Recursos de colección

Project Euclid (Hosted at Cornell University Library) (191.996 recursos)

Bulletin of the Belgian Mathematical Society-Simon Stevin

1. 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.

2. 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,...

3. 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.

4. Algebra depth in tensor categories

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...

5. 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.

6. 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.

7. 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.

8. 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...

9. 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.

10. 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.

11. 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.

12. Optimization through dense sets

Jayanarayanan, C. R.; Rao, T. S. S. R. K.
In this paper, we study two optimization problems where solutions on a dense set yield global solution. We study these problems for spaces of Bochner integrable functions and for spaces of continuous functions. The first one deals with expressing the length of a vector as a sum of the distance to a best approximation and minimal best approximation and the second one relates to approximating a subsequence of a minimizing sequence with a sequence of proximinal vectors.

13. On the Frobenius vector of some simplicial affine semigroups

We give a formula for a Frobenius vector of a Gorenstein simplicial affine semigroup $S$, and when the semigroup is Cohen-Macaulay we give an algorithm computing the set of minimal Frobenius vectors of $S$ for a special class of semigroups.

14. Flag-transitive point-primitive non-symmetric $2$-$(v,k,2)$ designs with alternating socle

Liang, Hongxue; Zhou, Shenglin
We prove that if $\mathcal{D}$ is a non-trivial non-symmetric $2$-$(v,k,2)$ design admitting a flag-transitive point-primitive automorphism group $G$ with $Soc(G)=A_{n}$ for $n\geq5$, then $\mathcal{D}$ is a $2$-$(6,3,2)$ or $2$-$(10,4,2)$ design.

15. Weak amenability of Banach algebras with respect to characters

Nasr-Isfahani, Rasoul; Shahmoradi, Somayeh; Soltani Renani, Sima
For a Banach algebra ${\cal A}$, we introduce and investigate weak amenability of ${\cal A}$ with respect to a character. We give some necessary conditions for the weak amenability of $\cal A$ with respect to a character and describe a class of Banach algebras that are not weakly amenable with respect to characters. Finally, we give examples of Banach algebras which are weakly amenable with respect to characters but neither weakly amenable nor amenable with respect to characters.

16. Fixed point properties for semigroups of non-expansive mappings in conjugate Banach spaces

In this paper we study common fixed point properties of non-linear actions of semi-topological semigroups on non-void weak* compact convex sets in dual Banach spaces. Among other things, we derive from our main result Theorem 1, the existence of a common fixed point property for semigroups of non-expansive mappings acting on non-empty weakly compact convex sets, generalizing a result of Hsu [13], Mitchell [25].

17. On linear extendability of isometrical embeddings

In this paper we first investigate linear extendability of an isometric embedding $T:\mathcal U \longrightarrow \mathcal Y$ from an open subset $\mathcal U$ of a real Banach space $\mathcal X$ into a real Banach space $\mathcal Y$ in the case where $\mathcal Y$ is either the space $C_\Bbb R(K)$ of continuous real-valued functions on a compact space $K$, or is a strictly convex Banach space. Then we obtain similar results for the case where $\mathcal Y$ is an arbitrary real Banach space and $T:\mathcal U \longrightarrow \mathcal Y$ is an isometry whose range satisfies some additional conditions.

18. Algebrability of some subsets of the disk algebra

Lourenço, Mary Lilian; Vieira, Daniela M.
We show that the subset of the disk algebra of the functions that are not in some Dales-Davie algebra is algebrable. In other words, the set $\Big\{f\in\mathcal{A}(D)\,:\,\sum_{n=0}^{\infty}\dfrac{\|f^{(n)}\|}{n!}=+\infty\Big\}$ is shown to be algebrable.

19. Faber Polynomial Coefficient Estimates for a Comprehensive Subclass of Analytic Bi-Univalent Functions Defined by Subordination

We introduce the notion of \emph{primitive arc} of a curve defined over a field $k$ and study criterions for the existence of such objects in terms of the geometry of the curve. We prove that this notion provides a criterion which determines when the normalization of an irreductible curve singularity $(X,x)$ induces an isomorphism between the formal neighborhoods of the associated arc schemes at the constant arc $x$ and its lifting $\bar x$ to the normalization $\bar X$. We also show that the existence of a primitive arc at $x\in X$ is equivalent to the smoothness of the analytically irreducible...