nLab RTT bialgebra

Idea

Given a solution of the matrix quantum Yang-Baxter equation $R\in End(V\otimes V)$ , which can be viewed as a certain endomorphism of $V\otimes V$ for a finite-dimensional vector space $V$ , one defines the RTT bialgebra $A(R)$ which is as an algebra free associative algebra $F$ on $n^2$ generators $T^i_j$ (these generators could be viewed as elements of the dual $End^*(V)$ ) forming a matrix $T = (T^i_j)\in F\otimes End(V)$ modulo the relations packed in a matrix form as

R (T\otimes Id) (Id\otimes T) = (Id\otimes T) (T\otimes Id) R,

where the tensor entries correspond to $V$ factors only (tensor factor in $F$ is not shown). The coalgebra structure on $A(R)$ is given on generators by $\Delta T^i_j = \sum_k T^i_k\otimes T^k_j$ and $\epsilon T^i_j = \delta^i_j$ (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 $Fun(SO_q(n))$ and alike. This is called the FRT construction or FRT approach to quantum groups.

Shahn Majid complemented this with another algebra $B(R)$ which is paired with $A(R)$ but the pairing is degenerate. Then in an minimal way one finds biideals in $A(R)$ and $B(R)$ 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.

Examples and variants

quantum linear groups
a class of generalizations appear as face algebras

Literature

FRT construction is introduced in

N. Yu. Reshetikhin, L. A. Takhtajan, L. D. Faddeev, Quantization of Lie groups and Lie algebras, Algebra i analiz 1, 178 (1989) (Russian), English transl. Leningrad Math. J. 1
L. D. Faddeev, N. Yu. Reshetikhin, L. A. Takhtajan, Quantum Groups. Braid Group, Knot Theory and Statistical Mechanics, in: Adv. Ser. Math. Phys. 9, World Sci. 1989, pp. 97–110
Shahn Majid, Quasitriangular Hopf algebras and Yang–Baxter equation, Intern. J. Mod. Phys. A 05, No. 01, pp. 1–91 (1990) doi

A slight generalization of the original procedure

Jacob Towber, Sara Westreich, Hopf algebras constructed by the FRT-construction, J. Pure & Applied Algebra 213:5 (2009) 772–782 doi

A relation between reflection equation algebras and FRT algebras is discussed in

Joseph Donin, Andrey Mudrov, Reflection equation- and FRT-type algebras, Czech J Phys 52, 1201–1206 (2002) doi