arXiv subjects
Publications and source records attributed to Yuta Kozakai.
For a tilting complex over a finite-dimensional symmetric algebra over an algebraically closed field, whose distinct graded components have no indecomposable projective direct summands in common, we explicitly (re)construct a two-sided tilting complex that corresponds to the tilting complex, starting from a bimodule inducing a stable equivalence of Morita type.
Let $Λ$ be a finite dimensional algebra with an action by a finite group $G$ and $A:= Λ*G$ the skew group algebra. One of our main results asserts that the canonical restriction-induction adjoint pair of the skew group algebra extension $Λ\subset A$ induces a poset isomorphism between the poset of $G$-stable support $τ$-tilting modules over $Λ$ and that of $(\!\!\!\mod G)$-stable support $τ$-tilting modules over $A$. We also establish a similar poset isomorphism of posets of appropriate classes of silting complexes over $Λ$ and $A$. These two results generalize and unify preceding results by Huang-Zhang, Breaz-Marcus-Modoi and the second and the third authors. Moreover, we give a practical condition under which $τ$-tilting finiteness and silting discreteness of $Λ$ are inherited to those of $A$. As applications we study $τ$-tilting theory and silting theory of the (generalized) preprojective algebras and the folded mesh algebras. Among other things, we determine the posets of support $τ$-tilting modules and of silting complexes over preprojective algebra $Π(\Bbb{L}_{n})$ of type $\Bbb{L}_{n}$.
According to Letourmy and Vendramin, a representation of a skew brace is a pair of representations on the same vector space, one for the additive group and the other for the multiplicative group, that satisfies a certain compatibility condition. Following their definition, we shall explain how some of the results from representation theory of groups, such as Maschke's theorem and Clifford's theorem, extend naturally to that of skew braces. We shall also give some concrete examples to illustrate that skew brace representations are more difficult to classify than group representations.
Demonet, Iyama and Jasso introduced a new class of finite dimensional algebras, $τ$-tilting finite algebras. It was shown by Eisele, Janssens and Raedschelders that tame blocks of group algebras of finite groups are always $τ$-tilting finite. Given the classical result that the representation type (representation finite, tame or wild) of blocks is determined by their defect groups, it is natural to ask what kinds of subgroups control $τ$-tilting finiteness of group algebras or their blocks. In this paper, as a positive answer to this question, we demonstrate that $τ$-tilting finiteness of a group algebra of a finite group $G$ is controlled by a $p$-hyperfocal subgroup of $G$ under some assumptions on $G$. We consider a group algebra of a finite group $P\rtimes H$ over an algebraically closed field of positive characteristic $p$, where $P$ is an abelian $p$-group and $H$ is an abelian $p'$-group acting on $P$, and show that $p$-hyperfocal subgroups determine $τ$-tilting finiteness of the group algebras in this case.
We explicitly construct two-sided tilting complexes corresponding to Membrillo-Hernández's tree-to-star tilting complexes for generalized Brauer tree algebras.
Let $G$ be a finite group, $N$ a normal subgroup of $G$, and $k$ a field of characteristic $p>0$. In this paper, we formulate the brick version of Clifford's theorem under suitable assumptions and prove it by using the theory of wide subcategories. As an application of our theorem, we consider the restrictions of semibricks and two-term simple-minded collections under the assumption that the index of the normal subgroup $N$ in $G$ is a $p$-power.
We discuss finiteness/infiniteness of $τ$-tilting modules over tensor products of two symmetric algebras. As an application, we discuss that over block algebras of direct products of finite groups.
Let $\tilde{G}$ be a finite group and $G$ a normal subgroup of $\tilde{G}$. In this paper, we give a necessary and sufficient condition for $\mathrm{Ind}_G^{\tilde{G}}M$ to be a support $τ$-tilting $k\tilde{G}$-module for a $kG$-module $M$. Moreover, we give the block version of the result.
Let $\tilde{G}$ be a finite group, $G$ a normal subgroup of $\tilde{G}$ and $k$ an algebraically closed field of characteristic $p>0$. The first main result in this paper is to show that support $τ$-tilting $k\tilde{G}$-modules satisfying some properties are support $τ$-tilting modules as $kG$-modules too. As the second main result, we give equivalent conditions for support $τ$-tilting $k\tilde{G}$-modules to satisfy the above properties, and show that the set of the support $τ$-tilting $k\tilde{G}$-modules with the properties is isomorphic to the set of $\tilde{G}$-invariant support $τ$-tilting $kG$-modules as partially ordered sets. As an application, we show that the set of $\tilde{G}$-invariant support $τ$-tilting $kG$-modules is isomorphic to the set of support $τ$-tilting $k\tilde{G}$-modules in the case that the index $G$ in $\tilde{G}$ is a $p$-power. As a further application, we give a feature of vertices of indecomposable $τ$-rigid $k\tilde{G}$-modules. Finally, we give the block versions of the above results.
Let $p$ be a prime number, $k$ an algebraically closed field of characteristic $p$, $\tilde{G}$ a finite group, and $G$ a normal subgroup of $\tilde{G}$ having a $p$-power index in $\tilde{G}$. Moreover let $B$ be a block of $kG$ with a cyclic defect group and $\tilde{B}$ be the unique block of $k\tilde{G}$ covering $B$. We study tilting complexes over the block $\tilde{B}$ and show that the block $\tilde{B}$ is a tilting-discrete algebra. Moreover we show that the set of all tilting complexes over $\tilde{B}$ is isomorphic to that over $B$ as partially ordered sets.
In this article we study support $τ$-tilting modules, semibricks and more over blocks of group algebras. Let $k$ be an algebraically closed field of characteristic $p>0$, $\tilde{G}$ a finite group and $G$ a normal subgroup of $\tilde{G}$. Moreover, let $\tilde{B}$ be a block of $k\tilde{G}$ and $B$ a block of $kG$ covered by $\tilde{B}$. We show that, under certain conditions for the factor group $\tilde{G}/G$ and $B$, induced modules and extending modules of support $τ$-tilting modules and semibricks over $B$ are also the ones over $\tilde{B}$, respectively.
Support $τ$-tilting modules correspond to some classes of categorical objects bijectively, such as two-term tilting complexes for any finite dimensional symmetric algebra. This fact motivates us to classify support $τ$-tilting modules over blocks of finite groups. Therefore we classify support $τ$-tilting modules over particular blocks of finite groups by using the modular representation theoretical approaches including Clifford theory, Green's indecomposability theorem and so on.