The ıquiver algebras were introduced recently by the authors to provide a Hall algebra realization of universal ıquantum groups, which is generalization Bridgeland's construction for (Drinfeld doubles of) quantum groups; here an group and corresponding Drinfeld-Jimbo form symmetric pair. In this paper, Dynkin are shown arise as new examples singular Nakajima-Keller-Scherotzke categories. Then w...

Two algebraic structures A and B are called categorically equivalent if there is a functor from the variety generated by A to the variety generated by B, carrying A to B, that is an equivalence of the varieties when viewed as categories. We characterize those algebras categorically equivalent to A when A is an algebra whose set of term operations is as large as possible subject to constraints p...

In the category of Hom-Leibniz algebras we introduce the notion of Hom-corepresentation as adequate coefficients to construct the chain complex from which we compute the Leibniz homology of Hom-Leibniz algebras. We study universal central extensions of Hom-Leibniz algebras and generalize some classical results, nevertheless it is necessary to introduce new notions of α-central extension, univer...

We give a geometric classification of n-dimensional nilpotent, commutative nilpotent and anticommutative algebras. prove that the corresponding varieties are irreducible, find their dimensions describe explicit generic families algebras which define each these varieties. show some applications results in study length

In this paper we introduce the notion of a congruence rich variety of algebras, and investigate which locally finite subvarieties of such a variety are relatively finitely based. We apply the results obtained to investigate the finite axiomatizability of an interesting variety generated by a particular five-element directoid. Roughly speaking, congruence rich varieties are those in which all la...