dorsal/arxiv
View SchemaTensor ideals in the category of tilting modules
| Authors | V. Ostrik |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9611033 |
| URL | https://arxiv.org/abs/q-alg/9611033 |
Abstract
We study the tensor category $\cQ$ of tilting modules over a quantum group $U_q$ with divided powers. The set $X_+$ of dominant weights is a union of closed alcoves $\oC_w$ numbered by the elements $w\in W^f$ of a certain subset of affine Weyl group $W$. G.Lusztig and N.Xi defined a partition of $W^f$ into canonical right cells and the right order $\le_R$ on the set of cells. For a cell $A\subset W^f$ we consider a full subcategory $\cQ_{<A}$ formed by direct sums of tilting modules $Q(\lambda)$ with highest weights $\lambda \in \bigcup_{w\in B<_RA} \oC_w$. We prove that $\cQ_{<A}$ is a tensor ideal in $\cQ$, generalizing H.Andersen's Theorem about the ideal of negligible modules which in our notations is nothing else then $\cQ_{<\{ e\}}$. The proof is an application of a recent result by W.Soergel who has computed the characters of tilting modules.
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"date_created": "2026-03-02T18:01:28.717000Z",
"date_modified": "2026-03-02T18:01:28.717000Z",
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"abstract": "We study the tensor category $\\cQ$ of tilting modules over a quantum group\n$U_q$ with divided powers. The set $X_+$ of dominant weights is a union of\nclosed alcoves $\\oC_w$ numbered by the elements $w\\in W^f$ of a certain subset\nof affine Weyl group $W$. G.Lusztig and N.Xi defined a partition of $W^f$ into\ncanonical right cells and the right order $\\le_R$ on the set of cells. For a\ncell $A\\subset W^f$ we consider a full subcategory $\\cQ_{\u003cA}$ formed by direct\nsums of tilting modules $Q(\\lambda)$ with highest weights $\\lambda \\in\n\\bigcup_{w\\in B\u003c_RA} \\oC_w$. We prove that $\\cQ_{\u003cA}$ is a tensor ideal in\n$\\cQ$, generalizing H.Andersen\u0027s Theorem about the ideal of negligible modules\nwhich in our notations is nothing else then $\\cQ_{\u003c\\{ e\\}}$. The proof is an\napplication of a recent result by W.Soergel who has computed the characters of\ntilting modules.",
"arxiv_id": "q-alg/9611033",
"authors": [
"V. Ostrik"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Tensor ideals in the category of tilting modules",
"url": "https://arxiv.org/abs/q-alg/9611033"
},
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