Near-group categories. (English) Zbl 1033.18004
The paper is a step in the classification of autonomous (= rigid = compact) \(R\)-linear monoidal categories for a commutative ring \(R\). The categories are assumed to have a finite set of simple objects that generate all objects under direct sum. In the case where the unit and counit for each simple object is invertible (that is, where the simple objects are pseudo-invertible), these structures are classified and called group-categories by F. Quinn [“Lectures on axiomatic topological quantum field theory”, in: Geometry and quantum field theory, IAS/Park City Math. Ser. 1, 323–453 (1995; Zbl 0901.18002)].
Here the author considers the case where all except exactly one simple object \(m\) have pseudo-inverses. The tensor square of \(m\) is a direct sum of a number of other simple objects and of a certain number \(k\) of copies of \(m\) itself. The case \(k=0\) was dealt with by D. Tambara and S. Yamagami [J. Algebra 209, 692–707 (1998; Zbl 0923.46052)], and the author [J. Siehler, “Braided near-group categories” (2000; arxiv.org/abs/math.QA/0011037)]. So the present paper is dedicated to strictly positive \(k\).
Here the author considers the case where all except exactly one simple object \(m\) have pseudo-inverses. The tensor square of \(m\) is a direct sum of a number of other simple objects and of a certain number \(k\) of copies of \(m\) itself. The case \(k=0\) was dealt with by D. Tambara and S. Yamagami [J. Algebra 209, 692–707 (1998; Zbl 0923.46052)], and the author [J. Siehler, “Braided near-group categories” (2000; arxiv.org/abs/math.QA/0011037)]. So the present paper is dedicated to strictly positive \(k\).
Reviewer: Ross H. Street (North Ryde)
MSC:
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 18B40 | Groupoids, semigroupoids, semigroups, groups (viewed as categories) |
References:
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| [7] | J Siehler, Braided near-group categories · Zbl 1033.18004 |
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