Column Ranks and Their Preservers of Matrices Over Max Algebra

The column rank of an m by n matrix A over max algebra is the weak dimension of the column space of A. We compare the column rank with rank of matrices over max algebra. We also characterize the linear operators which preserve the column rank of matrices over max algebra.     

Rank preservers of matrices over max algebra

The max algebra consists of the nonnegative real numbers equipped with two binary operations, maximization and multiplication. We consider the semimodules over max algebra and study the properties of the weak basis and weak dimension of the semi-modules. Moreover, we obtain the characterizations of those linear operators that preserve rank of matrices over max-algebra.     

Column Ranks and Their Preservers of Matrices Over Max Algebra

The column rank of an m by n matrix A over max algebra is the weak dimension of the column space of A. We compare the column rank with rank of matrices over max algebra. We also characterize the linear operators which preserve the column rank of matrices over max algebra.     

Spectral properties of matrix polynomials in the max algebra

We consider the spectral properties of matrix polynomials over the max algebra. In particular, we show how the Perron-Frobenius theorem for the max algebra extends to such polynomials and illustrate the relevance of this for multistep difference equations in the max algebra. We also present a number of inequalities for the largest max eigenvalue of a matrix polynomial.     

Weak Hopf Algebra in Yetter-Drinfeld Categories and Weak Biproducts

The Yetter-Drinfeld category of the Hopf algebra over a field is a pre braided category. In this paper we prove this result for the weak Hopf algebra. We study the smash product and smash coproduct, weak biproducts in the weak Hopf algebra over a field k. For a weak Hopf algebra A in left Yetter-Drinfeld category HHYD. we prove that the weak biproducts of A and H is a weak Hopf algebra.     

Resolving infeasibility in extremal algebras

Two extremal algebras =(B,) based on a linearly ordered set (B, ) are considered: in the maxmin algebra =max, = min and in the maxgroup algebra = max and is a group operation. If a system A x = b of linear equations over an extremal algebra is insolvable, then any subset of equations such that its omitting leads to a solvable subsystem is called a relieving set. We show that the problem of finding the minimum cardinality relieving set is NP-complete in the maxmin algebra already for bivalent systems, while it is polynomially solvable for bivalent systems in maxgroup algebra and also NP-complete for trivalent systems.     

Nichols algebras over weak Hopf algebras

In this paper, we study a Yetter-Drinfeld module V over a weak Hopf algebra H.Although the category of all left H-modules is not a braided tensor category, we can define a Yetter-Drinfeld module. Using this Yetter-Drinfeld modules V, we construct Nichols algebra B(V) over the weak Hopf algebra H, and a series of weak Hopf algebras. Some results of [8] are generalized.     

Invertible commutativity preservers of matrices over max algebra

The max algebra consists of the nonnegative real numbers equipped with two binary operations, maximization and multiplication. We characterize the invertible linear operators that preserve the set of commuting pairs of matrices over a subalgebra of max algebra.     

On commuting matrices in max algebra and in classical nonnegative algebra

This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector which directly leads to max analogs and nonnegative analogs of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the Perron roots of the components are distinct. For the case of max algebra, we show how the intersection of eigencones of commuting matrices can be described and we consider connections with Boolean algebra which enables us to prove that two commuting irreducible matrices in max algebra have a common eigennode.     

弱Hopf代数作用与冲积

本文研究了弱Hopf代数上的冲积并讨论了它约性质．设H是弱Hopf代数,A是左H-摸代数．我们给出了冲积A#H是弱双代数的一个充分条件以及A#H是A可分扩张的一个判定条件．另外,利用积分理论研究了Hopf模代数的有限性条件．     

Yetter-Drinfeld modules over weak bialgebras

We discuss properties of Yetter-Drinfeld modules over weak bialgebras over commutative rings. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules over a weak Hopf algebra are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. IfH is finitely generated and projective, then we introduce the Drinfeld double using duality results between entwining structures and smash product structures, and show that the category of Yetter-Drinfeld modules is isomorphic to the category of modules over the Drinfeld double. The category of finitely generated projective Yetter-Drinfeld modules over a weak Hopf algebra has duality.     

Hom-弱Hopf代数上的Hom-smash积

本文研究了在Hom-Hopf代数上引入Hom-弱Hop代数的问题.通过建立弱左H-模Hom-代数的方法,构造Hom-smash积,证明Hom-smash积是Hom-代数,且给出使之成为Hom-弱Hopf代数的充分条件,推广了由Bohm等人定义的弱Hop代数.     

Weak Module Amenability for Semigroup Algebras

We study the weak module amenability of Banach algebras which are Banach module over another Banach algebra with compatible actions. As an example we show that the semigroup algebra of a commutative inverse semigroup is always weakly amenable as a module over the semigroup algebra of its subsemigroup of idempotents.     

Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras

We extend the Larson–Sweedler theorem [Amer. J. Math. 91 (1969) 75] to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and invertible modules. We also reveal the connection of invertible modules to left and right grouplike elements in the dual weak Hopf algebra. Defining distinguished left and right grouplike elements, we derive the Radford formula [Amer. J. Math. 98 (1976) 333] for the fourth power of the antipode in a weak Hopf algebra and prove that the order of the antipode is finite up to an inner automorphism by a grouplike element in the trivial subalgebra A^T of the underlying weak Hopf algebra A.     

Extensions of the endomorphism algebra of weak comodule algebras

Let H be a weak Hopf algebra, and let A/B be a weak right H-Galois extension. In this paper, we mainly discuss the extension of the endomorphism algebra of a module over A. A necessary and sufficient condition for such an extension of the endomorphism algebra to be weak H-Galois is obtained by using Hopf-Galois theory and Morita theory.     

Hom-弱Hopf代数上的Hom-smash余积

本文研究了在Hom-Hopf代数上引入Hom-弱Hopf代数的问题.利用建立弱左H-Hom-余模双代数的方法,获得了Hom-smash余积的代数结构,并证明了Hom-smash余积是Hom-余代数和Hom-弱Hopf代数,推广了由Molnar定义的smash余积Hopf代数.     

Cotorsion Dimension of Weak Crossed Products

Let H be a finite-dimensional weak Hopf algebra over a field κ and A an associative algebra, and A#_σH a weak crossed product. In this paper, a spectral sequence for Ext is constructed which yields an estimate for cotorsion dimension of A#_σH in terms of the corresponding data for H and A.     

A maschke type theorem for weak Hopf algebras

In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. Also, from these results, we deduce a version of Maschke＇s Theorem for （H, B）-Hopf modules associated with a weak Hopf algebra H and a right H-comodule algebra B.     

Twisting theory for weak Hopf algebras

The main aim of this paper is to study the twisting theory of weak Hopf algebras and give an equivalence between the （braided） monoidal categories of weak Hopf bimodules over the original and the twisted weak Hopf algebra to generalize the result from Oeckl （2000）.     

Generic Flatness and the Jacobson Conjecture

A result of Artin, Small, and Zhang is used to show that a Noetherian algebra over a commutative, Noetherian Jacobson ring will be Jacobson if the algebra possesses a locally finite, Noetherian associated graded ring. This result is extended to show that if an algebra over a commutative Noetherian ring has a locally finite, Noetherian associated graded ring, then the intersection of the powers of the Jacobson radical is nilpotent. The proofs rely on a weak generalization of generic flatness and some observations about G-rings.