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<p>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 <i>T</i>-Koszul algebras, for which we obtain a higher version of classical Koszul duality. Our approach is motivated by and has applications for <i>n</i>-hereditary algebras. In particular, we characterize an important class of <i>n</i>-<i>T</i>-Koszul algebras of highest degree <i>a</i> in terms of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((na-1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi>a</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-representation infinite algebras. As a consequence, we see that an algebra is <i>n</i>-representation infinite if and only if its trivial extension is <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((n+1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Koszul with respect to its degree 0 part. Furthermore, we show that when an <i>n</i>-representation infinite algebra is <i>n</i>-representation tame, then the bounded derived categories of graded modules over the trivial extension and over the associated <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((n+1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-preprojective algebra are equivalent. In the <i>n</i>-representation finite case, we introduce the notion of almost <i>n</i>-<i>T</i>-Koszul algebras and obtain similar results.</p>

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Higher Koszul duality and connections with n-hereditary algebras

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  • 发表日期：

    2026年6月23日

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  • Johanne Haugland,
  • Mads Hustad Sandøy

摘要

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 \((na-1)\) ( n a - 1 ) -representation infinite algebras. As a consequence, we see that an algebra is n-representation infinite if and only if its trivial extension is \((n+1)\) ( 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 \((n+1)\) ( 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.