Search

This paper is devoted to studying the structures of the cell modules of the complexified Green algebra R(D(H4)), where D(H4) is the Drinfel'd quantum double of Sweedler's 4-dimensional Hopf algebra H4. We show that R(D(H4)) has one infinite dimensional cell module, one 4-dimensional cell module generated by all finite dimensional indecomposable projective modules of D(H4) and infinitely many 2-dimensional cell modules. More precisely, we obtain the decompositions of all finite dimensional cell modules into the direct sum of indecomposable submodules, and show that the infinite dimensional cell module can be written as the direct sum of two infinite dimensional indecomposable submodules. [ABSTRACT FROM AUTHOR]

In this article, we investigate the representation rings (or Green rings) of the Drinfeld doubles of the Taft algebras. It is shown that these Green rings are commutative rings generated by infinitely many elements subject to certain relations. The generators together with the subjecting relations are given. The stable Green rings of of the Drinfeld doubles of the Taft algebras are also described. [ABSTRACT FROM AUTHOR]

In this paper, we continue our study of the tensor product structure of category of weight modules over a Hopf–Ore extensions of a group algebra kG, where k is an algebraically closed field of characteristic zero. We first describe the tensor product decomposition rules for all indecomposable weight modules under the assumption that the orders of χ and are different. Then we describe the Green ring of the tensor category. It is shown that is isomorphic to the polynomial algebra over the group ring in one variable when , and that is isomorphic to the quotient ring of the polynomial algebra over the group ring in two variables modulo a principle ideal when. When , is isomorphic to the quotient ring of a skew group ring modulo some ideal, where is a polynomial algebra over in infinitely many variables. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the tensor structure of the category of finite- dimensional weight modules over the Hopf-Ore extensions kG(χ−1,a,0) of group algebras kG. The tensor product decomposition rules for all indecomposable weight modules are explicitly given under the assumptions that k is an algebraically closed field of characteristic zero, and the orders of χ and χ(a) are the same. [ABSTRACT FROM AUTHOR]

In this paper, we study the Green ring (or the representation ring) of Drinfeld quantum double D( H) of Sweedler's four-dimensional Hopf algebra H. We first give the decompositions of the tensor products of finite dimensional indecomposable modules into the direct sum of indecomposable modules over D( H). Then we describe the structure of the Green ring r( D( H)) of D( H) and show that r( D( H)) is generated, as a ring, by infinitely many elements subject to a family of relations. [ABSTRACT FROM AUTHOR]

Quiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one. [ABSTRACT FROM AUTHOR]

In this article, we study the representations of the new finite-dimensional pointed Hopf algebras in positive characteristic given in [6]. We find that these Hopf algebras are symmetric algebras. We determine the simple modules and their projective covers over these Hopf algebras. We show that these Hopf algebras are of wild representation type. [ABSTRACT FROM PUBLISHER]

Let H be a Hopf algebra over a field k, and (H, A, C) be a Doi-Hopf datum. The categories of comodules over C and Doi-Hopf modules are then the Grothendieck categories with enough injectives. We study the derived functors of the associated Hom functors, and of the coinvariants functor, and discuss spectral sequences that connect them. We also discuss when the coinvariants functor preserves injectives. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field. The algebraic and geometric classification of finite dimensional algebras over k with ch(k) ≠ 2 was initiated by Gabriel in [6], where a complete list of nonisomorphic 4-dimensional k-algebras was given and the number of irreducible components of the variety Alg4 was discovered to be 5. The classification of 5-dimensional k-algebras was done by Mazzola in [10]. The number of irreducible components of the variety Alg5 is 10. With the dimension n increasing, the algebraic and geometric classification of n-dimensional k-algebras becomes more and more difficult. However, a lower and a upper bound for the number of irreducible components of Algn can be given (see [11]). In this article, we classify 4-dimensional 2-graded (or super) algebras with a nontrivial grading over any field k with ch(k) ≠ 2, up to isomorphism. A complete list of nonisomorphic 2-graded algebras over an algebraically closed field k with ch(k) ≠ 2 is obtained. The main result in this article is twofold. On one hand, it completes the classification of 4-dimensional Yetter-Drinfeld module algebras over Sweedler's 4-dimensional Hopf algebra H4 initiated in [3]. On the other hand, it establishes the basis for the geometric classification of 4-dimensional super algebras. In approaching the geometric classification of n-dimensional 2-graded algebras, we define a new variety, Salgn, which possesses many different properties to Alg4. [ABSTRACT FROM AUTHOR]

Let k be a field and An(ω) be the Taft's n2-dimensional Hopf algebras. When n is odd, the Drinfeld quantum double D(An(ω)) of An(ω) is a Ribbon Hopf algebra. In the previous articles, we constructed an n4-dimensional Hopf algebra Hn(p, q) which is isomorphic to D(An(ω)) if p ≠ 0 and q = ω-1, and studied the finite dimensional representations of Hn(1, q). We showed that the basic algebra of any nonsimple block of Hn(1, q) is independent of n. In this article, we examine the infinite representations of H2(1, - 1), or equivalently of Hn(1, q)≃D(An(ω)) for any n ≥ 2. We investigate the indecomposable and algebraically compact modules over H2(1, - 1), describe the structures of these modules and classify them under the elementary equivalence. [ABSTRACT FROM AUTHOR]

Let H be a finite dimensional Hopf algebra over a field k. We show that H contains a unique maximal Hopf ideal Jw(H) contained in J(H), the Jacobson radical of H. We give various characterizations of Jw(H), for example Jw(H) = AnnH((H/J(H))⊗n) for all large enough n. The smallest positive integer n with this property is denoted by lw(H). We prove that lw(H) equals the smallest number n such that(H/J(H))⊗n contains every projective indecomposable H/Jw(H)-module as a direct summand. This also equals the minimal n such that the tensor product of n suitable simple H-modules contains the projective cover of the trivial H/Jw(H)-module as a direct summand. We define projective homomorphisms between H-modules, which are used to obtain various reciprocity laws for tensor products of simple H-modules and their projective indecomposable direct summands. We also discuss some consequences of our general results in case H = kG is a group algebra of a finite group G and k is a field of characteristic p. [ABSTRACT FROM AUTHOR]

In the previous papers[1,2] we constructed an n4-dimensional Hopf algebra H,(p,q) which is isomorphic to Drinfeld quantum double D(A,(co)) of Taft's Hopf algebra A,(co) if p 5@ 0 and co = q-1 is a primitive nth root of unity, and studied the irreducible representations of H,(1, q). In this paper, we continue our study and examine the irreducible representations of H,(1,q) when q is any nth root of 1. We give the structures of all simple H,(1, q)-modules, and then classify them. We show that there are only n2 distinct simple H, (1, q)modules up to isomorphism. Though H,(1,q) is not a bialgebra, we show that the tensor product of two simple H, (1, q)-modules possesses an H, (1, q)-module structure in a way similar to the case that q is a primitive nth root of 1. Then we give a sufficient and necessary condition for the tensor product of two simple H[sub n](1, q)-modules to be semisimple. More generally, we describe the structure of socal of the tensor product. [ABSTRACT FROM AUTHOR]

Considers Hopf modules, Miyashita-Ulbrich coactions and monoidal center constructions. Hopf algebra with a bijective antipode over a field k, and A a right H-Galois extension of B; Braided monoidal categories as basic tool in the construction and understanding of knot variants.

The double bicrossproduct D = A[SUP &phis;,SUB α] &Bowtie; [SUP ψ,SUB β] H of two bialgebras A and H is constructed in a braided tensor category and the necessary and sufficient conditions for D to be a bialgebra are given. The universal property of double bicrossproduct is obtained. [ABSTRACT FROM AUTHOR]

The utilization of pump as turbines (PATs) within water distribution systems for energy regulation and hydroelectricity generation purposes has increasingly attracted the energy field players' attention. However, its power production efficiency still faces difficulties due to PAT's lack of flow control ability in such dynamic systems. This has eventually led to the introduction of the so-called "variable operating strategy" or VOS, where the impeller rotational speed may be controlled to satisfy the system-required flow conditions. Taking from these grounds, this study numerically investigates PAT eventual flow structures formation mechanism, especially when subjected to varying impeller rotational speed. CFD-backed numerical simulations were conducted on PAT flow under four operating conditions (1.00 QBEP, 0.82 QBEP, 0.74 QBEP, and 0.55 QBEP), considering five impeller rotational speeds (110 rpm, 130 rpm, 150 rpm, 170 rpm, and 190 rpm). Study results have shown that both PAT's flow and pressure fields deteriorate with the machine influx decrease, where the impeller rotational speed increase is found to alleviate PAT pressure pulsation levels under high-flow operating conditions, while it worsens them under part-load conditions. This study's results add value to a thorough understanding of PAT flow dynamics, which, in a long run, contributes to the solution of the so-far existent technical issues. [ABSTRACT FROM AUTHOR]

Objectives: To establish a system for simultaneous detection of miR-888 and miR-891a by droplet digital PCR (ddPCR), and to evaluate its application value in semen identification., Methods: The hydrolysis probes with different fluorescence modified reporter groups were designed to realize the detection of miR-888 and miR-891a by duplex ddPCR. A total of 75 samples of 5 body fluids (including peripheral blood, menstrual blood, semen, saliva and vaginal secretion) were detected. The difference analysis was conducted by Mann-Whitney U test. The semen differentiation ability of miR-888 and miR-891a was evaluated by ROC curve analysis and the optimal cut-off value was obtained., Results: There was no significant difference between the dual-plex assay and the single assay in this system. The detection sensitivity was up to 0.1 ng total RNA, and the intra- and inter-batch coefficients of variation were less than 15%. The expression levels of miR-888 and miR-891a detected by duplex ddPCR in semen were both higher than those in other body fluids. ROC curve analysis showed that the AUC of miR-888 was 0.976, the optimal cut-off value was 2.250 copies/μL, and the discrimination accuracy was 97.33%; the AUC of miR-891a was 1.000, the optimal cut-off value was 1.100 copies/μL, and the discrimination accuracy was 100%., Conclusions: In this study, a method for detection of miR-888 and miR-891a by duplex ddPCR was successfully established. The system has good stability and repeatability and can be used for semen identification. Both miR-888 and miR-891a have high ability to identify semen, and the discrimination accuracy of miR-891a is higher.