The Yangian double DY(gN) is introduced for the classical types of gN=o2n+1, sp2n, and o2n. Via the Gauss decomposition of the generator matrix, the Yangian double is given the Drinfeld presentation. In addition, bosonization of level 1 realizations for the Yangian double DY(gN) of non-simply laced types are explicitly constructed.

1.
V. G.
Drinfeld
, “
A new realization of Yangians and of quantum affine algebras
,”
Dokl. Akad. Nauk SSSR
296
,
13
17
(
1987
) (Russian) [Soviet Math. Dokl. 36, 212-216 (1988)].
2.
V.
Chari
and
A.
Pressley
,
A Guide to Quantum Groups
(
Cambridge University Press
,
Cambridge
,
1994
).
3.
I. V.
Cherednik
, “
On the method of constructing factorized S-matrices in elementary functions
,”
Theor. Math. Phys.
43
,
117
119
(
1980
) (Russian).
4.
I. B.
Frenkel
and
N. Y.
Reshetikhin
, “
Quantum affine algebras and holonomic difference equations
,”
Commun. Math. Phys.
146
,
1
60
(
1992
).
5.
E.
Frenkel
and
E.
Mukhin
, “
Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras
,”
Commun. Math. Phys.
216
,
23
57
(
2001
).
6.
E.
Frenkel
and
N.
Reshetikhin
, “
Quantum affine algebras and deformations of the Virasoro and W -algebras
,”
Commun. Math. Phys.
178
,
237
264
(
1996
).
7.
N. Y.
Reshetikhin
, “
Quasitriangular Hopf algebras and invariants of links
,”
Algebra Anal.
1
,
169
188
(
1989
) (Russian) [Leningrad Math. J. 1, 491–513 (1990)].
8.
N.
Reshetikhin
and
V. G.
Turaev
, “
Invariants of 3-manifolds via link polynomials and quantum groups
,”
Invent. Math.
103
,
547
597
(
1991
).
9.
M.
Jimbo
, “
A q-difference analogue of U(g) and the Yang-Baxter equation
,”
Lett. Math. Phys.
10
,
63
69
(
1985
).
10.
P. P.
Kulish
and
N. Y.
Reshetikhin
, “
The quantum linear problem for the sine-Gordon equation and higher representations
,”
Zap. Nauch. Sem. LOMI
101
,
101
110
(
1981
) (Russian), Questions in quantum field theory and statistical physics, 2.
11.
M.
Jimbo
, “
Quantum R-matrix for the generalized Toda system
,”
Commun. Math. Phys.
102
,
537
547
(
1986
).
12.
A.
Molev
,
Yangians and Classical Lie Algebras
, Mathematical Surveys and Monographs Vol. 143 (
American Mathematical Society
,
Providence, RI
,
2007
).
13.
V. O.
Tarasov
, “
Structures of L-operators for the R-matrix of the XXZ-model
,”
Theor. Math. Phys.
61
,
1065
1072
(
1984
).
14.
V. G.
Drinfeld
, “
Hopf algebras and the quantum Yang-Baxter equation
,”
Dokl. Akad. Nauk SSSR
283
(
5
),
10601
1064
(
1985
) [Sov. Math. Dokl. 32, 254-258 (1985)].
15.
N.
Guay
,
V.
Regelskis
, and
C.
Wendlandt
, “
Vertex representations for Yangians of Kac-Moody algebras
,”
J. Éc. Polytech. Math.
6
,
665
706
(
2019
).
16.
A.
Tsymbaliuk
, “
PBW theorems and shuffle realizations for Uv(Lsln), Uv1,v2(Lsln), Uv(Lsl(m|n))
,” arXiv:1808.09536.
17.
H.
Zhang
, “
Representations of quantum affine superalgebras
,”
Math. Z.
278
,
663
703
(
2014
).
18.
J.
Brundan
and
A.
Kleshchev
, “
Parabolic presentations of the Yangian Y(gln)
,”
Commun. Math. Phys.
254
,
191
220
(
2005
).
19.
S. M.
Khoroshkin
and
V. N.
Tolstoy
, “
Yangian double
,”
Lett. Math. Phys.
36
,
373
402
(
1996
).
20.
X. M.
Ding
,
B. Y.
Hou
,
B.
Yuan Hou
, and
L.
Zhao
, “
Free boson realization of DYh(glN)k
,”
J. Math. Phys.
39
,
2273
(
1998
).
21.
D.
Arnaudon
,
A.
Molev
, and
E.
Ragoucy
, “
On the R-matrix realization of Yangians and their representations
,”
Ann. Henri Poincaré
7
,
1269
1325
(
2006
).
22.
N.
Jing
,
M.
Liu
, and
A.
Molev
, “
Isomorphism between the R-matrix and Drinfeld presentations of Yangian in types B, C and D
,”
Commun. Math. Phys.
361
,
827
872
(
2018
).
23.
I. B.
Frenkel
and
N.
Jing
, “
Vertex representations of quantum affine algebras
,”
Proc. Natl. Acad. Sci. U. S. A.
85
,
9373
9377
(
1988
).
24.
D.
Bernard
, “
Vertex operator representations of the quantum affine algebra Uq(Br(1))
,”
Lett. Math. Phys.
17
,
239
245
(
1989
).
25.
N.
Jing
and
Y.
Koyama
, “
Vertex operators of admissible modules of Uq(Cn(1))
,”
J. Algebra
205
,
294
316
(
1998
).
26.
N.
Jing
,
Y.
Koyama
, and
K. C.
Misra
, “
Bosonic realizations of Uq(Cn(1))
,”
J. Algebra
200
,
155
172
(
1998
).
27.
K.
Iohara
, “
Bosonic representations of Yangian double DY(g) with g=glN,slN
,”
J. Phys. A: Math. Gen.
29
,
4593
4621
(
1996
).
28.
S.
Kozic
, “
Commutative operators for double Yangian DY(sln)
,”
Glas. Mat. Ser. III
53
,
97
113
(
2018
).
29.
Y.
Xu
and
R. B.
Zhang
, “
Drinfeld realizations and vertex operator representations of quantum affine superalgebras
,” arXiv:1802.09702.
30.
A. B.
Zamolodchikov
and
Al. B.
Zamolodchikov
, “
Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models
,”
Ann. Phys.
120
,
253
291
(
1979
).
31.
C.
Kassel
,
Quantum Groups
, Graduate Texts in Mathematics Vol. 155 (
Springer-Verlag
,
New York
,
1995
).
32.
N.
Jing
,
S.
Kožić
,
A.
Molev
, and
F.
Yang
, “
Center of the quantum affine vertex algebra in type A
,”
J. Algebra
496
,
138
186
(
2018
).
33.
I. M.
Gelfand
and
V. S.
Retakh
, “
Determinants of matrices over noncommutative rings
,”
Funct. Anal. Appl.
25
,
91
102
(
1991
).
34.
N.
Guay
,
V.
Regelskis
, and
C.
Wendlandt
, “
Equivalences between three presentations of orthogonal and symplectic Yangians
,”
Lett. Math. Phys.
109
,
327
379
(
2019
).
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