Braid group actions and tensor products

By Vyjayanthi Chari. 2001

132 citations recorded by the source corpus.

We define an action of the braid group of a simple Lie algebra on the space of imaginary roots in the corresponding quantum affine algebra. We then use this action to determine an explicit condition for a tensor product of arbitrary irreducible finite--dimensional representations is cyclic. This allows us to determine the set of points at which the corresponding $R$--matrix has a zero.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4429298baf248136acec65bdcbcb9fc699ea871b

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Citations in the corpus

Quantum Geometry and Quiver Gauge Theories (2013, methodology): “Hernandez [97] constructed (q, t)-characters for gΓ-quantum affine algebra U q (gΓ) of generic symmetrizable Kac-Moody Lie algebra gΓ, developed representation theory in [134] and constructed quantum fusion tensor category in [135]; Chari et al [136, 137] constructed fundamental U q (g) q-characters”
Characters and blocks for finite-dimensional representations of quantum affine algebras (2004, methodology): “The proof for the other Lie algebras is similar working with the reduced expressions for w0 given in [ 5, Section 6 ]. �”
Quantum toroidal algebras and their representations (2008, background): “quantum ane algebras, certain tensor products of l-highest weight modules are lhighest weight (see [ Cha2 , Theorem 4] and [Kash2, Theorem 9.1]), but not all.”
Demazure Modules, Fusion Products and Q-Systems (2013, background): “Together with results established in [2], [6], it follows that the local Weyl module is the q = 1 limit of a tensor product of irreducible representations of quantum affine algebra.”
Simple tensor products (2009, methodology): “d cases, but it holds also for twisted cases. Given Vin F, let τ∗ a(V) be the object obtained by pulling Vback via the automorphism τa. Thefollowing isproved in [FM] (the highest term was computed in [C]) for untwisted cases. The proof is the same for twisted cases. Proposition 4.6. χq(τ∗ a(V)) is o”
Cluster algebras and quantum affine algebras (2009, background): “,...,Sk of CZ and some complex numbers a1,...,ak such that ai aj 6∈q2Z, (1 6i < j 6k). (This follows, for example, from the fact that such a tensor product satisﬁes the irreducibility criterion in [Cha2].) Therefore, the description of the simple objects of C essentially reduces to the descriptio”
Representations of Shifted Yangians and Finite W-algebras (2005, methodology): “ tableau: entries increase strictly up columns and weakly along rows. Let Std(λ) denote the set of all standard λ-tableaux A∈Col(λ). Our proof of the next theorem is based on an argument due to Chari [C] in the context of quantum aﬃne algebras; see §7.3. Theorem D (Highest weight standard modules). ”
The Kirillov-Reshetikhin conjecture and solutions of T-systems (2005, unknown): “2001, no. 12, 629 654 (2001) [4℄ V. Chari, Braid group a tions and tensor produ ts Int.”