Generalized Jordan N-Derivations of Unital Algebras with Idempotents

Let A be a unital algebra with idempotent e over a 2-torsionfree unital commutative ring ℛ and S : A ⟶ A be an arbitrary generalized Jordan n-derivation associated with a Jordan n-derivation J . We show that, under mild conditions, every generalized Jordan n-derivation S : A ⟶ A is of the form S x = λ x + J x in the current work. As an application, we give a description of generalized Jordan derivations for the condition n = 2 on classical examples of unital algebras with idempotents: triangular algebras, matrix algebras, nest algebras, and algebras of all bounded linear operators, which generalize some known results.

Characterizations of additive -Lie derivations on unital algebras

UDC 512.5 Let be a commutative ring with unity and be a unital algebra over (or field ).An -linear map is called a Lie derivation on if holds for all For scalar an additive map is called an additive -Lie derivation on if where holds for all In the present paper, under certain assumptions on it is shown that every Lie derivation (resp., additive -Lie derivation) on is of standard form, i.e., where is an additive derivation on and is a mapping vanishing at with in Moreover, we also characterize the additive -Lie derivation for by its action at zero product in a unital algebra over

Cyclic-Homology Chern–Weil Theory for Families of Principal Coactions

Viewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern–Weil homomorphism. To realize the thus constructed Chern–Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of Loday and Wodzicki. We work with families of principal coactions, and instantiate our noncommutative Chern–Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect in the spirit of Feng and Tsygan for the quantum-deformation family of the standard quantum Hopf fibrations.

Duplication methods for embeddings of real division algebras

We introduce two groups of duplication processes that extend the well known Cayley–Dickson process. The first one allows to embed every [Formula: see text]-dimensional (4D) real unital algebra [Formula: see text] into an 8D real unital algebra denoted by [Formula: see text] We also find the conditions on [Formula: see text] under which [Formula: see text] is a division algebra. This covers the most classes of known [Formula: see text]D real division algebras. The second process allows us to embed particular classes of [Formula: see text]D RDAs into [Formula: see text]D RDAs. Besides, both duplication processes give an infinite family of non-isomorphic [Formula: see text]D real division algebras whose derivation algebras contain [Formula: see text]

The isomorphism problem for group algebras: A criterion

Let R be a finite unital commutative ring. We introduce a new class of finite groups, which we call hereditary groups over R. Our main result states that if G is a hereditary group over R, then a unital algebra isomorphism between group algebras {RG\cong RH} implies a group isomorphism {G\cong H} for every finite group H. As application, we study the modular isomorphism problem, which is the isomorphism problem for finite p-groups over {R=\mathbb{F}_{p}}, where {\mathbb{F}_{p}} is the field of p elements. We prove that a finite p-group G is a hereditary group over {\mathbb{F}_{p}} provided G is abelian, G is of class two and exponent p, or G is of class two and exponent four. These yield new proofs for the theorems by Deskins and Passi–Sehgal.

f-Biderivations and Jordan biderivations of unital algebras with idempotents

The notion of [Formula: see text]-derivations was introduced by Beidar and Fong to unify several kinds of linear maps including derivations, Lie derivations and Jordan derivations. In this paper, we introduce the notion of [Formula: see text]-biderivations as a natural “biderivation” counterpart of the notion of “[Formula: see text]-derivations”. We first show, under some conditions, that any [Formula: see text]-biderivation is a Jordan biderivation. Then, we turn to study [Formula: see text]-biderivations of a unital algebra with an idempotent. Our second main result shows, under some conditions, that every Jordan biderivation can be written as a sum of a biderivation, an antibiderivation and an extremal biderivation. As a consequence, we show that every Jordan biderivation on a triangular algebra is a biderivation.

Linear maps on *-algebras acting on orthogonal elements like derivations or anti-derivations

Let U be a unital *-algebra and ? : U ? U be a linear map behaving like a derivation or an anti-derivation at the following orthogonality conditions on elements of U: xy = 0, xy* = 0, xy = yx = 0 and xy* = y*x = 0. We characterize the map ? when U is a zero product determined algebra. Special characterizations are obtained when our results are applied to properly infinite W*-algebras and unital simple C*-algebras with a non-trivial idempotent.

Characterizations of Lie n-derivations of unital algebras with nontrivial idempotents

Let A be a unital algebra with a nontrivial idempotent e, and f = 1-e. Suppose that A satisfies that exe ? eAf = {0} = fAe ? exe implies exe = 0, and that eAf ? fxf = {0} = fxf ?fAe implies fxf = 0 for each x in A. For a Lie n-derivation ? on A, we obtain the necessary and sufficient conditions for ? to be standard, i.e., ? = d + ?, where d is a derivation on A, and is a linear mapping from A into the centre Z(A) vanishing on all (n-1)-th commutators of A. Furthermore, we also consider the sufficient conditions under which each Lie n-derivation on A can be standard.

Nontriviality results for the characteristic algebra of a DGA

Assume that we are given a semifree noncommutative differential graded algebra (DGA for short) whose differential respects an action filtration. We show that the canonical unital algebra map from the homology of the DGA to its characteristic algebra, i.e. the quotient of the underlying algebra by the two-sided ideal generated by the boundaries, is a monomorphism. The main tool that we use is the weak division algorithm in free noncommutative algebras due to P. Cohn.