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We prove a number of results having to do with equipping type-I\mathrm{C}^*-algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the\mathrm{C}^*-algebra in question is an extension of a non-zero finite direct sum of elementary\mathrm{C}^*-algebras by a commutative unital\mathrm{C}^*-algebra then it must be finite-dimensional.
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A category $\mathcal{O}$ for oriented matroids
We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite-dimensional algebra, whose representation theory is analogous to blocks of Bernstein—Gelfand—Gelfand category\mathcal{O}. When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden—Licata—Proudfoot—Webster. Applying our construction to non-linear oriented matroid programs provides a large new class of algebras. For Euclidean oriented matroid programs, the resulting algebras are quasi-hereditary and Koszul, as in the linear setting. In the non-Euclidean case, we obtain algebras that are not quasi-hereditary and not known to be Koszul, but still have a natural class of standard modules and satisfy numerical analogues of quasi-heredity and Koszulity on the level of graded Grothendieck groups.
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- Award ID(s):
- 1802299
- PAR ID:
- 10550365
- Publisher / Repository:
- EMS Press
- Date Published:
- Journal Name:
- Journal of Combinatorial Algebra
- Volume:
- 7
- Issue:
- 1
- ISSN:
- 2415-6302
- Page Range / eLocation ID:
- 159 to 226
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4171/JCA/71
