Higher Koszul duality and connections with $n$-hereditary algebras
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| Format: | Preprint |
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2021
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| _version_ | 1866909539966648320 |
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| author | Haugland, Johanne Sandøy, Mads Hustad |
| author_facet | Haugland, Johanne Sandøy, Mads Hustad |
| contents | We establish a connection between two areas of independent interest in representation theory, namely Koszul duality and higher homological algebra. This is done through a generalization of the notion of $T$-Koszul algebras, for which we obtain a higher version of classical Koszul duality. Our approach is motivated by and has applications for $n$-hereditary algebras. In particular, we characterize an important class of $n$-$T$-Koszul algebras of highest degree $a$ in terms of $(na-1)$-representation infinite algebras. As a consequence, we see that an algebra is $n$-representation infinite if and only if its trivial extension is $(n+1)$-Koszul with respect to its degree $0$ part. Furthermore, we show that when an $n$-representation infinite algebra is $n$-representation tame, then the bounded derived categories of graded modules over the trivial extension and over the associated $(n+1)$-preprojective algebra are equivalent. In the $n$-representation finite case, we introduce the notion of almost $n$-$T$-Koszul algebras and obtain similar results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_12743 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Higher Koszul duality and connections with $n$-hereditary algebras Haugland, Johanne Sandøy, Mads Hustad Representation Theory 16G20, 16S37, 16W50, 18E30, 18G20 We establish a connection between two areas of independent interest in representation theory, namely Koszul duality and higher homological algebra. This is done through a generalization of the notion of $T$-Koszul algebras, for which we obtain a higher version of classical Koszul duality. Our approach is motivated by and has applications for $n$-hereditary algebras. In particular, we characterize an important class of $n$-$T$-Koszul algebras of highest degree $a$ in terms of $(na-1)$-representation infinite algebras. As a consequence, we see that an algebra is $n$-representation infinite if and only if its trivial extension is $(n+1)$-Koszul with respect to its degree $0$ part. Furthermore, we show that when an $n$-representation infinite algebra is $n$-representation tame, then the bounded derived categories of graded modules over the trivial extension and over the associated $(n+1)$-preprojective algebra are equivalent. In the $n$-representation finite case, we introduce the notion of almost $n$-$T$-Koszul algebras and obtain similar results. |
| title | Higher Koszul duality and connections with $n$-hereditary algebras |
| topic | Representation Theory 16G20, 16S37, 16W50, 18E30, 18G20 |
| url | https://arxiv.org/abs/2101.12743 |