{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:20:42Z","timestamp":1740122442891,"version":"3.37.3"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2021,4,3]],"date-time":"2021-04-03T00:00:00Z","timestamp":1617408000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,4,3]],"date-time":"2021-04-03T00:00:00Z","timestamp":1617408000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["1801011","2001128"],"award-info":[{"award-number":["1801011","2001128"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]},{"name":"DFG","award":["PE 980-7\/1","PE 980-8\/1"],"award-info":[{"award-number":["PE 980-7\/1","PE 980-8\/1"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Appl Categor Struct"],"published-print":{"date-parts":[[2021,10]]},"abstract":"Abstract<\/jats:title>Let$$V_*\\otimes V\\rightarrow {\\mathbb {C}}$$<\/jats:tex-math>V<\/mml:mi>\u2217<\/mml:mo><\/mml:mrow><\/mml:msub>\u2297<\/mml:mo>V<\/mml:mi>\u2192<\/mml:mo>C<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>be a non-degenerate pairing of countable-dimensional complex vector spacesV<\/jats:italic>and$$V_*$$<\/jats:tex-math>V<\/mml:mi>\u2217<\/mml:mo><\/mml:mrow><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The Mackey Lie algebra$${\\mathfrak {g}}=\\mathfrak {gl}^M(V,V_*)$$<\/jats:tex-math>g<\/mml:mi>=<\/mml:mo>gl<\/mml:mi><\/mml:mrow>M<\/mml:mi><\/mml:msup>(<\/mml:mo>V<\/mml:mi>,<\/mml:mo>V<\/mml:mi>\u2217<\/mml:mo><\/mml:mrow><\/mml:msub>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>corresponding to this pairing consists of all endomorphisms$$\\varphi $$<\/jats:tex-math>\u03c6<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>ofV<\/jats:italic>for which the space$$V_*$$<\/jats:tex-math>V<\/mml:mi>\u2217<\/mml:mo><\/mml:mrow><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is stable under the dual endomorphism$$\\varphi ^*: V^*\\rightarrow V^*$$<\/jats:tex-math>\u03c6<\/mml:mi>\u2217<\/mml:mo><\/mml:msup>:<\/mml:mo>V<\/mml:mi>\u2217<\/mml:mo><\/mml:msup>\u2192<\/mml:mo>V<\/mml:mi>\u2217<\/mml:mo><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We study the tensor Grothendieck category$${\\mathbb {T}}$$<\/jats:tex-math>T<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>generated by the$${\\mathfrak {g}}$$<\/jats:tex-math>g<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-modulesV<\/jats:italic>,$$V_*$$<\/jats:tex-math>V<\/mml:mi>\u2217<\/mml:mo><\/mml:mrow><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and their algebraic duals$$V^*$$<\/jats:tex-math>V<\/mml:mi>\u2217<\/mml:mo><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and$$V^*_*$$<\/jats:tex-math>V<\/mml:mi>\u2217<\/mml:mo><\/mml:mrow>\u2217<\/mml:mo><\/mml:msubsup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The category$${{\\mathbb {T}}}$$<\/jats:tex-math>T<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is an analogue of categories considered in prior literature, the main difference being that the trivial module$${\\mathbb {C}}$$<\/jats:tex-math>C<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is no longer injective in$${\\mathbb {T}}$$<\/jats:tex-math>T<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We describe the injective hullI<\/jats:italic>of$${\\mathbb {C}}$$<\/jats:tex-math>C<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>in$${\\mathbb {T}}$$<\/jats:tex-math>T<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and show that the category$${\\mathbb {T}}$$<\/jats:tex-math>T<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is Koszul. In addition, we prove thatI<\/jats:italic>is endowed with a natural structure of commutative algebra. We then define another category$$_I{\\mathbb {T}}$$<\/jats:tex-math>I<\/mml:mi><\/mml:msub>T<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>of objects in$${\\mathbb {T}}$$<\/jats:tex-math>T<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>which are free asI<\/jats:italic>-modules. Our main result is that the category$${}_I{\\mathbb {T}}$$<\/jats:tex-math>I<\/mml:mi><\/mml:msub>T<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is also Koszul, and moreover that$${}_I{\\mathbb {T}}$$<\/jats:tex-math>I<\/mml:mi><\/mml:msub>T<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is universal among abelian$${\\mathbb {C}}$$<\/jats:tex-math>C<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-linear tensor categories generated by two objectsX<\/jats:italic>,Y<\/jats:italic>with fixed subobjects$$X'\\hookrightarrow X$$<\/jats:tex-math>X<\/mml:mi>\u2032<\/mml:mo><\/mml:msup>\u21aa<\/mml:mo>X<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>,$$Y'\\hookrightarrow Y$$<\/jats:tex-math>Y<\/mml:mi>\u2032<\/mml:mo><\/mml:msup>\u21aa<\/mml:mo>Y<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and a pairing$$X\\otimes Y\\rightarrow {\\mathbf{1 }}$$<\/jats:tex-math>X<\/mml:mi>\u2297<\/mml:mo>Y<\/mml:mi>\u2192<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>where1<\/jats:bold>is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories$${\\mathbb {T}}$$<\/jats:tex-math>T<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and$${}_I{\\mathbb {T}}$$<\/jats:tex-math>I<\/mml:mi><\/mml:msub>T<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s10485-021-09640-2","type":"journal-article","created":{"date-parts":[[2021,4,3]],"date-time":"2021-04-03T05:02:46Z","timestamp":1617426166000},"page":"915-950","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Universal Tensor Categories Generated by Dual Pairs"],"prefix":"10.1007","volume":"29","author":[{"given":"Alexandru","family":"Chirvasitu","sequence":"first","affiliation":[]},{"given":"Ivan","family":"Penkov","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,4,3]]},"reference":[{"key":"9640_CR1","doi-asserted-by":"publisher","first-page":"466","DOI":"10.1090\/S0002-9947-1960-0157984-8","volume":"95","author":"H Bass","year":"1960","unstructured":"Bass, H.: Finitistic dimension and a homological generalization of semi-primary rings. 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