algebras is investigated. From the view point of the corresponding Gel’fand–Dorfman bialgebras, some su?cient conditions and necessary conditions to ensure simplicity of quadratic Lie conformal algebras are presented. By these observations, we present several new classes of infinite simple Lie conformal algebras. These results will be useful for classification purposes.     

Solvable complete lie algebras. I ∗

In this paper, we will discuss the properties of solvable complete Lie algebra, describe the structures of the root spaces of solvable complete Lie algebra, prove that solvable Lie algebras of maximal rank are com-plete, and construct some new complete Lie algebras from Kac-Moody algebras.