The Gaudin model based on the sl2-invariant r-matrix with an extra Jordanian term depending on the spectral parameters is considered. The appropriate creation operators defining the Bethe states of the system are constructed through a recurrence relation. The commutation relations between the generating function t(λ) of the integrals of motion and the creation operators are calculated and therefore the algebraic Bethe ansatz is fully implemented. The energy spectrum as well as the corresponding Bethe equations of the system coincide with the ones of the sl2-invariant Gaudin model. As opposed to the sl2-invariant case, the operator t(λ) and the Gaudin Hamiltonians are not Hermitian. Finally, the inner products and norms of the Bethe states are studied.

2.
M.
Gaudin
,
La Fonction d’Onde de Bethe
(
Masson
,
Paris
,
1983
), Chap. 13.
3.
E. K.
Sklyanin
,
L. A.
Takhtajan
, and
L. D.
Faddeev
,
Theor. Math. Phys.
40
,
688
(
1980
).
4.
L. A.
Takhtajan
and
L. D.
Faddeev
,
Russ. Math. Surveys
34
,
11
(
1979
).
5.
P. P.
Kulish
and
E. K.
Sklyanin
,
Lect. Notes Phys.
151
,
61
(
1982
).
6.
V. E.
Korepin
,
N. M.
Bogoliubov
, and
A. G.
Izergin
,
Quantum Inverse Scattering Method and Correlation Functions
(
Cambridge University Press
,
Cambridge, England
,
1993
).
7.
M. A.
Semenov-Tian-Shansky
, “
Quantum and classical integrable systems
,” in
Integrability of Nonlinear Systems
(Pondicherry, 1996),
Lecture Notes in Physics Vol. 495
(
Springer
,
Berlin
,
1997
), pp.
314
.
8.
L. D.
Faddeev
, “
How the algebraic Bethe ansatz works for integrable models
,” in
Quantum Symmetries/Symetries Quantiques
, in Proceedings of the Les Houches Summer School, Session LXIV, edited by
A.
Connes
,
K.
Gawedzki
, and
J.
Zinn-Justin
, (
North-Holland
,
Amsterdam
,
1998
), pp.
149
.
9.
E. K.
Sklyanin
,
J. Sov. Math.
47
,
2473
(
1989
).
10.
E. K.
Sklyanin
and
T.
Takebe
,
Phys. Lett. A
219
,
217
(
1996
).
11.
E. K.
Sklyanin
,
Lett. Math. Phys.
47
,
275
(
1999
).
12.
H. M.
Babujian
and
R.
Flume
,
Mod. Phys. Lett. A
9
,
2029
(
1994
).
13.
B.
Feigin
,
E.
Frenkel
, and
N.
Reshetikhin
,
Commun. Math. Phys.
166
,
27
(
1994
).
14.
N.
Reshetikhin
and
A.
Varchenko
, “
Quasi-classical asymptotics of solutions to the KZ equations
,” in
Geometry, Topology and Physics, Conference Proceedings and Lecture Notes in Geometry and Topology
(
International Press
,
Cambridge, MA
,
1995
), pp.
293
.
15.
P. P.
Kulish
and
N.
Manojlović
,
J. Math. Phys.
42
(
10
),
4757
(
2001
).
16.
P. P.
Kulish
and
N.
Manojlović
,
J. Math. Phys.
44
(
2
),
676
(
2003
).
17.
A.
Lima-Santos
and
W.
Utiel
,
Nucl. Phys. B
600
,
512
(
2001
).
18.
T.
Skrypnyk
,
J. Math. Phys.
50
,
033504
(
2009
).
19.
A. A.
Belavin
and
V. G.
Drinfeld
,
Funct. Anal. Appl.
16
(
3
),
159
(
1982
)
A. A.
Belavin
and
V. G.
Drinfeld
, [
Funkc. Anal. Priloz.
16
(
3
),
1
(
1982
) (in Russian)].
20.
A. A.
Stolin
,
Math. Scand.
69
,
81
(
1991
).
21.
A. A.
Stolin
,
Math. Scand.
69
,
57
(
1991
).
22.
P. P.
Kulish
and
A. A.
Stolin
,
Czech. J. Phys.
12
,
1207
(
1997
).
23.
S.
Khoroshkin
,
A.
Stolin
and
V.
Tolstoy
,
Commun. Algebra
26
,
1041
(
1998
).
24.
P. P.
Kulish
, in
Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years after NEEDS’79, Gallipoli, 1999
(
World Scientific
,
River Edge, NJ
,
2000
), pp.
304
.
25.
V.
Lyakhovsky
,
A.
Mirolubov
, and
M.
del Olmo
,
J. Phys. A
34
,
1467
(
2001
).
26.
N. Cirilo
António
and
N.
Manojlović
,
J. Math. Phys.
46
(
10
),
102701
(
2005
).
27.
P. P.
Kulish
, “
Twisted sl(2) Gaudin model
,” preprint PDMI 08/2002.
28.
P.
Kulish
,
N.
Manojlović
,
M.
Samsonov
, and
A.
Stolin
,
Proceedings of the Estonian Academy of Sciences
59
(
4
),
326
(
2010
).
29.
Y.
Kosmann-Schwarzbach
,
Groups and Symmetries. From Finite Groups to Lie Groups
(
Springer-Verlag
,
Berlin
,
2009
).
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