The Quantized Walled Brauer Algebra and Mixed Tensor Space

In this paper we investigate a multi-parameter deformation of the walled Brauer algebra which was previously introduced by Leduc ( 1994 ). We construct an integral basis of consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of on mix...

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Bibliographic Details
Published in:Algebras and representation theory 2014-04, Vol.17 (2), p.675-701
Main Authors: Dipper, R., Doty, S., Stoll, F.
Format: Article
Language:English
Subjects:
Associative Rings and Algebras
Commutative Rings and Algebras
Mathematics
Mathematics and Statistics
Non-associative Rings and Algebras
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Summary:In this paper we investigate a multi-parameter deformation of the walled Brauer algebra which was previously introduced by Leduc ( 1994 ). We construct an integral basis of consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of on mixed tensor space and prove that the kernel is free over the ground ring R of rank independent of R . As an application, we prove one side of Schur–Weyl duality for mixed tensor space: the image of in the R -endomorphism ring of mixed tensor space is, for all choices of R and the parameter q , the endomorphism algebra of the action of the (specialized via the Lusztig integral form) quantized enveloping algebra U of the general linear Lie algebra on mixed tensor space. Thus, the U -invariants in the ring of R -linear endomorphisms of mixed tensor space are generated by the action of .
ISSN:1386-923X
1572-9079
DOI:10.1007/s10468-013-9414-2