Tensor ideals, Deligne categories and invariant theory. (English) Zbl 1404.18017
Broadly speaking, the (classical) First and Second Fundamental Theorems of invariant theory respectively describe a set of generators and a complete set of relations for the space \(X^U\) of invariants of a group, Lie algebra, associative algebra, or some other algebraic structure \(U\), acting linearly on a space \(X\).
For \(\Bbbk\) an algebraically closed field of characteristic zero and any \(\delta\in\Bbbk\), P. Deligne [Studies in Mathematics. Tata Institute of Fundamental Research 19, 209–273 (2007; Zbl 1165.20300)] has defined universal categories \(\underline{\mathsf{Rep}}_0(\mathsf{GL}_\delta)\) and \(\underline{\mathsf{Rep}}_0(\mathsf{O}_\delta)\) (and their pseudo-abelian envelopes \(\underline{\mathsf{Rep}}(\mathsf{GL}_\delta)\) and \(\underline{\mathsf{Rep}}(\mathsf{O}_\delta)\)) which interpolate the classical representation categories of the general linear and orthogonal group. They are universal in the sense that, for example, for every \(\Bbbk\)-linear monoidal category \(\mathcal{C}\) such that \(\mathsf{End}_{\mathcal{C}}(\mathbf{1})=\Bbbk\) and for every dualizable object \(X\) of \(\mathcal{C}\) of dimension \(\delta\) there exists a unique \(\Bbbk\)-linear monoidal functor \(\underline{\mathsf{Rep}}_0(\mathsf{GL}_\delta) \to \mathcal{C}\) mapping the generator of \(\underline{\mathsf{Rep}}_0(\mathsf{GL}_\delta)\) to \(X\). There are cases in which, by their universal properties, these categories come with full monoidal functors to the representation categories of the general linear supergroup \(\mathsf{GL}(m|n)\) and the orthosymplectic supergroup \(\mathsf{OSp}(m|2n)\) (a modern formulation of the FFT).
In the present paper, the author develop some general tools to study tensor ideals in monoidal categories (we point out Theorem 3.1.1, which is a suitable reformulation of a result by Y. André and B. Kahn, Theorem 4.3.1 and its special cases 4.3.4 and 4.4.4) and use them to classify tensor ideals in Deligne’s universal categories \(\underline{\mathsf{Rep}}(\mathsf{GL}_\delta), \underline{\mathsf{Rep}}(\mathsf{O}_\delta), \underline{\mathsf{Rep}}(P)\) (taking advantage of the previously mentioned FFTs), \(\underline{\mathsf{Rep}}(\mathsf{S}_t)\) and in the categories of tilting modules for \(\mathsf{SL}_2(\Bbbk)\) with \(\mathsf{char}(\Bbbk)>0\) and for \(U_q(\mathfrak{sl}_2)\) with \(q\) a root of unity. Moreover, these results are applied to obtain new insights into the SFT for some algebraic supergroups.
For \(\Bbbk\) an algebraically closed field of characteristic zero and any \(\delta\in\Bbbk\), P. Deligne [Studies in Mathematics. Tata Institute of Fundamental Research 19, 209–273 (2007; Zbl 1165.20300)] has defined universal categories \(\underline{\mathsf{Rep}}_0(\mathsf{GL}_\delta)\) and \(\underline{\mathsf{Rep}}_0(\mathsf{O}_\delta)\) (and their pseudo-abelian envelopes \(\underline{\mathsf{Rep}}(\mathsf{GL}_\delta)\) and \(\underline{\mathsf{Rep}}(\mathsf{O}_\delta)\)) which interpolate the classical representation categories of the general linear and orthogonal group. They are universal in the sense that, for example, for every \(\Bbbk\)-linear monoidal category \(\mathcal{C}\) such that \(\mathsf{End}_{\mathcal{C}}(\mathbf{1})=\Bbbk\) and for every dualizable object \(X\) of \(\mathcal{C}\) of dimension \(\delta\) there exists a unique \(\Bbbk\)-linear monoidal functor \(\underline{\mathsf{Rep}}_0(\mathsf{GL}_\delta) \to \mathcal{C}\) mapping the generator of \(\underline{\mathsf{Rep}}_0(\mathsf{GL}_\delta)\) to \(X\). There are cases in which, by their universal properties, these categories come with full monoidal functors to the representation categories of the general linear supergroup \(\mathsf{GL}(m|n)\) and the orthosymplectic supergroup \(\mathsf{OSp}(m|2n)\) (a modern formulation of the FFT).
In the present paper, the author develop some general tools to study tensor ideals in monoidal categories (we point out Theorem 3.1.1, which is a suitable reformulation of a result by Y. André and B. Kahn, Theorem 4.3.1 and its special cases 4.3.4 and 4.4.4) and use them to classify tensor ideals in Deligne’s universal categories \(\underline{\mathsf{Rep}}(\mathsf{GL}_\delta), \underline{\mathsf{Rep}}(\mathsf{O}_\delta), \underline{\mathsf{Rep}}(P)\) (taking advantage of the previously mentioned FFTs), \(\underline{\mathsf{Rep}}(\mathsf{S}_t)\) and in the categories of tilting modules for \(\mathsf{SL}_2(\Bbbk)\) with \(\mathsf{char}(\Bbbk)>0\) and for \(U_q(\mathfrak{sl}_2)\) with \(q\) a root of unity. Moreover, these results are applied to obtain new insights into the SFT for some algebraic supergroups.
Reviewer: Paolo Saracco (Bruxelles)
MSC:
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 17B45 | Lie algebras of linear algebraic groups |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 15A72 | Vector and tensor algebra, theory of invariants |
Keywords:
monoidal (super)category; tensor ideal; thick tensor ideal; Deligne category; algebraic (super)group; second fundamental theorem of invariant theory; tilting modules; quantum groupsCitations:
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