Abstract
Let e = ±1 and let i \( \in \) I. The symmetry T’ ie : U→ U (resp. T" I,e : U → U) induces for each λ’, λ" a linear isomorphism \(\lambda \prime{\rm{U}}\lambda \prime\prime \to s_i (\lambda \prime){\rm{U}}s_i (\lambda \prime\prime)\) (notation of 23.1.1; s i : X → X is as in 2.2.6). Taking direct sums, we obtain an algebra automorphism \(T\prime_{i,e} :{\rm{\dot U}} \to {\rm{\dot U}}\) (resp. \(T\prime\prime_{i,e} :{\rm{\dot U}} \to {\rm{\dot U}}\)) such that \(T\prime_{i,e} (1_\lambda ) = 1_{s_i (\lambda )} \) (resp. \(T\prime\prime_{i,e} (1_\lambda ) = 1_{s_i (\lambda )} \)) for all λ and \(T\prime_{i,e} (uxx\prime u\prime) = T\prime_{i,e} (u)T\prime_{i,e} (x)T\prime_{i,e} (x\prime)T\prime_{i,e} (u\prime)\) (resp. \(T\prime\prime_{i,e} (uxx\prime u\prime) = T\prime\prime_{i,e} (u)T\prime\prime_{i,e} (x)T\prime\prime_{i,e} (x\prime)T\prime\prime_{i,e} (u\prime)\)) for all \(u,u\prime \in {\rm{\dot U}}\) and \(x,x\prime \in {\rm{\dot U}}\). Then T’ ie is an automorphism of the algebra U with inverse T" i-e . These automorphisms satisfy braid group relations just like those of U.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Lusztig, G. (2010). Integrality Properties of the Symmetries. In: Introduction to Quantum Groups. Modern Birkhäuser Classics. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4717-9_41
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4717-9_41
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4716-2
Online ISBN: 978-0-8176-4717-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)