Long-range deformations of arbitrary homogeneous Yang-Baxter integrable spin chains are realized as twists of the quantum group, with the Drinfeld associator encoding the long-range interaction terms up to first order in the deformation parameter.
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For cocommutative Hopf dialgebras the set-like rack is naturally isomorphic to the conjugation rack of the group-like digroup, and every finite generalized digroup arises as the group-like elements of its digroup algebra.
For a local vector bundle V over M, the bundle S^⊠(S^⊗(V)) is the free commutative 2-algebra generated by V, and skew-symmetric maps V ⊠ V to the unit induce compatible Poisson brackets on the resulting equivariant 2-algebra bundle over configuration spaces.
Diagrams of Eilenberg-MacLane spaces of any height are formal over Q, implying spectral sequence collapse at page 2 for any diagram over any small category.
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Cocommutative Hopf Dialgebras and Rack Combinatorics
For cocommutative Hopf dialgebras the set-like rack is naturally isomorphic to the conjugation rack of the group-like digroup, and every finite generalized digroup arises as the group-like elements of its digroup algebra.