The augmented necklace Lie bialgebra, whose bracket and cobracket insert rather than remove involution pairs of arrows, is claimed to satisfy the IBL axioms and is witnessed by quartic Poisson/BV structures on representation varieties.
Matrix De Rham complex and quantum A-infinity algebras
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I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum A-infinity-algebras, introduced in "Modular operads and Batalin-Vilkovisky geometry" IMRN, Vol. 2007, doi: 10.1093/imrn/rnm075, is represented via de Rham differential acting on the matrix spaces related with Bernstein-Leites simple associative algebras with odd trace q(N), and with gl(N|N). I also show that the Lagrangians of the matrix integrals from "Noncommmutative Batalin-Vilkovisky geometry and Matrix integrals", Comptes Rendus Mathematique, vol 348 (2010), pp. 359-362, arXiv:0912.5484, are equivariantly closed differential forms.
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Quartic BV structures in supercategories and modified necklace Lie bialgebras
The augmented necklace Lie bialgebra, whose bracket and cobracket insert rather than remove involution pairs of arrows, is claimed to satisfy the IBL axioms and is witnessed by quartic Poisson/BV structures on representation varieties.