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.
Batalin - Vilkovisky structures on moduli spaces of flat connections
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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.