A Note on Quiver Quantum Toroidal Algebra

Abstract

Recently, Li and Yamazaki proposed a new class of infinite-dimensional algebras, quiver Yangian, which generalizes the affine Yangian. The characteristic feature of the algebra is the action on BPS states for non-compact toric Calabi-Yau threefolds, which are in one-to-one correspondence with the crystal melting models. These algebras can be bootstrapped from the action on the crystals and have various truncations.In this paper, we propose a q-deformed version of the quiver Yangian, referred to as the quiver quantum toroidal algebra (QQTA). We examine some of the consistency conditions of the algebra. In particular, we show that QQTA is a Hopf superalgebra with a formal super coproduct, like known quantum toroidal algebras. QQTA contains an extra central charge. When it is trivial, QQTA has the three-dimensional representation acting on the three-dimensional crystals, like Li-Yamazaki's quiver Yangian. When Calabi-Yau threefolds have compact 4-cycles, the action seems to have some inconsistency, whose nature is not clear at this moment.

Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations

“…Quiver Yangian acts on three-dimensional BPS crystals [61], while shifted quiver Yangian acts on the subcrystal of the mother BPS crystal. A trigonometric deformation of quiver Yangian, which we call quiver quantum toroidal algebra (QQTA), was introduced in our previous paper [62]. QQTA acts on the same three-dimensional BPS crystal as the quiver Yangian.…”

“…The shifted quiver quantum toroidal algebra we define has a structure similar to the Hopf superalgebra structure of QQTA [62]. It has a generalized coproduct, counit, and antipode.…”

“…This paper is organized as follows. In section 2, we review the definition of the quiver quantum toroidal algebra defined in [62]. In section 3, we review the properties and representations of the quantum toroidal gl 1 .…”

“…2 Review: Quiver quantum toroidal algebra Let us review the quiver quantum toroidal algebra defined in [62]. The algebra is defined from the quiver diagram, which is derived from the toric Calabi-Yau geometry.…”

“…, where Q 0 is a set of vertices, Q 1 is a set of arrows between vertices, and Q 2 is the set of closed loops surrounded by Q 1 (see section 2 and Appendix B in [62]). We use There are two types of quivers, which can be derived from the geometry: "symmetric" quiver and "asymmetric" quiver.…”