Theoretical analysis demonstrates infinitesimal 2-braidings from 2-shifted Poisson structures, suggesting concrete representations for higher quantum groups.
It is shown that every $2$-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra A defines a very explicit infinitesimal $2$-braiding on the homotopy $2$-category of the symmetric monoidal dg-category of finitely generated semi-free A-dg-modules. This provides a concrete realization, to first order in the deformation parameter , of the abstract deformation quantization results in derived algebraic geometry due to Calaque, Pantev, To\"en, Vaqui\'e and Vezzosi. Of particular interest is the case when A is the Chevalley-Eilenberg algebra of a higher Lie algebra, where the braided monoidal deformations developed in this paper may be interpreted as candidates for representation categories of `higher quantum groups'.
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Kemp et al. (2024) studied this question.
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