2 1. YANGIAN FO R glN

Taking the coefficients of u~rv~s on both sides gives

r

U(r)

t

(sh

=

V^ ft(a-l)t(r+s-a) _ ^(r+s_a)^(a-l)\

L

%J

i

Kb

- I / j \

KJ it KJ 11

I

a=l

This agrees with (1.4) in the case r ^ s. Finally, if r s observe that

£

(tS-1)4p+-0)-C*"0)4a-1,)=°.

completing the proof. D

We will often use formal series to define or describe maps between various

algebras. If a(u) and b(u) are formal power series in

u~x

with coefficients in certain

algebras, then assignments of the type a{u) i-» b(u) are understood in the sense

that every coefficient of a(u) is mapped to the corresponding coefficient of b(u).

Many applications of the Yangian are based on the following simple observation.

PROPOSITION

1.1.3. The assignment

(1.5) 7TJV : Uj(u) H- 5ij + EijU-1

defines a surjective homomorphism Y(giN) — JJ(giN). Moreover, the assignment

(1.6)

Ei3 -

tg

defines an embedding U(glN) ^- Y ^ l ^ ) .

PROOF .

By Definition 1.1.1, we need to verify the equality

(u - v)[Eij,

Ekilu^v'1

= (Skj + EkjU'^iSu + £ ; ^

- 1

) - (4 j + EkjV-^iSu +

Sttw"1).

But this clearly holds due to the commutation relations in glN, which proves the

first part of the proposition. In order to prove the second part, put r = s = 1 in

(1.4) which gives

\Pij -lkl \ — °kjlU °illkj '

Thus, (1.6) is an algebra homomorphism. Its injectivity follows from the observation

that by applying (1.6) and then (1.5), we get the identity map on XJ(glN). •

The homomorphism 7T/v is called the evaluation homomorphism. By its virtue,

any representation of the Lie algebra giN can be regarded as a representation of

Y(giN). Any irreducible representation of giN remains irreducible over Y(giN), by

surjectivity of 7TJV-

Note that an alternative form of the defining relations (1.1) is also common in

the literature. It is obtained by swapping the indices r and s on the right-hand side

of (1.1); see also Remark 1.2.3 below.

1.2. Matrix form of the defining relations

Introduce the N x N matrix T(u) whose ij-th entry is the series Uj(u). One

can regard T(u) as an element of the algebra EndC ^ 0

Y(glAr)[[^_1]].

Then

N

(1.7) T(u)= Y^ eij®Uj(u),

i5J = l