In this paper,we introduce the notion of a generalized weak group-like algebra which is designed to cover finite groupoid algebras and group-like algebras.Then we construct a list of examples and show that a generaliz...In this paper,we introduce the notion of a generalized weak group-like algebra which is designed to cover finite groupoid algebras and group-like algebras.Then we construct a list of examples and show that a generalized weak group-like algebra can be regarded as a generalized weak bi-Frobenius algebra.Moreover,we discuss when a generalized weak bi-Frobenius algebra is a groupoid algebra.Finally,we give the classifications of low-dimensional generalized weak group-like algebras.展开更多
With the purpose of providing a categorical treatment of weak multiplier bialgebras （introduced by BShm, Gomez-Torrecillas and Ldpez-Centella in 2015）, an ap- propriate notion of morphism for these algebraic objects...With the purpose of providing a categorical treatment of weak multiplier bialgebras （introduced by BShm, Gomez-Torrecillas and Ldpez-Centella in 2015）, an ap- propriate notion of morphism for these algebraic objects is proposed. This allows us to define a category wmb of （regular） weak multiplier bialgebras （with a right full comultipli- cation）, containing as a full subcategory the category wba of weak bialgebras defined by BShm, Gomez-Torrecillas and Lopez-Centella in 2014. We present a great source of ex- amples of these morphisms proving that, under some assumption, a functor between small categories induces a morphism of this kind between the natural weak multiplier bialgebra structures carried by the linear spans of the arrow sets of the categories. We explore the notion of elements of group-like type in a weak multiplier bialgebra, proposing a definition in the line of the one by the aforementioned authors for weak bialgebras. We show a big number of its properties and provide more general versions of many results known in the context of weak bialgebras. In particular, in analogy with the classical bialgebra setting （where the set of group-like elements is a monoid）, we prove that the set of these elements possesses a structure of category.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.12001174)the Natural Science Foundation of Shandong Province(Grant No.ZR2023MA008)。
文摘In this paper,we introduce the notion of a generalized weak group-like algebra which is designed to cover finite groupoid algebras and group-like algebras.Then we construct a list of examples and show that a generalized weak group-like algebra can be regarded as a generalized weak bi-Frobenius algebra.Moreover,we discuss when a generalized weak bi-Frobenius algebra is a groupoid algebra.Finally,we give the classifications of low-dimensional generalized weak group-like algebras.
文摘With the purpose of providing a categorical treatment of weak multiplier bialgebras （introduced by BShm, Gomez-Torrecillas and Ldpez-Centella in 2015）, an ap- propriate notion of morphism for these algebraic objects is proposed. This allows us to define a category wmb of （regular） weak multiplier bialgebras （with a right full comultipli- cation）, containing as a full subcategory the category wba of weak bialgebras defined by BShm, Gomez-Torrecillas and Lopez-Centella in 2014. We present a great source of ex- amples of these morphisms proving that, under some assumption, a functor between small categories induces a morphism of this kind between the natural weak multiplier bialgebra structures carried by the linear spans of the arrow sets of the categories. We explore the notion of elements of group-like type in a weak multiplier bialgebra, proposing a definition in the line of the one by the aforementioned authors for weak bialgebras. We show a big number of its properties and provide more general versions of many results known in the context of weak bialgebras. In particular, in analogy with the classical bialgebra setting （where the set of group-like elements is a monoid）, we prove that the set of these elements possesses a structure of category.
