We construct a universal solution of the generalized coboundary equation in the case of quantum affine algebras, which is an extension of our previous work to |$U_q(A^{(1)}_r)$|Uq(Ar(1)). This universal solution has a simple Gauss decomposition which is constructed using Sevostyanov's characters of twisted quantum Borel algebras. We show that in the evaluation representations it gives a vertex-face transformation between a vertex type solution and a face type solution of the quantum dynamical Yang-Baxter equation. In particular, in the evaluation representation of |$U_q(A_1^{(1)})$|Uq(A1(1)), it gives Baxter's well-known transformation between the 8-vertex model and the interaction-round-faces (IRF) height model.

1.
G. E.
Andrews
,
R. J.
Baxter
, and
P. J.
Forrester
, “
Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities
,”
J. Stat. Phys.
35
,
193
(
1984
).
2.
D.
Arnaudon
,
E.
Buffenoir
,
E.
Ragoucy
, and
Ph.
Roche
, “
Universal solutions of quantum dynamical Yang-Baxter equations
,”
Lett. Math. Phys.
44
,
201
(
1998
);
3.
O.
Babelon
, “
Universal exchange algebra for Bloch waves and Liouville theory
,”
Commun. Math. Phys.
139
,
619
643
(
1991
).
4.
J.
Balog
,
L.
Dabrowski
, and
L.
Feher
, “
Classical r-matrix and exchange algebra in WZNW and Toda theories
,”
Phys. Lett. B
244
(
2
),
227
(
1990
).
5.
R. J.
Baxter
,
Exactly Solved Models in Statistical Physics
(
Academic
,
London
,
1986
).
6.
R. J.
Baxter
, “
Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. II. Equivalence to a generalized ice-type lattice
,”
Ann. Phys.
76
,
25
47
(
1973
).
7.
J.
Beck
, “
Convex bases of PBW type for quantum affine algebras
,”
Commun. Math. Phys.
165
,
193
200
(
1994
);
8.
A. A.
Belavin
, “
Dynamical symmetries of integrable quantum systems
,”
Nucl. Phys. B
180
,
189
200
(
1981
).
9.
A. A.
Belavin
and
V. G.
Drinfeld
, “
Triangle equations and simple Lie algebras
,”
Sov. Sci. Rev., Sect. C, Math. Phys. Rev.
4
,
93
165
(
1984
).
10.
E.
Buffenoir
,
Ph.
Roche
, and
V.
Terras
, “
Quantum dynamical coboundary equation for finite dimensional simple Lie algebras
,”
Adv. Math.
214
,
181
229
(
2007
);
11.
V.
Chari
and
A.
Pressley
, “
Quantum affine algebras and their representations
,”
Canadian Math. Soc. Conf. Proc.
16
,
59
78
(
1995
);
12.
I.
Damiani
, “
La R-matrice pour les algèbres quantiques de type affine non tordu (The R-matrix for non-twisted affine quantum algebras)
,”
Ann. Sci. Ec. Normale Super.
31
(
4
),
493
523
(
1998
).
13.
P.
Etingof
, “
On the dynamical Yang-Baxter equation
,” in
Proceedings of the International Congress of Mathematicians, Beijing, 2002
;
14.
P.
Etingof
, “
Whittaker functions on quantum groups and q-deformed Toda operators
,”
Am. Math. Soc. Transl. Ser. 2
194
,
9
25
(
1999
);
15.
P.
Etingof
and
D.
Kazhdan
, “
Quantization of Lie bialgebras III
,”
Sel. Math., New Ser.
4
,
233
269
(
1998
);
16.
P.
Etingof
and
D.
Nikshych
, “
Vertex-IRF transformations and quantization of dynamical r-matrices
,”
Math. Res. Lett.
8
(
3
),
331
346
(
2001
);
17.
P.
Etingof
and
O.
Schiffmann
, “
Twisted traces of intertwiners and quantum dynamical R-matrices corresponding to generalized Belavin-Drinfeld triples
,”
Commun. Math. Phys.
205
,
19
52
(
1999
);
18.
P.
Etingof
and
O.
Schiffmann
, “
Lectures on the dynamical Yang-Baxter equation
,” Quantum Groups and Lie Theory (Durham, 1999),
London Math. Soc. Lecture Notes
290
,
89
129
(
2001
);
19.
P.
Etingof
and
A.
Varchenko
, “
Geometry and classification of solutions of the classical dynamical Yang-Baxter equation
,”
Commun. Math. Phys.
192
(
1
),
77
120
(
1998
);
20.
P.
Etingof
,
T.
Schedler
, and
O.
Schiffmann
, “
Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras
,”
J. Am. Math. Soc.
13
,
595
609
(
2000
);
21.
P.
Etingof
,
O.
Schiffmann
, and
A.
Varchenko
, “
Traces of intertwiners for quantum groups and difference equations, II
,”
Lett. Math. Phys.
62
(
2
),
143
158
(
2002
);
22.
G.
Felder
, “
Elliptic quantum groups
,” in
Proceedings of the International Congress of Mathematical Physics, Paris, 1994
(International,
1995
), pp.
211
218
.
23.
G.
Felder
and
A.
Varchenko
, “
Algebraic Bethe ansatz for the elliptic quantum groupEτ,η(sl2)
,”
Nucl. Phys. B
480
,
485
503
(
1996
);
24.
M.
Jimbo
,
H.
Konno
,
S.
Odake
, and
J.
Shiraishi
, “
Quasi-Hopf twistors for elliptic quantum groups
,”
Transform. Groups
4
(
4
),
303
327
(
1999
);
25.
M.
Jimbo
,
T.
Miwa
, and
M.
Okado
, “
Solvable lattice models related to the vector representation of classical simple Lie algebras
,”
Commun. Math. Phys.
116
,
507
525
(
1988
).
26.
M.
Jimbo
,
T.
Miwa
, and
M.
Okado
, “
Solvable lattice models whose states are dominant integral weights of |$A_r^{(1)}$|Ar(1)
,”
Lett. Math. Phys.
14
,
123
(
1987
).
27.
S.
Khoroshkin
,
A. A.
Stolin
, and
V. N.
Tolstoy
, “
Gauss decomposition of trigonometric R-matrices
,”
Mod. Phys. Lett.
A10
,
1375
1392
(
1995
);
28.
E.
Karolinsky
,
A.
Stolin
, and
V.
Tarasov
, “
Equivariant quantization of Poisson homogeneous spaces and Kostant's problem
,” e-print arXiv:math.QA/0908.0349.
29.
S.
Khoroshkin
and
V. N.
Tolstoy
, “
Universal R-matrix for quantized (super) algebras
,”
Commun. Math. Phys.
141
,
559
(
1991
).
30.
S.
Khoroshkin
and
V. N.
Tolstoy
, “
The universal R-matrix for quantum non-twisted affine Lie algebras
,”
Funkc. Anal. Priloz.
26
(
1
),
85
88
(
1992
).
31.
O.
Schiffmann
, “
On classification of dynamical r-matrices
,”
Math. Res. Lett.
5
,
13
30
(
1998
);
32.
A.
Sevostyanov
, “
Quantum deformation of Whittaker modules and Toda lattice
,”
Duke Math. J.
105
(
2
),
211
238
(
2000
);
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