PBW theory for quantum affine algebras · arXivDesk
2011.14253Nov 29, 202063 pages. This is a full paper of the announcement: PBW theoretic approach to the module category of quantum affine algebras, arXiv:2005.04838v2. v2: minor changes
PBW theory for quantum affine algebras
Masaki Kashiwara, Myungho Kim, Se-jin Oh, Euiyong Park
Let Uq′(g) be a quantum affine algebra of arbitrary type and let Cg be Hernandez-Leclerc's category. We can associate the quantum affine Schur-Weyl duality functor FD
Nearby in the stack
to a duality datum
D
in
Cg
. We introduce the notion of a strong (complete) duality datum
D
and prove that, when
D
is strong, the induced duality functor
FD
sends simple modules to simple modules and preserves the invariants
Λ
and
Λ∞
introduced by the authors. We next define the reflections
Sk
and
Sk−1
acting on strong duality data
D
. We prove that if
D
is a strong (resp.\ complete) duality datum, then
Sk(D)
and
Sk−1(D)
are also strong (resp.\ complete ) duality data. We finally introduce the notion of affine cuspidal modules in
Cg
by using the duality functor
FD
, and develop the cuspidal module theory for quantum affine algebras similarly to the quiver Hecke algebra case.