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128 A.P. Isaev et al. / Nuclear Physics B 904 (2016) 124–147
L
a
b
(u) =
∞
k=0
(L
(k)
)
a
b
u
k
,L
(0)
=I, (3.12)
where u is called spectral parameter. The function L(u) is called L-operator and the defining
relations of Y(G) are represented [6] as RLL-relations
R
a
1
a
2
b
1
b
2
(u −v)L
b
1
c
1
(u)L
b
2
c
2
(v) =L
a
2
b
2
(v)L
a
1
b
1
(u)R
b
1
b
2
c
1
c
2
(u −v) ⇔
R
12
(u −v)L
1
(u) L
2
(v) =L
2
(v) L
1
(u) R
12
(u −v) . (3.13)
Here R
a
1
a
2
b
1
b
2
(u −v) is the Yang–Baxter R-matrix (2.8) and in the second line of (3.13) we use the
standard matrix notations of [13]. Recall that in this notations the matrices R
12
(u) (2.8) and P
12
,
K
12
(2.9) are operators in V
f
⊗V
f
, where V
f
denotes a n-dimensional vector space, while L
1
and L
2
are matrices ||L
a
b
|| which act nontrivially only in the first and in the second factors of
V
f
⊗V
f
, respectively, and have algebra valued matrix elements.
The defining relations (3.13) are homogeneous in L and one can do in (3.12), (3.13) the
redefinition L(u) → f(
u) L(u + b
0
) with any scalar function f(u) = 1 + b
1
/u + b
2
/u
2
+
..., where b
i
are parameters. Now it is clear that the Yangian (2.8), (3.13) possesses the set of
automorphisms
L(u) →
(u −a)
k
u
k
L(u) , (k =1, 2,...),
where a is a constant (in general a is a central element in Y(G)). In particular for k = 1we obtain
that the generators L
(j)
are transforming as
L
(1)
→L
(1)
−aI
n
,L
(2)
→L
(2)
−aL
(1)
,L
(3)
→L
(3)
−aL
(2)
, ..., (3.14)
Taking a =
1
n
Tr(L
(1)
) one can fix L
(1)
such that Tr(L
(1)
) = 0 (below we show that Tr(L
(1)
) is
central element in Y(G)).
We represent the fundamental R-matrix (2.8) in the concise form
R
12
(u) =u(u +β)I +(u +β)P
12
−uK
12
, (3.15)
where β = (
n
2
−). Further, after the shift of the spectral parameter u →u −v, we write it as
1
u
2
v
2
R(u −v) =(
1
v
−
1
u
)(
1
v
−
1
u
+
β
uv
) −(
1
uv
2
−
1
u
2
v
+
β
u
2
v
2
)P −(
1
uv
2
−
1
u
2
v
)K. (3.16)
Then we substitute (3.16) and (3.12) into (3.13) and obtain, as a coefficient at u
−k
v
−j
, the explicit
quadratic relations for the generators (L
(k)
)
a
b
of the Yangians Y(G)
[L
(k)
1
,L
(j−2)
2
]−2[L
(k−1)
1
,L
(j−1)
2
]+[L
(k−2)
1
,L
(j)
2
]+
+β([L
(k−1)
1
,L
(j−2)
2
]−[L
(k−2)
1
,L
(j−1)
2
]) +
+P
L
(k−1)
1
L
(j−2)
2
−L
(k−2)
1
L
(j−1)
2
+βL
(k−2)
1
L
(j−2)
2
−
−
L
(j−2)
2
L
(k−1)
1
−L
(j−1)
2
L
(k−2)
1
+βL
(j−2)
2
L
(k−2)
1
P +
+
K(L
(k−2)
1
L
(j−1)
2
−L
(k−1)
1
L
(j−2)
2
) −(L
(j−1)
2
L
(k−2)
1
−L
(j−2)
2
L
(k−1)
1
)K
=0 ,
(3.17)
where the operators K, P are given in (2.9), =+1for G = so(n) and =−1for G = sp(2m).
For the special value k =1we obtain from (3.17) the set of relations