Trigonometric solutions of the associative

This equation was introduced in [Agu00, Agu01] and independently in [Pol00]. The algebraic meaning of this equation, explained in [Agu00, Agu01], is as follows. An associative algebra A is called an infinitesimal bialgebra if it is equipped with a coassociative coproduct which is a derivation, i.e. ∆(ab) = (a⊗ 1)∆(b)+∆(a)(1⊗b). This notion was introduced by Joni and Rota [RJ79] and is useful in combinatorics. Now, given an associative algebra A and a solution r ∈ A⊗A of the AYBE, one can define a comultiplication by ∆(a) = (a⊗1)r−r(1⊗a). (This comultiplication is a derivation for any r, and is coassociative if r satisfies the AYBE). Thus, (A,∆) is an infinitesimal bialgebra. One may also consider the AYBE with spectral parameter,