Vertex Lie algebras and cyclotomic coinvariants

Benoit Vicedo, Charles A. S. Young

    Research output: Contribution to journalArticlepeer-review

    5 Citations (Scopus)
    53 Downloads (Pure)

    Abstract

    Given a vertex Lie algebra $\mathscr L$ equipped with an action by automorphisms of a cyclic group $\Gamma$, we define spaces of cyclotomic coinvariants over the Riemann sphere. These are quotients of tensor products of smooth modules over `local' Lie algebras $\mathsf L(\mathscr L)_{z_i}$ assigned to marked points $z_i$, by the action of a `global' Lie algebra ${\mathsf L}^{\Gamma}_{\{z_i \}}(\mathscr L)$ of $\Gamma$-equivariant functions. On the other hand, the universal enveloping vertex algebra $\mathbb V (\mathscr L)$ of $\mathscr L$ is itself a vertex Lie algebra with an induced action of $\Gamma$. This gives `big' analogs of the Lie algebras above. From these we construct the space of `big' cyclotomic coinvariants, i.e. coinvariants with respect to ${\mathsf L}^{\Gamma}_{\{z_i \}}(\mathbb V(\mathscr L))$. We prove that these two definitions of cyclotomic coinvariants in fact coincide, provided the origin is included as a marked point. As a corollary we prove a result on the functoriality of cyclotomic coinvariants which we require for the solution of cyclotomic Gaudin models in arXiv:1409.6937. At the origin, which is fixed by $\Gamma$, one must assign a module over the stable subalgebra $\mathsf L(\mathscr L)^{\Gamma}$ of $\mathsf L(\mathscr L)$. This module becomes a $\mathbb V(\mathscr L)$-quasi-module in the sense of Li. As a bi-product we obtain an iterate formula for such quasi-modules.
    Original languageEnglish
    Number of pages62
    JournalCommunications in Contemporary Mathematics
    Volume19
    Issue number2
    DOIs
    Publication statusPublished - 30 Mar 2016

    Keywords

    • math.QA
    • vertex algebras
    • vertex Lie algebras
    • cyclotomic coinvariants
    • infinite dimensional Lie algebras

    Fingerprint

    Dive into the research topics of 'Vertex Lie algebras and cyclotomic coinvariants'. Together they form a unique fingerprint.

    Cite this