中科院数学与系统科学研究院
数学研究所
学术报告
表示论研讨班
报告人： Mohammad Hossein 博士 (华东师范大学)
题 目：Dominant and codominant dimensions for quiver representations
时 间：2023.09.22（星期五）15:30-16:30
地 点：数学院南楼N913
摘 要：Let M be a module category and Q a rooted quiver. In this talk, we study the dominant (resp. codominant) dimension of the category Rep(Q,M) of M-valued representations of Q. To do this, we first study injective envelopes and projective covers that play important roles in homological algebra and give explicit formulas for them in the category Rep(Q,M) that their origins go back to the classical representation theory of a finite quiver over a field. Then, by using such descriptions, we compute the dominant (resp. codominant) dimension of Rep(Q,M).
We show that the dominant dimension of Rep(Q,M) is at most one for every nonzero module category M and any right rooted quiver with at least one arrow.
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报告人：Tiago Cruz 博士 (Max Planck Institute for Mathematics)
题 目：Relative Auslander-Gorenstein pairs
时 间：2023.09.22（星期五）16:40-17:40
地 点：数学院南楼N913
摘 要：A famous result in representation theory is the Auslander’s correspondence which connects finite-dimensional algebras of finite representation-type with Auslander algebras. Over the years, many generalisations of Auslander algebras have been proposed: for instance n-Auslander algebras (by Iyama), n-minimal Auslander–Gorenstein algebras (by Iyama and Solberg), among others. All of the concepts above require the existence of a faithful projective-injective module and use classical dominant dimension. Now replace the faithful projective-injective module with a self-orthogonal module and classical dominant dimension with relative dominant dimension with respect to a module and you get a relative Auslander-Gorenstein pair.
In this talk, we introduce relative Auslander-Gorenstein pairs. Further, we will characterise relative Auslander pairs (those whose underlying algebras have finite global dimension) by the existence and uniqueness of tilting-cotilting modules having higher values of relative dominant and codominant dimension with respect to the self-orthogonal module. At the end, we discuss explicit examples of relative Auslander pairs. (This is joint work with Chrysostomos Psaroudakis.)
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