Quantum cluster algebras and the dual canonical basis
Quantum cluster algebras and the dual canonical basis
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DissertationDate of Exam
28.02.2011Date of Publication
14.03.2011Advisor
Schröer, JanCo-Referee
Stroppel, CatharinaMetadata
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Abstract
Let Q be either a Dynkin quiver of type A with alternating orientation or the Kronecker quiver. With the indecomposable injective modules over the path algebra of Q and their Auslander-Reiten translates we associate an element w in the Weyl group of corresponding type. The thesis verifies that the subalgebra Uv(w) of the quantized universal enveloping algebra attached to w carries the structure of a quantum cluster algebra in the sense of Berenstein-Zelevinsky. The quantum cluster algebra is a v-deformation of the cluster algebra A(w) Geiß-Leclerc-Schröer attached to w. Furthermore, we show that all quantum cluster variables are elements in the dual of Lusztig's canonical basis (up to a power of v).
Subjects
Darstellungstheorie, Quantenalgebra, Kombinatorik, Köcher, Cluster, Representation theory, Quantum algebra, Combinatorics, Quiver
Classification (DDC)
510 MathematikLampe, Philipp: Quantum cluster algebras and the dual canonical basis. - Bonn, 2011. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5N-24553
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5N-24553
@phdthesis{handle:20.500.11811/4949,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5N-24553,
author = {{Philipp Lampe}},
title = {Quantum cluster algebras and the dual canonical basis},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2011,
month = mar,
note = {Let Q be either a Dynkin quiver of type A with alternating orientation or the Kronecker quiver. With the indecomposable injective modules over the path algebra of Q and their Auslander-Reiten translates we associate an element w in the Weyl group of corresponding type. The thesis verifies that the subalgebra Uv(w) of the quantized universal enveloping algebra attached to w carries the structure of a quantum cluster algebra in the sense of Berenstein-Zelevinsky. The quantum cluster algebra is a v-deformation of the cluster algebra A(w) Geiß-Leclerc-Schröer attached to w. Furthermore, we show that all quantum cluster variables are elements in the dual of Lusztig's canonical basis (up to a power of v).},
url = {https://hdl.handle.net/20.500.11811/4949}
}
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5N-24553,
author = {{Philipp Lampe}},
title = {Quantum cluster algebras and the dual canonical basis},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2011,
month = mar,
note = {Let Q be either a Dynkin quiver of type A with alternating orientation or the Kronecker quiver. With the indecomposable injective modules over the path algebra of Q and their Auslander-Reiten translates we associate an element w in the Weyl group of corresponding type. The thesis verifies that the subalgebra Uv(w) of the quantized universal enveloping algebra attached to w carries the structure of a quantum cluster algebra in the sense of Berenstein-Zelevinsky. The quantum cluster algebra is a v-deformation of the cluster algebra A(w) Geiß-Leclerc-Schröer attached to w. Furthermore, we show that all quantum cluster variables are elements in the dual of Lusztig's canonical basis (up to a power of v).},
url = {https://hdl.handle.net/20.500.11811/4949}
}
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