Cosemisimple Coextensions and Homological Dimension of Smash Coproducts

Let H be a finite dimensional cosemisimple Hopf algebra, C a left H-comodule coalgebra and let C = C/C（H^＊）^＋ be the quotient coalgebra and the smash coproduct of C and H. It is shown that if C/C is a eosemisimple coextension and C is an injective right C-comodule, then gl. dim（the smash coproduct of C and H） = gl. dim（C） = gl. dim（C）, where gl. dim（C） denotes the global dimension of coalgebra C.     

On Constantive Simple and Order-Primal Algebras

A finite algebra is said to be order-primal if its clone of all term operations is the set of all operations defined on A which preserve a given partial order ≤ on A. In this paper we study algebraic properties of order-primal algebras for connected ordered sets (A; ≤). Such order-primal algebras are constantive, simple and have no non-identical automorphisms. We show that in this case cannot have only unary fundamental operations or only one at least binary fundamental operation. We prove several properties of the varieties and the quasi-varieties generated by constantive and simple algebras and apply these properties to order-primal algebras. Further, we use the properties of order-primal algebras to formulate new primality criteria for finite algebras.* Research supported by the Hungarian research grant No. TO34137 and by the János Bolyai grant.** Research supported by the Thailand Research Fund.     

Twisted hopf comodule algebras

For k a commutative ring, H a k‐bialgebra and A a right H‐comodule k‐algebra, we define a new multiplication on the H‐comodule A to obtain a twisted algebra” A^T, T sumHom(H,End (A)). If ^T is convolution invertible, the categories of relative right Hopf modules over A and A^Tare isomorphic. Similarly a convolution invertible left twisting gives an isomorphism of the categories of relative left Hopf modules. We show that crossed products are invertible twistings of the tensor product, and obtain, as a corollary, a duality theorem for crossed products     

Smash Products of H-Simple Module Algebras

Let H be a finite-dimensional Hopf algebra and A a finite-dimensional H-simple left H-module algebra. We show that the smash product A#H is isomorphic to End _A′(V ? H*), where V ≠ 0 is a finite-dimensional left A-module and (A′, V′) the stabilizer of (A, V). As an application it is proved that A#H is isomorphic to a full matrix algebra over A′ when H is semisimple and dim V|dim A.     

Leonard Pairs and Leonard Triples of q-Racah Type from the Quantum Algebra U q (sl 2)

In this article, we describe the construction of Leonard pairs and Leonard triples that have q-Racah type from U _q (sl ₂)-modules by using equitable generators of U _q (sl ₂). Our result solves an open problem proposed by Terwilliger.     

ν-Stable τ-Tilting Modules

Inspired by τ-tilting theory [3 Adachi , T. , Iyama , O. , Reiten , I. ( 2014 ). τ-tilting theory . Compos. Math. 150 ( 3 ): 415 – 452 .[Crossref], [Web of Science ®] , [Google Scholar]], we introduce the notion of ν-stable support τ-tilting modules. For any finite dimensional selfinjective algebra Λ, we give bijections between two-term tilting complexes in K ^b (proj Λ), ν-stable support τ-tilting Λ-modules, and ν-stable functorially finite torsion classes in modΛ. Moreover, these objects correspond bijectively to selfinjective cluster tilting objects in 𝒞 if Λ is a 2-CY tilted algebra associated with a Hom-finite 2-CY triangulated category 𝒞. We also study some properties of support τ-tilting modules over 2-CY tilted algebras, and we give a necessary condition such that algebras are 2-CY tilted in terms of support τ-tilting modules.     

Precovering and Preenveloping Ideals

In [8 Salce , L. ( 1979 ). Cotorsion theories for abelian groups . Symposia Mathematica 23 : 11 – 23 . [Google Scholar]] L. Salce introduced the notion of a cotorsion pair (?, 𝒞) in the category of abelian groups. But his definitions and basic results carry over to more general abelian categories and have proven to be useful in a variety of settings. A significant result of cotorsion theory proven by Eklof and Trlifaj is that if a pair (?, 𝒞) of classes of R-modules is cogenerated by a set, then it is complete [1 Eklof , P. C. , Trlifaj , J. ( 2001 ). How to make Ext vanish . Bull. London Math. Soc. 33 : 31 – 41 .[Crossref], [Web of Science ®] , [Google Scholar]]. Recently Fu, Herzog, Guil, and Torrecillas developed the ideal approximation theory [6 Herzog , I. ( 2014 ). Phantom morphisms and Salce's Lemma . Contemp. Math. 607 : 57 – 83 .[Crossref] , [Google Scholar]], [4 Fu X. H. , Guil Asensio , P. A. , Herzog , I. , Torrecillas , B. ( 2013 ). Ideal Approximation Theory . Adv. in Math. 244 : 750 – 790 .[Crossref], [Web of Science ®] , [Google Scholar]]. In this article we look at a result motivated by the Eklof and Trlifaj argument for an ideal ? when it is generated by a set of homomorphisms.     

Monoid Valuations and Value Ordered Supervaluations

We complement two papers on supertropical valuation theory ([11 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 . Preprint at arXiv:1003.1101 .[Crossref], [Web of Science ®] , [Google Scholar]], [12 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2013 ). Dominance and transmissions in supertropical valuation theory . Commun. Algebra 41 ( 7 ): 2736 – 2782 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]) by providing natural examples of m-valuations (= monoid valuations), and afterwards of supervaluations and transmissions between them. These supervaluations have values in totally ordered supertropical semirings, and the transmissions discussed respect the orderings. We develop the basics of the theory of such semirings and transmissions.