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Abstract We describe a connection between the subjects of cluster algebras, polynomial identity algebras, and discriminants. For this, we define the notion of root of unity quantum cluster algebras and prove that they are polynomial identity algebras. Inside each such algebra we construct a (large) canonical central subalgebra, which can be viewed as a far reaching generalization of the central subalgebras of big quantum groups constructed by De Concini, Kac, and Procesi and used in representation theory. Each such central subalgebra is proved to be isomorphic to the underlying classical cluster algebra of geometric type. When the root of unity quantum cluster algebra is free over its central subalgebra, we prove that the discriminant of the pair is a product of powers of the frozen variables times an integer. An extension of this result is also proved for the discriminants of all subalgebras generated by the cluster variables of nerves in the exchange graph. These results can be used for the effective computation of discriminants. As an application we obtain an explicit formula for the discriminant of the integral form over of each quantum unipotent cell of De Concini, Kac, and Procesi for arbitrary symmetrizable Kac–Moody algebras, where is a root of unity.
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Property RD and Hypercontractivity for Orthogonal Free Quantum Groups
Abstract We prove that the twisted property RD introduced in [ 2] fails to hold for all non-Kac type, non-amenable orthogonal free quantum groups. In the Kac case we revisit property RD, proving an analogue of the $$L_p-L_2$$ non-commutative Khintchine inequality for free groups from [ 29]. As an application, we give new and improved hypercontractivity and ultracontractivity estimates for the generalized heat semigroups on free orthogonal quantum groups, both in the Kac and non-Kac cases.
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- Award ID(s):
- 1700267
- PAR ID:
- 10185211
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- 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/10.1093/imrn/rnaa118
