Quantum Algebras and Cyclic Quiver Varieties
 Title:
 Quantum Algebras and Cyclic Quiver Varieties
 Author(s):
 Negut, Andrei
 Thesis Advisor(s):
 Okounkov, Andrei
 Date:
 2015
 Type:
 Dissertations
 Department(s):
 Mathematics
 Persistent URL:
 http://dx.doi.org/10.7916/D8J38RGF
 Notes:
 Ph.D., Columbia University.
 Abstract:
 The purpose of this thesis is to present certain viewpoints on the geometric representation theory of Nakajima cyclic quiver varieties, in relation to the MaulikOkounkov stable basis. Our main technical tool is the shuffle algebra, which arises as the Ktheoretic Hall algebra of the double cyclic quiver. We prove the isomorphism between the shuffle algebra and the quantum toroidal algebra U_[q,t](sl_n), and identify the quotients of Verma modules for the shuffle algebra with the Ktheory groups of Nakajima cyclic quiver varieties, which were studied by Nakajima and VaragnoloVasserot.
The shuffle algebra viewpoint allows us to construct the universal Rmatrix of the quantum toroidal algebra U_[q,t](sl_n), and to factor it in terms of pieces that arise from subalgebras isomorphic to quantum affine groups U_q(gl_m), for various m. This factorization generalizes constructions of KhoroshkinTolstoy to the toroidal case, and matches the factorization that MaulikOkounkov produce via the stable basis in the Ktheory of Nakajima quiver varieties. We connect the two pictures by computing formulas for the root generators of U_[q,t](sl_n) acting on the stable basis, which provide a wide extension of MurnaghanNakayama and Pieri type rules from combinatorics.
 Subject(s):
 Mathematics
 Item views
 189
 Metadata:

text  xml
 Suggested Citation:
 Andrei Negut, 2015, Quantum Algebras and Cyclic Quiver Varieties, Columbia University Academic Commons, http://dx.doi.org/10.7916/D8J38RGF.