Perverse sheaves and quantum Grothendieck rings

By M. Varagnolo, E. Vasserot. 2001

74 citations recorded by the source corpus.

We give a geometric construction of a deformation of the Grothendieck ring of finite-dimensional representations of quantized affine algebras. It yields a positivity result for products of some canonical element. In the classical case, related partial results are given on tensor categories of representations of simple Lie algebras.

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

View Paper: https://www.semanticscholar.org/paper/76652b41899c8add43bdde5d9984e83fff0ccaa0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Quiver varieties and cluster algebras (2009, methodology): “) The t-deformed multiplication was originally given in terms of t-analogue of q-characters, but Varagnolo and Vasserot [59] later introduced a convolution diagram on M0(W ) which gives the multiplication in a more direct and geometric way.”
Representations of Algebras and Related Topics (2011, methodology): “he canonical direct sum decomposition of the corresponding generic quiver representation y of EW into indecomposable summands. The proof uses the geometric realization given by Varagnolo and Vasserot [41] of the t-deformed product of (q,t)-characters in terms of convolution of perverse sheaves. Fina”
t-Analogs of q-Characters of Quantum Affine Algebras of Type E6, E7, E8 (2006, methodology): “At the first stage we set amP (t) = 1 and amP ,i(t) = 0 for all i ∈ I for the l–highest weight monomial mP .”
Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras (2013, methodology): “ces a ring isomorphism φFe: K(TJ)/(q−1)K(TJ)−→∼K(CJ). Therefore TJ may be regarded as a Z-graded lifting of the rigid tensor category CJ. Recall that the ring K(CJ) admits interesting t-deformations ([37, 33, 11]). In [13], Hernandez and Leclerc gave a presentation of Kt, one of those t-deformations”
Quantum Grothendieck rings and derived Hall algebras (2011, background): “ty we can assume that p <s. Then, (6) simpliﬁes as N (i,p; j,s)=Ceij(s−p+1)−Ceij(s−p−1), (p <s). (8) We regard the noncommutative ring (Yt,∗)as a quantum torus of inﬁnite rank. 10 Remark 3.1 In [VV1] and [N3], the construction of a t-deformed Grothendieck ring is based on a slightly different ”