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Let H be a Hopf algebra. The concept of symmetry of a quasitriangu-lar structure on H is introduced and some properties of a symmetric quasitriangular structure are characterized. We apply this to study quasitriangular structures on small quantum groups and abelian extensions. In particular, the quasitriangular structures on uq(sl2) (if they exist) are proved to be symmetric and all quasi-triangular structures on K(8n, σ, τ), which is a generalization of the well-known 8-dimensional Kac algebra, are determined. 2010 Mathematics Subject Classification. 16T05, 16T25, 17B37.

The aim of this paper is to study quasitriangular structures on a class of semisimple Hopf algebras kG#σ,τkZ2 constructed through abelian extensions of kZ2 by kG for an abelian group G. We prove th...

The Asian Working Group for Sarcopenia (AWGS) 2019 recommends different measurement protocols for handgrip strength (HGS). We aimed to explore (1) whether these protocols induce a significant difference in HGS; (2) whether these differences be clinically meaningful; and (3) whether these protocols affect the identification of HGS weakness or asymmetry.Cross-sectional study.A total of 356 community-dwelling older adults (mean age 67.9 years; 146 women).Maximal HGS was measured according to protocols from the National Health and Nutrition Examination Survey (Method A, standing with full elbow extended) and the American Society of Hand Therapists (Method B, sitting with elbow flexed at 90°). HGS was analyzed using the maximal value of 2 or 3 repetitions of the dominant hand or 4 or 6 repetitions of both hands.The difference in maximal HGS between Method A and Method B might have clinical meaning in approximately half of the participants. When measured using Method A, respective proportions of individuals with differences in HGS between the 6 repetitions group and the other repetition groups beyond the noninferiority threshold were 42%, 20%, and 25% in men and 39%, 21%, and 17% in women. Using Method B, the corresponding percentages were 25%, 18%, and 6% in men, and 27%, 20%, and 5% in women, respectively. Different protocols did not significantly affect the identification of HGS weakness, as different protocols reached diagnostic accuracies of 0.910 to 0.967 in men and 0.911 to 0.986 in women when using Method A (6 repetitions) as the reference standard. However, different protocols significantly affected the identification of HGS asymmetry, as different protocols had diagnostic accuracies of 0.667 to 0.886 in men and 0.658 to 0.863 in women.The different protocols recommended by the AWGS 2019 update significantly affect maximal HGS values and the identification of HGS asymmetry but not HGS weakness.

Let $\mathcal{C}$ be a fusion category over an algebraically closed field $\mathbb{k}$ of arbitrary characteristic. Two numerical invariants of $\mathcal{C}$ , that is, the Casimir number and the determinant of $\mathcal{C}$ are considered in this paper. These two numbers are both positive integers and admit the property that the Grothendieck algebra $(\mathcal{C})\otimes_{\mathbb{Z}}K$ over any field K is semisimple if and only if any of these numbers is not zero in K. This shows that these two numbers have the same prime factors. If moreover $\mathcal{C}$ is pivotal, it gives a numerical criterion that $\mathcal{C}$ is nondegenerate if and only if any of these numbers is not zero in $\mathbb{k}$ . For the case that $\mathcal{C}$ is a spherical fusion category over the field $\mathbb{C}$ of complex numbers, these two numbers and the Frobenius–Schur exponent of $\mathcal{C}$ share the same prime factors. This may be thought of as another version of the Cauchy theorem for spherical fusion categories.

BACKGROUND/AIM Colorectal cancer (CRC) is a disease of poor prognosis. An advantageous connection between vitamin D supplementation and prognostic improvement was depicted in CRC patients. However, the effects of circulating vitamin D on cancer outcomes are unclear for advanced CRC patients, especially for those receiving chemotherapy. MATERIALS AND METHODS The review was registered on PROSPERO (register number: CRD42021243547). PUBMED, EMBASE, Cochrane Library, and Web of Science were searched for English-language publications using relevant keywords. Two reviewers independently selected articles, assessed quality, and extracted data. We applied RevMan5.4 and Stata14 for meta-analysis. RESULTS We included an RCT and three prospective cohort studies, which were of high overall quality. Higher circulating 25(OH)D level was related with better disease outcomes in advanced CRC patients undergoing chemotherapy: progression-free survival (HR=0.85, 95% CI=0.71-0.99; I2=34.4%), overall survival (OR=0.56, 95% CI=0.38-0.82; I2=0%). CONCLUSION High circulating 25(OH)D content is beneficial for improving prognosis of advanced CRC receiving chemotherapy.

In this note, we find a combinatorial identity which is closely related to the multi-dimensional integral $\gamma_{m}$ in the study of divisor functions. As an application, we determine the finite dual of the group algebra of infinite dihedral group., Comment: Comments are welcome!

In this paper, we study the antipode of a finite-dimensional Hopf algebra H with the dual Chevalley property and obtain an annihilation polynomial for the antipode. This generalizes an old result given by Taft and Wilson in 1974. As consequences, we show that 1) the quasi-exponent of H is the same as the exponent of its coradical, that is, qexp ( H ) = exp ⁡ ( H 0 ) ; 2) qexp ( H ⋊ k 〈 S 2 〉 ) = qexp ( H ) .

The classification of affine prime regular Hopf algebras of GK-dimension one is completed. As consequences, 1) we give a negative answer to an open question posed by Brown-Zhang and 2) we show that there do exist prime regular Hopf algebras of GK-dimension one which are not pointed., 39pages

In this paper, we classify all prime Hopf algebras $H$ of GK-dimension one satisfying the following two conditions: 1) $H$ has a 1-dimensional representation of order PI.deg$(H)$ and 2) the invariant components of $H$ with respect to this 1-dimensional representation are domains (see Section 2 for related definitions). As consequences, 1) a number of new Hopf algebras of GK-dimension one are found and some of them are not pointed, 2) we give a partial answer to a question posed by Brown and Zhang and 3) some new kinds of finite-dimensional Hopf algebras are found., 92 pages. In page 2, the graph should be "cusp $\in ^{\Z_6}_{Z_6}\mathcal{YD}$ ". Your comments are very welcome! Comparing with the first version, some typos were corrected

The quasi-Frobenius-Lusztig kernel ${\operatorname{Q}}\mathbf{u}_{q}(\mathfrak{sl}_{2})$ associated with $\mathfrak{sl}_{2}$ has been constructed in [9]. In this paper we study the representations of this small quasi-quantum group. We give a complete list of non-isomorphic indecomposables and the tensor product decomposition rules for simples and projectives. A description of the Grothendieck ring is also provided.

We study coquasitriangular pointed Majid algebras via the quiver approaches. The class of Hopf quivers whose path coalgebras admit coquasitriangular Majid algebras is classified. The quiver setting for general coquasitriangular pointed Majid algebras is also provided. Through this, some examples and classification results are obtained., Comment: 13 pages

The goal of the present paper is to classify an interesting class of elementary quasi-Hopf algebras, or equivalently, finite-dimensional pointed Majid algebras. By a Tannaka-Krein type duality, this determines a big class of pointed finite tensor categories. Based on some interesting observations of normalized 3-cocycles on finite abelian groups, we elucidate an explicit connection between our objective pointed Majid algebras and finite-dimensional pointed Hopf algebras over finite abelian groups. With a help of this connection and the successful theory of diagonal Nichols algebras over abelian groups, we provide a conceptual classification of finite-dimensional graded pointed Majid algebras of diagonal type. Some efficient methods of construction are also given., Comment: 35pages. All comments are welcome

Let κ be an algebraically closed field of characteristic p > 3 and g a restricted Lie superalgebra over κ. We introduce the definition of restricted cohomology for g and show its cohomology ring is finitely generated provided g is a basic classical Lie superalgebra. As a consequence, we show that the restricted enveloping algebra of a basic classical Lie superalgebra g is always wild except g = sl 2 or g = osp ( 1 | 2 ) or g = C ( 2 ) .

In this paper, we give the classification of finite supergroup schemes of finite representation type. Moreover, their Auslander-Reiten quivers are determined., All comments are welcome!

In continuation of the articles (Liu J Algebra 299:841–853, 2006; Huang, J Algebra 321:2650–2669, 2009) we classify all finite-dimensional basic Hopf algebras of tame type over an algebraically closed field of characteristic 0 in this paper. As consequences, we show the following statements: (1) the representation dimension of a tame basic Hopf algebra is exactly 3, (2) for a basic Hopf algebra H, if \(\textrm{C}(H)\geq 3\) then it is wild. These conclusions verify a folklore conjecture and one of Rickard’s statements for the class of finite-dimensional basic Hopf algebras.

Let H be a finite-dimensional Hopf algebra and assume that both H and H* are semisimple. The main result of this paper is to show that the representation dimension is an invariant under cleft extensions of H, that is, rep.dim(A) = rep.dim(A#σH). Some of the applications of this equality are also given.

In continuation of the article \cite{L} we classify all radically graded basic Hopf algebras of tame type over an algebraically closed field of characteristic 0., Comment: 25 pages

Let k be an algebraically closed field. The main goal of this paper is to classify the finite-dimensional pointed Hopf algebras over k of finite corepresentation type. To do so, we give a necessary and sufficient condition for a basic Hopf algebra over k to be of finite representation type firstly. Explicitly, we prove that a basic Hopf algebra over k is of finite representation type if and only if it is Nakayama. By this conclusion, we classify all finite-dimensional pointed Hopf algebras over k of finite corepresentation type.

In this paper, we introduce the deflnition of groupoid graded rings. Group graded rings, (skew) groupoid rings, artinian semisimple rings, matrix rings and others can be regarded as special kinds of groupoid graded rings. Our main task is to classify strongly groupoid graded rings by cohomology of groupoids. Some classical results about group graded rings are generalized to groupoid graded rings. In particular, the Clifiord Theorem for a strongly groupoid graded ring is given. 2000 Mathematics Subject Classiflcation: 16W50

In this paper, firstly, we give a complete list of tame local graded Frobenius algebras, then, by the method of bosonization and this list, we determine the structure of tame (radically) graded basic Hopf algebras completely.

In this paper, we study the structures of monomial Hopf algebras over a field of positive characteristic. A necessary and sufficient condition for the monomial coalgebra C d (n) to admit Hopf structures is given here, and if it is the case, all graded Hopf structures on C d (n) are completely classified. Moreover, we construct a Hopf algebras filtration on C d (n) which will help us to discuss a conjecture posed by Andruskiewitsch and Schneider. Finally combined with a theorem by Montgomery, we give the structure theorem for all monomial Hopf algebras.

The class of graded elementary quasi-Hopf algebras of tame type is classified. Combining with our previous work [19], this completes the trichotomy for such class of algebras according to their representation types. In addition, new examples of genuine elementary quasi-Hopf algebras, and accordingly finite pointed tensor categories, are provided., 20 pages

Let H be a finite dimensional Hopf algebra over a field k, and A an H-module algebra. If H and H* are semisimple, then we prove that gl.dim(A#H) = gl.dim(A). The relationship between this result and Kaplansky's Fifth Conjecture is discussed.

In this paper, we construct the quasi-Frobenius-Lusztig kernel for any simple Lie algebra $\mathfrak{g}$., a few changes made

We define total Frobenius-Schur indicator for each object in a spherical fusion category $C$ as a certain canonical sum of its higher indicators. The total indicators are invariants of spherical fusion categories. If $C$ is the representation category of a semisimple quasi-Hopf algebra $H$, we prove that the total indicators are non-negative integers which satisfy a certain divisibility condition. In addition, if $H$ is a Hopf algebra, then all the total indicators are positive. Consequently, the positivity of total indicators is a necessary condition for a quasi-Hopf algebra being gauge equivalent to a Hopf algebra. Certain twisted quantum doubles of finite groups and some examples of Tambara-Yamagami categories are discussed for the sufficiency of this positivity condition., Comment: 20pages Latex

We study finite quasi-quantum groups in their quiver setting developed recently by the first author in arXiv:0902.1620 and arXiv:0903.1472. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation type, or equivalently a classification of elementary quasi-Hopf algebras of finite representation type, over the field of complex numbers. By the Tannaka-Krein duality principle, this provides a classification of the finite tensor categories in which every simple object has Frobenius-Perron dimension 1 and there are finitely many indecomposable objects up to isomorphism. Some interesting information of these finite tensor categories is given by making use of the quiver representation theory., 22 pages

This paper is devoted to the study of the quasitriangularity of Hopf algebras via Hopf quiver approaches. We give a combinatorial description of the Hopf quivers whose path coalgebras give rise to coquasitriangular Hopf algebras. With a help of the quiver setting, we study general coquasitriangular pointed Hopf algebras and obtain a complete classification of the finite-dimensional ones over an algebraically closed field of characteristic 0., Comment: 19 pages