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