Zoran Skoda Jordanian twist

Introduced in Ogievetsky 1989 and in

V. Coll, M. Gerstenhaber, A. Giaquinto, An explicit deformation formula with non-commuting derivations, Ring Theory, Vol. 1, Israel Mathematical Conference Proceedings (1989) 396-403

MR1029329…a generalization of some work of Gerstenhaber [Ann. of Math. (2) 88 (1968), 1–34; MR0240167]. The main result is, if A is a k-algebra and if φ and ψ are k-derivations of A such that [φ,ψ]=λφ for some λ∈k, then φ∪ψ may be integrated to yield a deformation of A. An explicit formula for the derivation is given, as are a number of examples.

It is related to Jordanian R-matrix which is a special case of an explicit family of matrices introduced by D. Gurevich in mid 80-s and quoted by Lyubashenko.

Analytic version

P. Stachura, On the quantum ‘ax+b’ group, J. Geom. Phys., 73 (2013) 125-149 doi
P. Bieliavsky, V. Gayral, S. Neshveyev, L. Tuset, Quantization of subgroups of the affine group, J. Funct. Analysis 280:4, (2021) 108844, doi

Generalizations are for deformations of Yangians

Alexander Stolin, Petr P. Kulish, New rational solutions of Yang-Baxter equation and deformed Yangians, Quantum groups and integrable systems, II (Prague, 1996). Czechoslovak J. Phys. 47:1 (1997) 123-129 MR1456489 doi
P. P. Kulish, A. A. Stolin, Deformed Yangians and integrable models, Quantum groups and integrable systems, II (Prague, 1997). Czechoslovak J. Phys. 47 (1997), no. 12, 1207-1212 MR1608809 doi
S. M. Khoroshkin, A. A. Stolin, V. N. Tolstoy, Deformation of Yangian Y(sl2), Comm. Algebra 26:4 (1998) 1041-1055 doi
S. M. Khoroshkin, A. A. Stolin, V. N. Tolstoy,
$q$ -power function over q-commuting variables and deformed XXX and XXZ chains, Phys. Atomic Nuclei 64 (2001), no. 12, 2173-2178 MR1883225; translated from Yadernaya Fiz. 64 (2001), no. 12, 2262–2267
Khoroshkin, S. M.; Pop, I. I.; Samsonov, M. E.; Stolin, A. A.; Tolstoy, V. N. On some Lie bialgebra structures on polynomial algebras and their quantization, Comm. Math. Phys. 282 (2008), no. 3, 625-662 MR2426139