Given a solution of the matrix quantum Yang-Baxter equation , which can be viewed as a certain endomorphism of for a finite-dimensional vector space , one defines the RTT bialgebra which is as an algebra free associative algebra on generators (these generators could be viewed as elements of the dual ) forming a matrix modulo the relations packed in a matrix form as
where the tensor entries correspond to factors only (tensor factor in is not shown). The coalgebra structure on is given on generators by and (hence it is an example of a matrix bialgebra). These RTT relations come from quantum inverse scattering method where a version with spectral parameter appears; Yangians and some other related structures satisfy versions of RTT equations.
To get Hopf algebras one further quotients the RTT algebra by further relations listed for classical series by Faddeev, Reshetikhin and Takhtajan, so obtain quantum function algebras like and alike. This is called the FRT construction or FRT approach to quantum groups.
Shahn Majid complemented this with another algebra which is paired with but the pairing is degenerate. Then in an minimal way one finds biideals in and such that the quotients become Hopf algebras in a nondegenerate pairing, which may be viewed as the quantum groups of the function and of the universal enveloping algebra type.
FRT construction is introduced in
A slight generalization of the original procedure
A relation between reflection equation algebras and FRT algebras is discussed in
Last revised on October 1, 2024 at 13:55:35. See the history of this page for a list of all contributions to it.