Mostrando recursos 1 - 20 de 1.324

  1. Vanishing theorems of the basic harmonic forms on a complete foliated Riemannian manifold

    Jung, Seoung Dal; Liu, Huili
    A well-known result by M. Min-Oo et al. states that there are no nontrivial basic harmonic $r (0< r

  2. Solvability of a fractional integral equation with the concept of measure of noncompactness

    Sadarangani, Kishin; Samet, Bessem
    We study the existence of solutions for a new class of integral equations involving a fractional integral with respect to another function. Our techniques are based on the measure of non-compactness concept combined with a generalized version of Darbo's theorem. Some examples are presented to illustrate the obtained results.

  3. Complex of injective words revisited

    Gan, Wee Liang
    We give a simple proof that (a generalization of) the complex of injective words has vanishing homology in all except the top degree.

  4. A criterion for $p$-henselianity in characteristic $p$

    Chatzidakis, Zoé; Perera, Milan
    Let $p$ be a prime. In this paper we give a proof of the following result: A valued field $(K,v)$ of characteristic $p > 0$ is $p$-henselian if and only if every element of strictly positive valuation is of the form $x^p - x$ for some $x \in K$.

  5. A simplicial approach to multiplier bimonoids

    Böhm, Gabriella; Lack, Stephen
    Although multiplier bimonoids in general are not known to correspond to comonoids in any monoidal category, we classify them in terms of maps from the Catalan simplicial set to another suitable simplicial set; thus they can be regarded as (co)monoids in something more general than a monoidal category (namely, the simplicial set itself). We analyze the particular simplicial maps corresponding to that class of multiplier bimonoids which can be regarded as comonoids.

  6. Three natural subgroups of the Brauer-Picard group of a Hopf algebra with applications

    Lentner, Simon; Priel, Jan
    In this article we construct three explicit natural subgroups of the Brauer-Picard group of the category of representations of a finite-dimensional Hopf algebra. In examples the Brauer Picard group decomposes into an ordered product of these subgroups, somewhat similar to a Bruhat decomposition. Our construction returns for any Hopf algebra three types of braided autoequivalences and correspondingly three families of invertible bimodule categories. This gives examples of so-called (2-)Morita equivalences and defects in topological field theories. We have a closer look at the case of quantum groups and Nichols algebras and give interesting applications. Finally, we briefly discuss the three families of group-theoretic extensions.

  7. Partial actions: what they are and why we care

    Batista, Eliezer
    We present a survey of recent developments in the theory of partial actions of groups and Hopf algebras.

  8. A finite-dimensional Lie algebra arising from a Nichols algebra of diagonal type (rank 2)

    Andruskiewitsch, Nicolás; Angiono, Iván; Rossi Bertone, Fiorela
    Let ${\mathcal{B}}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q} \in {\mathbf{k}}^{\theta \times \theta}$. Let ${\mathcal{L}}_{\mathfrak{q}}$ be the Lusztig algebra associated to ${\mathcal{B}}_{\mathfrak{q}}$. We present ${\mathcal{L}}_{\mathfrak{q}}$ as an extension (as braided Hopf algebras) of ${\mathcal{B}}_{\mathfrak{q}}$ by ${\mathfrak Z}_{\mathfrak{q}}$ where ${\mathfrak Z}_{\mathfrak{q}}$ is isomorphic to the universal enveloping algebra of a Lie algebra ${\mathfrak{n}}_\mathfrak{q}$. We compute the Lie algebra ${\mathfrak{n}}_{\mathfrak{q}}$ when $\theta = 2$.

  9. From finite groups to finite-dimensional Hopf algebras

    Cohen, Miriam; Westreich, Sara

  10. Representations of the small quasi-quantum group ${\operatorname{Q}}\mathbf{u}_{q}(\mathfrak{sl}_{2})$

    Liu, Gongxiang; Van Oystaeyen, Fred; Zhang, Yinhuo
    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.

  11. A note on the restricted universal enveloping algebra of a restricted Lie-Rinehart Algebra

    Schauenburg, Peter
    Lie-Rinehart algebras, also known as Lie algebroids, give rise to Hopf algebroids by a universal enveloping algebra construction, much as the universal enveloping algebra of an ordinary Lie algebra gives a Hopf algebra, of infinite dimension. In finite characteristic, the universal enveloping algebra of a restricted Lie algebra admits a quotient Hopf algebra which is finite-dimensional if the Lie algebra is. Rumynin has shown that suitably defined restricted Lie algebroids allow to define restricted universal enveloping algebras that are finitely generated projective if the Lie algebroid is. This note presents an alternative proof and possibly fills a gap that might,...

  12. Picard-Vessiot and categorically normal extensions in differential-difference Galois theory

    Janelidze, G.
    The connection between categorical and differential Galois theories established by the author (published in 1989) is extended to the context that includes difference Galois theory.

  13. Algebra depth in tensor categories

    Kadison, Lars
    Study of the quotient module of a finite-dimensional Hopf subalgebra pair in order to compute its depth yields a relative Maschke Theorem, in which semisimple extension is characterized as being separable, and is therefore an ordinary Frobenius extension. We study the core Hopf ideal of a Hopf subalgebra, noting that the length of the annihilator chain of tensor powers of the quotient module is linearly related to the depth, if the Hopf algebra is semisimple. A tensor categorical definition of depth is introduced, and a summary from this new point of view of previous results are included. It is shown...

  14. Curved Rota-Baxter systems

    Brzeziński, Tomasz
    Rota-Baxter systems are modified by the inclusion of a curvature term. It is shown that, subject to specific properties of the curvature form, curved Rota-Baxter systems $(A,R,S,\omega)$ induce associative and (left) pre-Lie products on the algebra $A$. It is also shown that if both Rota-Baxter operators coincide with each other and the curvature is $A$-bilinear, then the (modified by $R$) Hochschild cohomology ring over $A$ is a curved differential graded algebra.

  15. The Projective Class Rings of a family of pointed Hopf algebras of Rank two

    Chen, Hui-Xiang; Mohammed, Hassen Suleman Esmael; Lin, Weijun; Sun, Hua
    In this paper, we compute the projective class rings of the tensor product $\mathcal{H}_n(q)=A_n(q)\ot A_n(q^{-1})$ of Taft algebras $A_n(q)$ and $A_n(q^{-1})$, and its cocycle deformations $H_n(0,q)$ and $H_n(1,q)$, where $n>2$ is a positive integer and $q$ is a primitive $n$-th root of unity. It is shown that the projective class rings $r_p(\mathcal{H}_n(q))$, $r_p(H_n(0,q))$ and $r_p(H_n(1,q))$ are commutative rings generated by three elements, three elements and two elements subject to some relations, respectively. It turns out that even $\mathcal{H}_n(q)$, $H_n(0,q)$ and $H_n(1,q)$ are cocycle twist-equivalent to each other, they are of different representation types: wild, wild and tame, respectively.

  16. A criterion for reflectiveness of normal extensions

    Montoli, Andrea; Rodelo, Diana; Van der Linden, Tim
    We give a new sufficient condition for the normal extensions in an admissible Galois structure to be reflective. We then show that this condition is indeed fulfilled when $\mathbb{X}$ is the (protomodular) reflective subcategory of $\mathcal{S}$-special objects of a Barr-exact $\mathcal{S}$-protomodular category $\mathbb{C}$, where $\mathcal{S}$ is the class of split epimorphic trivial extensions in $\mathbb{C}$. Next to some concrete examples where the criterion may be applied, we also study the adjunction between a Barr-exact unital category and its abelian core, which we prove to be admissible.

  17. Coalgebras governing both weighted Hurwitz products and their pointwise transforms

    Garner, Richard; Street, Ross
    We give further insights into the weighted Hurwitz product and the weighted tensor product of Joyal species. Our first group of results relate the Hurwitz product to the pointwise product, including the interaction with Rota--Baxter operators. Our second group of results explain the first in terms of convolution with suitable bialgebras, and show that these bialgebras are in fact obtained in a particularly straightforward way by freely generating from pointed coalgebras. Our third group of results extend this from linear algebra to two-dimensional linear algebra, deriving the existence of weighted Hurwitz monoidal structures on the category of species using convolution with freely generated bimonoidales. Our final group of results relate Hurwitz...

  18. Transitive parallelism of residues in buildings

    Clais, Antoine
    We study the buildings in which parallelism of residues is an equivalence relation. If the building admits a group action, we describe how parallel residues are related to residues with equal stabilizers. This permits to retrieve the fact that in a Coxeter group or in a graph product, intersections of parabolic subgroups are parabolic.

  19. Injective mappings in $\mathbb{R}^\mathbb{R}$ and lineability

    Jiménez-Rodríguez, P.; Maghsoudi, S.; Muñoz-Fernández, G.A.; Seoane-Sepúlveda, J.B.
    It is known that there is not a two dimensional linear space in $\mathbb R^\mathbb R$ every non-zero element of which is an injective function. Here, we generalize this result to arbitrarily large dimensions. We also study the convolution of non-differentiable functions which gives, as a result, a differentiable function. In this latter case, we are able to show the existence of linear spaces of the largest possible dimension formed by functions enjoying the previous property. By doing this we provide both positive and negative results to the recent field of lineability. Some open questions are also provided.

  20. Weighted composition operators on algebras of differentiable functions

    Amiri, S.; Golbaharan, A.; Mahyar, H.
    Let $X$ be a perfect compact plane set, $n\in \mathbb{N}$ and $D^n(X)$ be the algebra of complex-valued functions on $X$ with continuous $n$-th derivative. In this paper we study weighted composition operators on algebras $D^n(X)$. We give a necessary and sufficient condition for these operators to be compact. As a consequence, we characterize power compact composition operators on these algebras. Then we determine the spectra of Riesz weighted composition operators on these algebras.

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