$\tau $-tilting theory and silting theory of skew group algebra extensions

Kimura, Yuta; Koshio, Ryotaro; Kozakai, Yuta; Minamoto, Hiroyuki; Mizuno, Yuya

doi:10.5802/art.31

Kimura, Yuta ¹; Koshio, Ryotaro ²; Kozakai, Yuta ³ ; Minamoto, Hiroyuki ⁴ ; Mizuno, Yuya ⁵

¹ Department of Mechanical Systems Engineering, Faculty of Engineering, Hiroshima Institute of Technology, 2-1-1 Miyake, Saeki-ku Hiroshima 731-5143, Japan
² Tokyo University of Science 1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan
³ Department of Mathematics and Science Education, Graduate School of Science, Tokyo University of Science 1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan
⁴ Department of Mathematics, Graduate School of Science/ Faculty of Science, Osaka Metropolitan University 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan
⁵ Faculty of Liberal Arts, Sciences and Global Education / Graduate School of Science, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan

Annals of Representation Theory, Volume 2 (2025) no. 4, pp. 599-637.

Abstract

Let $\Lambda $ be a finite dimensional algebra with an action by a finite group $G$ and $A:= \Lambda *G$ the skew group algebra. One of our main results asserts that the canonical restriction-induction adjoint pair induced by the skew group algebra extension $\Lambda \subset A$ induces a poset isomorphism between the poset of $G$-stable support $\tau $-tilting modules over $\Lambda $ and that of ($\operatorname{mod}G$)-stable support $\tau $-tilting modules over $A$. We also establish a similar poset isomorphism between posets of appropriate classes of silting complexes over $\Lambda $ and $A$. These two results generalize and unify the preceding results by Zhang–Huang, Breaz–Marcus–Modoi and the second and the third authors. Moreover, we give a practical condition under which $\tau $-tilting finiteness and silting discreteness of $\Lambda $ are inherited by $A$. As applications we study $\tau $-tilting theory and silting theory of the (generalized) preprojective algebras and the folded mesh algebras. Among other things, we determine the posets of support $\tau $-tilting modules and of silting complexes over preprojective algebra $\Pi (\mathbb{L}_{n})$ of type $\mathbb{L}_{n}$.

Received: 2024-10-27
Revised: 2025-03-13
Accepted: 2025-04-14
Published online: 2025-09-16

DOI: 10.5802/art.31

Classification: 16G10, 16B50, 16E35, 16S35
Keywords: $\tau $-tilting theory, silting complexes, skew group algebras

Author's affiliations:

Kimura, Yuta ¹; Koshio, Ryotaro ²; Kozakai, Yuta ³; Minamoto, Hiroyuki ⁴; Mizuno, Yuya ⁵

¹ Department of Mechanical Systems Engineering, Faculty of Engineering, Hiroshima Institute of Technology, 2-1-1 Miyake, Saeki-ku Hiroshima 731-5143, Japan
² Tokyo University of Science 1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan
³ Department of Mathematics and Science Education, Graduate School of Science, Tokyo University of Science 1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan
⁴ Department of Mathematics, Graduate School of Science/ Faculty of Science, Osaka Metropolitan University 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan
⁵ Faculty of Liberal Arts, Sciences and Global Education / Graduate School of Science, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {$\tau $-tilting theory and silting theory of skew group algebra extensions},
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Kimura, Yuta; Koshio, Ryotaro; Kozakai, Yuta; Minamoto, Hiroyuki; Mizuno, Yuya. $\tau $-tilting theory and silting theory of skew group algebra extensions. Annals of Representation Theory, Volume 2 (2025) no. 4, pp. 599-637. doi : 10.5802/art.31. https://art.centre-mersenne.org/articles/10.5802/art.31/

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