CHAPTER 1
Yangian for gl
N
As we pointed out in the Preface, although the discovery of the Yangians was
motivated by the quantum inverse scattering theory, the Yangian defining relations
can be "observed" from a purely algebraic viewpoint. We regard (0.3) as an alge-
braic motivation for the definition of the Yangian for the general linear Lie algebra
$lN. We demonstrate that the defining relations can be written in a matrix form
which provides a starting point for special algebraic techniques to study the Yan-
gian structure. These techniques play an essential role in the construction of the
quantum determinant and description of the center of the Yangian.
1.1. Defining relations
DEFINITION
1.1.1. The Yangian for giN is a unital associative algebra over
C with countably many generators t\-\ t\, ,.. . where i,j = l,...,iV, and the
defining relations
/-, i\ f i H l ) i(«)i \Ar) ,(s+lh _ ,(r)(s) As),(r)
V1'1) llij Zkl \ llij lkl \—lkjlil lkjlil '
where r, s = 0 , 1 , . . . and t\j = Sij. This algebra is denoted by Y(glfy).
Introducing the formal generating series
(1.2) Uj{u) = Sij + t g V
1
+
t^u~2
+ -.-G Y(QlN)[[u-%
we can write (1.1) in the form
(1.3) (u - v) [Uj(u), tki(v)\ = tkj(u) tu(v) - tkj(v) tu(u);
the indeterminates u and v are considered to be commuting with each other and
with the elements of the Yangian.
The following is an equivalent form of the defining relations of the algebra
PROPOSITION
1.1.2. The system of relations (1.1) is equivalent to the system
min{r,s}
(1.4) [$,$}= £
(tfrX*-a)-%+'-a)ti-1))-
a=l
PROOF .
Observe that the multiplication of both sides of (1.3) by the formal
series Y^%Lou~P~lvP y i e ^ s a n equivalent relation
oo
[Uj(u),tki(v)] = (tkj(u)tu(v) -tkj(v)tu(u)j ^ju~p~1vp.
p=0
http://dx.doi.org/10.1090/surv/143/01
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