Quantum Matrix Algebras via compatible R-matrices

The notion of compatible R-matrices was introduced by Isaev-Ogievetsky-Pyatov.
It enables one to introduced a large family of new quantum algebras and the
corresponding Bethe subalgebras. Also, this notion  is useful for defining
the so-called half-quantum algebras (Isaev-Ogievetsky) which generalize the
notions of the Manin and q-Manin matrices. In my talk I plan to introduce  all
these  objects and to exhibit the role of  the mentioned algebras in
studying the  Yangians and their different generalizations.