Comparison conjecture for the Maulik–Okounkov and cuspidal quiver Lie algebras

Let QQ be a quiver. Let gQ\mathbf{g}_Q be the Z\mathbb{Z}-graded Borcherds Lie algebra constructed from stable envelopes in Nakajima quiver varieties, and let g~Q\widetilde{\mathfrak{g}}_Q be the Z\mathbb{Z}-graded Borcherds--Kac--Moody algebra associated with the graded Cartan matrix defined using the polynomials CQ,dabsC^{abs}_{Q,\mathbf{d}}.

Comparison conjecture. The Z\mathbb{Z}-graded Lie algebras gQ\mathbf{g}_Q and g~Q\widetilde{\mathfrak{g}}_Q are isomorphic.

The source describes this as essentially a restatement of Okounkov's character conjecture. It is conditional in motivation on the positivity conjecture, and no proof or disproof is supplied.

Sources & referencesView supporting material

Primary source

T. Bozec and O. Schiffmann, “Counting absolutely cuspidals for quivers”, arXiv:1710.03036 (2019).

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