28 research outputs found
quantum groups of split type via derived Hall algebras
A quantum symmetric pair consists of a quantum group and its
coideal subalgebra (called an
quantum group) with parameters . In this
note, we use the derived Hall algebras of 1-periodic complexes to realize the
quantum groups of
split type.Comment: 14 pages. arXiv admin note: text overlap with arXiv:2110.0257
Quantum Borcherds-Bozec algebras via semi-derived Ringel-Hall algebras II: braid group actions
Based on the realization of quantum Borcherds-Bozec algebra
and quantum generalized Kac-Moody algebra
via semi-derived Ringel-Hall algebra of a quiver
with loops, we deduce the braid group actions of
introduced by Fan and Tong recently and establish braid group actions for
by applying the BGP reflection functors to
semi-derived Ringel-Hall algebras.Comment: 18 pages, minor changes, accepted by Bulletin of the London
Mathematical Society. arXiv admin note: substantial text overlap with
arXiv:2107.0316
Hall algebra of Jordan quiver and Hall-Littlewood functions
We show that the Hall algebra of the Jordan quiver is a polynomial
ring in infinitely many generators and obtain transition relations among
several generating sets. We establish a ring isomorphism from this Hall
algebra to the ring of symmetric functions in two parameters , which
maps the Hall basis to a class of (modified) inhomogeneous
Hall-Littlewood (HL) functions. The (modified) HL functions
admit a formulation via raising and lowering operators. We formulate and prove
Pieri rules for (modified) HL functions. The modified HL
functions specialize at to the modified HL functions; they
specialize at to the deformed universal characters of type C, which
further specialize at to the universal characters of type C.Comment: v2,41 pages,references adde