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On generators and defining relations of Yangians
JOURNAI
Journal of Geometry and Physics 12 (1993) 1—11 North-Holland
OF
GEOMETRYAND
PHYSICS
On generators and defining relations of Yangians S.Z. Levendorskii Budennovskii 94, ap. 64, Rostov-on-Don, 344018 Russian Federation Received 3 August 1992
So far, two equivalent definitions ofYangians are known. The first one involves a finite set of generators and relations but some of the latter are too complicated, while the second one exploits infinite sequences ofgenerators and relations of more convenient form. It is shown in this paper that only a finite part of thesesequences suffices to define the structure ofYangians. In addition, we construct generators for the “Cartan part” of Yangian, which enjoy properties more similar to those ofgenerators ofa Cartan subalgebra of a simple Lie algebra.
Keywords: Yangian, generators, defining relations 1991 MSC: 17B37, 81 R 50, 82B23
Drinfeld [3] introduced the notion of a quantum group, the most important examples being quantized enveloping algebras and Yangians. Both these types of quantum group are closely related to solutions of the quantum Yang—Baxter equation, hence thorough study of them is of interest. So far, two equivalent definitions of Yangians are known. The first one, given by Drinfeld [3], involves a finite set of generators and relations but some of the latter are too complicated. Later, Drinfeld [4] (see also ref. [51) suggested a more convenient set of generators and relationsbut this set is infinite. In the main theorem of the present paper we show that only a finite part ofthe last set suffices to define the structure of Yangians. In addition, we construct the generators of the “Cartan part” of Yangian, which enjoy properties similar to those of the generators ofa Cartan subalgebra ofa simple Lie algebra. It should be noted that so far Yangians are introduced for simple Lie algebras only whereas it seems reasonable to introduce them in the case of affine Lie algebras as well. Still, it can be shown that one cannot expect simple infinite sequences of relations similar to those of ref. [4] to exist in the case of affine Lie algebras. Therefore it is important to be able to handle Yangians by using only a finite set ofthe relations introduced in ref. [4] (or their analogs). The results on “affine Yangians” will appear elsewhere. For various results on the representation theory ofYangians and their relations 0393-0440/1993/$6.OO © 1993
Elsevier Science Publishers B.V. All rights reserved
S. Z. Levendorskii / On generators and defining relations of Yanglans
2
with the theory of integrable quantum systems, see Drinfeld [3—51 and Chari and Pressley [1,21. 1. Main theorems Let g be a finite-dimensional complex simple Lie algebra, let g = n ±~ ~ n be a triangular decomposition of g, let A~be the set of positive roots and {a1 a~},n=rank g, the corresponding set of simple roots, and let (a0) be the Cartan matrix of g. Fix a non-zero invariant symmetric bilinear form ( ) on g, and for each positive root a of g, choose root vectors x~in the ±a root spaces such that (x~, x~)=1.Thenset h~=[x~,x~],hj—ha,,x~ =X~. ,
Definition 1.1. Denote by Y(g) the algebra over C with generators x~,x~’,h,0,
~
(1 ~i~n) and defining relations
[/~~,/~~1=O, [/~~,Ii~~]=O, [
,
[/~i,~i1=O;
(1.1)
xJ~]= ±( a1, (1.2)
[x~,x~]=ô~0,
[x~,x~]=(i1+~ii~);
~
(1.3) (1.4)
where b0= (a1, a1)/2; [x~, [x~,..., [x~,x1fl~]]=O, (1
—
ij,
(1.5)
a0 commutators); [~1,x~],x~fl+[x~,
[/~,,x~]]=O.
(1.6)
Set for keZ~, xfk±l=±(aI,al)[h,l,xlkl,
hlk=[x~,x,~}.
Theorem 1.2. The algebra ?(g) is isomorphic to the Yangian Y(g), the isomorph-
ism being defined by t
—
1.
In other words, in
~
EY(g)
“tik .3
g) thefollowing relations hold: (1.7)
3
S. Z. Levendorskii/ On generators anddefining relations of Yan glans
~
;
(1.8) (1.9) (1.10) (1.11)
~ ~ ~
[x~
12>,
[x~L(,~), xJ~’]“]
...,
1=0
,
(1.12)
for i j, where m = 1 a0 and the sum is taken over all the permutations a of{ 1, m}. Relations(1.7)—(1.12)implyrelations (1.1)—(1.6). —
Remark 1.3. Relations (1.7)—(1.12) are just the relations introduced by Drinfeld in ref. [4]. The proof of theorem 1.2 is based on the followinglemma. To state it we introduce the unital algebra A with the generators h~,~ (Je7L+) and the defining relations [hk,h~]=0;
(1.13) (1.14)
where h1
=
I and yelR is independent of k, 1. Set h(t)=
~
hkt~~ x(’r)= ,
k~—I
~ X~Tt 1 t~O
and define h’kEA (kEl~)by the equality ~ hkt~~=lnh(t).
(1.15)
k~O
Here the right-hand side is the expansion in powers of t in the vicinity of t=
+cc.
Lemma 1.4. Let (1.13) holdforj~
[Rk,xtl=2yxk+l+2
~ O~sek—2 k+s even
~.+ 1
C~~1x1~.
(1.16)
Proof Due to (1.15), Ek hk is a polynomial in h0, h1, ..., hkI. Hence, while deriving (1.16), we may and shall assume that (1.13), (1.14) hold for all k, lel~.Now (1.14) can be rewritten as —
4
5. Z. Levendorskit / On generators and defining relations of Yan glans
h(t)x(t) =x(r)h(t)
t—i+p t—t—y
therefore [R(t), x(r)]
=
[ln h(t), x(r)] =ln tt+Yx(T)
(1.17)
Simple calculations give in
=ln( 1 —t~ (r—y) )—ln( 1 =
—
—1~ (~+y))
~
k~1
t~ = k~1
k
~
C~t’y’((—l)’~+l);
O~j~k
hence, ( 1. 17) gives ( 1. 16 ). Corollary 1.5. Let (1.8) hold for leZ~, 1 ~
~ ~t~=ln(1+ k>0
h1~t~~)
(118)
k~O
Thenfork~p,le7L~,1~i,j~n, [~k,
x~I
=
± (a1, aI)xJ~+k ±
~
21
O~ss~k—2 k± seven
(aj,aj)~1_1Ck XJ~±s. + 1
(1.19)
Remark 1.6. Theorem 1.2 being proved, the commutation relations (1.19) hold for all indices. Note that these relations, modulo terms of smaller second indices, are similar to the usual commutation relations in g: t [h1,x7]=±(a1,a1)x] In addition, the RJk behave a bit nicer under the action of the coproduct in the
Yangian Y(g) (for the definition of the coproduct in Y(g), see refs. [1—5]). Proposition 1.7. Ahjk=kk®1+1®h,k
(mod YH®HY~),
(1.20)
S.Z. Levendorskill On generators and defining relations of Yangians
5
where Y~(H) is a subalgebra of Y(g), generated by x~(by 1 and hfl). Proof An easy induction shows (cf. ref. [1], proposition 1.6, and ref. [2], proposition 3.2) that ~
h1,1~®h~..1
(1.21)
1-~j~k
modulo YH®HY~, and (1.20) follows easily from (1.18), (1.19) and (1.21). Remark 1.8. Formula (1.20) allows us to formulate the highest weight modules theory for Yangians similarity to the one for simple Lie algebras (cf. the highest weight modules theory for Yangians based on (1.21), in refs. [4,1,2]).
2. Proof of theorem 1.2 First we show that relations (1.7)—( 1.12) imply relations (1.1 )—( 1.6); next we deduce from (l.l)—(1.6) some simple relations in (1.7)—(l.12); after that we prove by induction the relations (1.7)— (1.12) for j =1~and finally we prove (1.7)—(1.12) for ij. By comparing (1.3)2 with (1.9), we see that (1.1)— (1.5) follow from (1.7)— (1.12) once we show that (1.2)2 holds with i~ _1. hi1 —Ui!
11,2
—
~hiiO
From (1.10) and (1.8), we deduce rl~ L’~i1
±i.....rj.
X30
j
—
±1 ‘
X10
j
1F1.2 —
±
~ L’~iO, X10
=[h10,xJfl±b0(h~0x1~ +x1~h~o)~b0(h1oxJ~ +xJ~h~o) =
±(a1, a~)xJ~
and (1.2)2with h11=h~1—h~/2 holds. Thus, (l.1)—(l.5) hold. Formula (1.6) can be rewritten as (a,, a1)~([x~, x~]—[x~, x~])=0,
and hence follows from (1.7), (1.9). Nowwestartderiving (l.7)—(l.l2) from (1.1)—(1.6).From (1.1), (1.2) and the definition of ~ h,k, it follows that [h,o,x~~]=±(a1,a~)x3~ [h10,h1~]=0;
;
(2.1) (2.2) (2.3)
6
S.Z. Levendorskit /On generators and definingrelations of Yangians
h11 = [x~ , x~]= [x~, x,1] .
(2.4)
By commutating both sides of (2.4) with h,1 we obtain h,2= [x~,x~]=
[x~,x~]=
[x~,x~]
.
(2.5)
Now we see that (1.6) is equivalent to [h,2,h,~]=0.
(2.6)
As above we deduce from (1.2) [h11,xJ~]=[h,o,xJj±b~,(h,ox~+x~h,0), and commutation with h11, due to (1.1)2, gives ~
+x~h10).
(2.7)
Tostarttheproofof(l.7), (l.9)—(l.ll) fori=j,wenotefirstthat (1.7) with k, l~>2follows from (2.2) and (2.6), formula (1.10) with k=0 is (2.7), and formula (1.11) with k=l=0 is just (1.4), which we can rewrite as follows: (2.8) Lemma 2.1. Lets, pelt, let
let
Jirn,
1t,.i ,...,
[h11,hlk] =0,
for k+l~
(2.9)
E~be defined by (1.18), let (1.19) holdfor k~
(2.10)
Then (1.11) holdsfork=p+sandl=p: [x~±
i,
x~]= [x~~±
5,x~± ~]
±~(a1,
a1)(x~~±~x~ +x~x~0±1)
Proof Note that (1. 19) is of the form [~k,xfl=±(a,,a/)x~/+k±
~
d’~rX,t7±r,
(2.11)
Osr~k—2
set
=
h10, and define inductively ~ O~r~k—2
d~rI~ir, k=l,2,...,s.
Then
~ and
x~]= ±(a1, aj)x,±k±/,k=0, 1,..., s , le~~ ,
(2.12)
S.Z. Levendorskill On generators and defining relations of Yangians
/~ik=
h,~+polynomial in h,0, h11,
7
..., h,,kl,
hlk=,c’k+ (another) polynomial in h10, h,1,
...,
h1,k!
.
(2.13)
Using (2.12) and (2.10), we obtain x~]=
±(a,,
a,) ‘[ ~
~]±(a1,
=
[~, [x~~±1,x~]] 1 [~, x~2(±~ a [x~~±5,x~+1]±(a1, a1) 1))]
=
[x~~±5,x~±1] ±~(a1, a,)(x~~±5x~ +x~x~~±5)
=
[x~~±5,~
x~~±1], xfl a1)
This proves the lemma. Due to (2.2), (2.7) and (2.8), the conditions oflemma 2.1 hold forsi~l,p=O. Hence, the conclusion of lemma 2.1 gives [x~,x~]=
±~(a~, a1)(x,~x~ +x~x,~)
(2.14)
.
Takethe + sign and commutate with x~: [h12,x~]+[x~, h10]=~(a1,a,)(h,1x~+x~h11+x~h10+h,0x~) =
[h11,x~] [h10,x~] + ~(a1, a1) (h11x~+x~5h,1) —
Hence, [h12,x~]= [h11,x~]+~(a1, a,)(h~1x~5 +x~h11),
(2.15)
and analogously, [h12,x~]=[h11,x~]—~(a1,a,)(h,1x~
+x~h11)
.
(2.16)
Since [h1,h12]=0, we deduce from (2.15), (2.16): [h12,x,~J]=[h11,x,~1±1 ]±~(a1, a,)(h11x~+x~’h11); in addition, we see that (2.ll)—(2.13) hold for kz~2. Lemma 2.2. Formula(1.11)holdsfork, le7L±,j~i. Proof Set ~ ~
+x~x~),
and note that (2.12) with k= 1, 2 gives the following implications:
(2.17)
S. Z. Levendorskil/ On generators anddefining relations of Yangians
8
X~(i,p,l)=0 (i, p+ 1, 1) +X~(i,
1+ 1) =0
X~(i,p+2,l)+2X~—(i,p+1,l+l)+X~(i,p,l+2)=0; (2.18) X~(i,p,l)=0
~.
X~(i,p+2,l)+X~(i,p,l+2)=0.
(2.19)
Hence, X~(i,p,l)=0
X~(i,p+1,l+1)=0.
(2.20)
Now (2.8), (2.14) and (2.20) give X’~(i,k,l)=0 for0~l
(2.21)
Suppose, (2.21) holds for 0~
k+l~~1;
(2.22)
(2.23) [h
±
1k, Xit
± =
i.k—!
, X,~+
±~(aI,aI)(hIklx,~+x~hIkI),k~>l,l~>0,k+l~
Since [h,~,h,1] =0 for l~p,we deduce from (2.24)~: [h10,x~,]=[h11,_1,x~m±i] ± ~(a1, a,)(h1,~_1x~~ +x~,h,~1) formcZ~hence,(2.11)—(2.l3)holdwithk=pand/el~,and 0= [h,~,h,~]=[h,1,,~]= =
—
[x~,x~],
~]
(a,, a1){ [x~0, x~]— [x~, x~]}.
(2.25)
Further, by commutating ~ with both sides of (2.23) with s=2p— 1, we obtain [x~0, x~]— [x~,
, x~]
(2.26)
By comparing (2.25) and (2.26), we conclude that (2.23) holds for s= r+ 1 =2p and that for q-
S. Z. LevendorskillOn generatorsand defining relations of Yangians
[hi,rq, hjq±1] = [X~~q, _r —
x~], ~,q±
± —1 LX,,r+ 1, X, 0 j
—
9
~]
r + — LXi,r_q, X,q+ I
— —
Hence, (2.22) holds for s=r+ 1. Finally, by commutating E~with (2.24) ±with s=r and taking into account that (2.22) holds for s~
+
—1
j
1~X~0 =
1 —
—
+
LX,.2p, X11
xzj I — [x~~1, x~] =
~
~
(2.27)
.
Due to (2.23), [h,~,h,1] =0 for l’~
h1~]= [h,,~+~, h,p] = [x~~+ =
—
)~,
[see (2.11)—
q~X~], h,~]
i
(a1, aj)([x~p+i...q, x~]
—
[X~p+i...q, X~p±q]) .
(2.28)
On the other hand, using (2.24) ±several times with k+ l~2p gives 11.
1,
1...FFL,
L”i,p± 1, ‘tlp J
—
L L “l,p+ 1,
=
[h,0, ~ ~
±
1
Xj,p
ii,
—1J..F
Xi1 i
‘
+
—
FL,
LX1,p_ I ,
L”i,p±I X11
x,j] (I,
L O”kj±p
~
~
± 1,p—j
~
i,p±j— I
—
1, l,p+j— 1
i,p—jJ,
it
[h1o,x~~±2]] ~
(h1,~1x~1
O~j±p =
(a1, a~)(~ ~
+
xj ]
—
~
i,
x~~+2])
{ [h1,~_1x~±1..1+xt,±1_ih1,~..1,xj]
O~j~kp
+~
h1~1x~.±1 +x~.÷1h1,~1]}.
(2.29)
In Y(g), the left-hand side in (2.29) is equal to zero and so is the first term on the right-hand side; therefore the sum over j is zero as well. But this sum can be represented as a sum of monomials with the property that the sum of the second indices is not greater than 2p; since by the induction hypothesis the sets of such vanishing sums in Y(g) and Y(g) coincide, we see that in ?(g), (2.29) may be rewritten as
S. Z. Levendorskit / On generators and defining relations of Yangians
10
By comparing this equality with (2.28), we find [h1~±1,h,~]= 0, and (2.23) holds for s=2p+ 1 =r+ 1. Now, the proof of (2.22) and (2.24) for s=r+ 1 is concluded just as in the case r odd above. Thus, we have proved all the relations (l.7)-.-( 1.1 1) forj= i; below, we prove (l.7)—(l.l2) for i~tj. First we prove (1.11). For k=l=0 it is just (1.4). If (a1, a~)=0, then by commutating successively with fi.~we obtain (1.11) for all k>_~0 and 1=0; then, by commutating with i~1,weobtain (1.11) for all k, leL. If (a,, a1) 0, we commutate X~(i,j; k, 1) := [x~±1 , x~] — [xi,
x~±~ I ~b,1(x~x1~ +x~xj)
with /~and independently with /~•~ to obtain: ifX~(i,j; k, l)=0, then (X~(i,j; k+ 1, 1), X~(i,j; k, 1+ 1)) is a solution to the homogeneous system of two equations with the determinant (a1,a1) —(a1,a1) (ct~,a,) —(a1,a1•)
~o
(230)
Hence, X~(i,j; k, 1) =0
X~(i,j; k+ 1,!) =0,
X~(i,j; k, 1+1) =0
SinceX~(i,j; 0,0) =0, we deduce (1.11) for all k, 1. Relation (1.9) is proved similarly: if (a1, a~)=0then we commutate [x~,x~]=0 with i~and obtain [x~ , x~]= 0 for kEZ±;after that, we commutate with /~and obtain (1.9) for all k, le7L~. If (a,, a1)0, we commutate [x~, x~]= 0 with ~ and independently with i~to obtain [x~~1, x~]=0, [x,~.,xj~± 1=0. Hence, the equality [x~, x10]=0 (it]) yields (1.9) for all k, 1. Now, we prove (1.10) for the + sign; the proof for the — sign is similar: N,
±1
‘ti,k±1~ X11 J
11
±
—1
±lrr
+
=
I L~i,k+ I , X10 j, X11 j — [ ~
=
It. ± 1 I “1k~Xjt± ii
1,11.
U11
±
~ft,kX1/
±1 i, X~j j,
—
X10
+1,
X11 ~
To prove (1.7) we first note that the h1k satisfy (1. 19) not only for i j (as was shown before) but for i j as well. Indeed, if we substitute commutation 1hlk for in hlkthe and ((a relations involving (i, i)~ ((a1, a1) / (a,, a1) )k± 1, a1)/ (a1, a1)) ~ for x~,we will get the commutation relations involving i =1. But 1hlk• hence, this substitution, due to (1.17), replaces h•k with ((a,, a~)/ (a,, a1) )k±
S.Z. Levendorskit / On generators and defining relations of Yangians
11
(1.19) with i=jgives (1.19) with iitj. By using (1.18), (1.19), we can construct /c,1,k (k=0, 1,...) such that (2.31)
[~J,k,x~]=±(ai,aj)x~k±t, 7,j,k =h,k
+ polynomial in h,
0,
1
...,
h,,kj
(2.32)
(cf. proof of lemma 2.2). Using (l.2)~and (2.1), we get
[h10,h1~]= [h,0, [x1~, xj
I] =
(a1, a1)’ (h11—h11) =0;
therefore (1.7) is proved for k=0 and all lEZ~.Suppose (1.7) holds for k=r. We want to show that it holds for k=r+ 1. Due to (2.31) and the induction hypothesis we may substitute ~ij,k for h1k; after that (2.32) allows us to prove (1.7) like we did in the case k=0. Now all that is left to do is to prove (1.12). If (a,, a~)=0,then (1.5) is [x~, x~]=0, and by commutating successively with /~and E~,we obtain [xi, x~] =0, for all k, lE7~. This means that (1.12) holds. Now, let (a1, a1) ~0. For lE~~and a non-increasing tuple k= (k1, k2, k,~), where k~e7L± and m= 1—a,1, denote by X~(k; 1) the left-hand side of relation ...,
(1.12).
First we show thatX~(0; 1) =0 for all lc7L~,i.e., (1.12) holds for k= (0, ..., 0) and all le7L~.For 1=0, it is just (1.4). Assume X~(0; /)=0 for ls~rand commutate with Ji~.1and with We get a system of two homogeneous linear equations with the unknowns X~((1, 0, ..., 0); r), X~(0; r+ 1) and the determinant (2.30). Hence, X~(0; 1) =0 for ~ Finally, we use induction on the number s of the first zero entry k, of k. For s=0, we have X~(k; 1) =X~(0; 1) =0. Assume X-~(k; 1) =0 for all k with s=s(k) ~ r and all leZ~.By commutating with h,~and using the induction hypothesis, we get X~(k’; 1) =0 for all land k’=k’(k, p) which are obtained from (k1, kr, p, 0, ..., 0) by reordering. Hence, X~(k; 1) =0 for all k with s=s(k)~r+1 and all leZ~,and (1.12) is proved. )~.
...,
The author expresses his gratitude to V.G. Drinfeld for discussion ofthe paper. References [I] [2] [3] [4]
[5]
V. Chari and A.N. Pressley, Yangians and R-matrices, L’Enseignement Mathematique 36 (1990) 267—302. V. Chari and AN. Pressley, Fundamental representations of Yangians, Tata preprint (1990). V.G. Drinfeld, Hopf algebras and the quantum Yang—Baxter equation, DokI. Akad. Nauk SSSR 283 (1985) 254—258 [Soy.Math. DokI. 32 (1985)]. V.G. Drinfeld, A new realization of Yangians and quantized affine algebras, DokI. Akad. Nauk SSSR296 (1987) 212—216 [Soy. Math. DokI. 36(1988)]. V.G. Drinfeld, Quantum groups, in: Proc. Intern. Congr. Math. (Berkeley, 1988), Vol. 1, pp. 798—820.
Journal of Geometry and Physics 12 (1993) 1—11 North-Holland
OF
GEOMETRYAND
PHYSICS
On generators and defining relations of Yangians S.Z. Levendorskii Budennovskii 94, ap. 64, Rostov-on-Don, 344018 Russian Federation Received 3 August 1992
So far, two equivalent definitions ofYangians are known. The first one involves a finite set of generators and relations but some of the latter are too complicated, while the second one exploits infinite sequences ofgenerators and relations of more convenient form. It is shown in this paper that only a finite part of thesesequences suffices to define the structure ofYangians. In addition, we construct generators for the “Cartan part” of Yangian, which enjoy properties more similar to those ofgenerators ofa Cartan subalgebra of a simple Lie algebra.
Keywords: Yangian, generators, defining relations 1991 MSC: 17B37, 81 R 50, 82B23
Drinfeld [3] introduced the notion of a quantum group, the most important examples being quantized enveloping algebras and Yangians. Both these types of quantum group are closely related to solutions of the quantum Yang—Baxter equation, hence thorough study of them is of interest. So far, two equivalent definitions of Yangians are known. The first one, given by Drinfeld [3], involves a finite set of generators and relations but some of the latter are too complicated. Later, Drinfeld [4] (see also ref. [51) suggested a more convenient set of generators and relationsbut this set is infinite. In the main theorem of the present paper we show that only a finite part ofthe last set suffices to define the structure of Yangians. In addition, we construct the generators of the “Cartan part” of Yangian, which enjoy properties similar to those of the generators ofa Cartan subalgebra ofa simple Lie algebra. It should be noted that so far Yangians are introduced for simple Lie algebras only whereas it seems reasonable to introduce them in the case of affine Lie algebras as well. Still, it can be shown that one cannot expect simple infinite sequences of relations similar to those of ref. [4] to exist in the case of affine Lie algebras. Therefore it is important to be able to handle Yangians by using only a finite set ofthe relations introduced in ref. [4] (or their analogs). The results on “affine Yangians” will appear elsewhere. For various results on the representation theory ofYangians and their relations 0393-0440/1993/$6.OO © 1993
Elsevier Science Publishers B.V. All rights reserved
S. Z. Levendorskii / On generators and defining relations of Yanglans
2
with the theory of integrable quantum systems, see Drinfeld [3—51 and Chari and Pressley [1,21. 1. Main theorems Let g be a finite-dimensional complex simple Lie algebra, let g = n ±~ ~ n be a triangular decomposition of g, let A~be the set of positive roots and {a1 a~},n=rank g, the corresponding set of simple roots, and let (a0) be the Cartan matrix of g. Fix a non-zero invariant symmetric bilinear form ( ) on g, and for each positive root a of g, choose root vectors x~in the ±a root spaces such that (x~, x~)=1.Thenset h~=[x~,x~],hj—ha,,x~ =X~. ,
Definition 1.1. Denote by Y(g) the algebra over C with generators x~,x~’,h,0,
~
(1 ~i~n) and defining relations
[/~~,/~~1=O, [/~~,Ii~~]=O, [
,
[/~i,~i1=O;
(1.1)
xJ~]= ±( a1, (1.2)
[x~,x~]=ô~0,
[x~,x~]=(i1+~ii~);
~
(1.3) (1.4)
where b0= (a1, a1)/2; [x~, [x~,..., [x~,x1fl~]]=O, (1
—
ij,
(1.5)
a0 commutators); [~1,x~],x~fl+[x~,
[/~,,x~]]=O.
(1.6)
Set for keZ~, xfk±l=±(aI,al)[h,l,xlkl,
hlk=[x~,x,~}.
Theorem 1.2. The algebra ?(g) is isomorphic to the Yangian Y(g), the isomorph-
ism being defined by t
—
1.
In other words, in
~
EY(g)
“tik .3
g) thefollowing relations hold: (1.7)
3
S. Z. Levendorskii/ On generators anddefining relations of Yan glans
~
;
(1.8) (1.9) (1.10) (1.11)
~ ~ ~
[x~
12>,
[x~L(,~), xJ~’]“]
...,
1=0
,
(1.12)
for i j, where m = 1 a0 and the sum is taken over all the permutations a of{ 1, m}. Relations(1.7)—(1.12)implyrelations (1.1)—(1.6). —
Remark 1.3. Relations (1.7)—(1.12) are just the relations introduced by Drinfeld in ref. [4]. The proof of theorem 1.2 is based on the followinglemma. To state it we introduce the unital algebra A with the generators h~,~ (Je7L+) and the defining relations [hk,h~]=0;
(1.13) (1.14)
where h1
=
I and yelR is independent of k, 1. Set h(t)=
~
hkt~~ x(’r)= ,
k~—I
~ X~Tt 1 t~O
and define h’kEA (kEl~)by the equality ~ hkt~~=lnh(t).
(1.15)
k~O
Here the right-hand side is the expansion in powers of t in the vicinity of t=
+cc.
Lemma 1.4. Let (1.13) holdforj~
[Rk,xtl=2yxk+l+2
~ O~sek—2 k+s even
~.+ 1
C~~1x1~.
(1.16)
Proof Due to (1.15), Ek hk is a polynomial in h0, h1, ..., hkI. Hence, while deriving (1.16), we may and shall assume that (1.13), (1.14) hold for all k, lel~.Now (1.14) can be rewritten as —
4
5. Z. Levendorskit / On generators and defining relations of Yan glans
h(t)x(t) =x(r)h(t)
t—i+p t—t—y
therefore [R(t), x(r)]
=
[ln h(t), x(r)] =ln tt+Yx(T)
(1.17)
Simple calculations give in
=ln( 1 —t~ (r—y) )—ln( 1 =
—
—1~ (~+y))
~
k~1
t~ = k~1
k
~
C~t’y’((—l)’~+l);
O~j~k
hence, ( 1. 17) gives ( 1. 16 ). Corollary 1.5. Let (1.8) hold for leZ~, 1 ~
~ ~t~=ln(1+ k>0
h1~t~~)
(118)
k~O
Thenfork~p,le7L~,1~i,j~n, [~k,
x~I
=
± (a1, aI)xJ~+k ±
~
21
O~ss~k—2 k± seven
(aj,aj)~1_1Ck XJ~±s. + 1
(1.19)
Remark 1.6. Theorem 1.2 being proved, the commutation relations (1.19) hold for all indices. Note that these relations, modulo terms of smaller second indices, are similar to the usual commutation relations in g: t [h1,x7]=±(a1,a1)x] In addition, the RJk behave a bit nicer under the action of the coproduct in the
Yangian Y(g) (for the definition of the coproduct in Y(g), see refs. [1—5]). Proposition 1.7. Ahjk=kk®1+1®h,k
(mod YH®HY~),
(1.20)
S.Z. Levendorskill On generators and defining relations of Yangians
5
where Y~(H) is a subalgebra of Y(g), generated by x~(by 1 and hfl). Proof An easy induction shows (cf. ref. [1], proposition 1.6, and ref. [2], proposition 3.2) that ~
h1,1~®h~..1
(1.21)
1-~j~k
modulo YH®HY~, and (1.20) follows easily from (1.18), (1.19) and (1.21). Remark 1.8. Formula (1.20) allows us to formulate the highest weight modules theory for Yangians similarity to the one for simple Lie algebras (cf. the highest weight modules theory for Yangians based on (1.21), in refs. [4,1,2]).
2. Proof of theorem 1.2 First we show that relations (1.7)—( 1.12) imply relations (1.1 )—( 1.6); next we deduce from (l.l)—(1.6) some simple relations in (1.7)—(l.12); after that we prove by induction the relations (1.7)— (1.12) for j =1~and finally we prove (1.7)—(1.12) for ij. By comparing (1.3)2 with (1.9), we see that (1.1)— (1.5) follow from (1.7)— (1.12) once we show that (1.2)2 holds with i~ _1. hi1 —Ui!
11,2
—
~hiiO
From (1.10) and (1.8), we deduce rl~ L’~i1
±i.....rj.
X30
j
—
±1 ‘
X10
j
1F1.2 —
±
~ L’~iO, X10
=[h10,xJfl±b0(h~0x1~ +x1~h~o)~b0(h1oxJ~ +xJ~h~o) =
±(a1, a~)xJ~
and (1.2)2with h11=h~1—h~/2 holds. Thus, (l.1)—(l.5) hold. Formula (1.6) can be rewritten as (a,, a1)~([x~, x~]—[x~, x~])=0,
and hence follows from (1.7), (1.9). Nowwestartderiving (l.7)—(l.l2) from (1.1)—(1.6).From (1.1), (1.2) and the definition of ~ h,k, it follows that [h,o,x~~]=±(a1,a~)x3~ [h10,h1~]=0;
;
(2.1) (2.2) (2.3)
6
S.Z. Levendorskit /On generators and definingrelations of Yangians
h11 = [x~ , x~]= [x~, x,1] .
(2.4)
By commutating both sides of (2.4) with h,1 we obtain h,2= [x~,x~]=
[x~,x~]=
[x~,x~]
.
(2.5)
Now we see that (1.6) is equivalent to [h,2,h,~]=0.
(2.6)
As above we deduce from (1.2) [h11,xJ~]=[h,o,xJj±b~,(h,ox~+x~h,0), and commutation with h11, due to (1.1)2, gives ~
+x~h10).
(2.7)
Tostarttheproofof(l.7), (l.9)—(l.ll) fori=j,wenotefirstthat (1.7) with k, l~>2follows from (2.2) and (2.6), formula (1.10) with k=0 is (2.7), and formula (1.11) with k=l=0 is just (1.4), which we can rewrite as follows: (2.8) Lemma 2.1. Lets, pelt, let
let
Jirn,
1t,.i ,...,
[h11,hlk] =0,
for k+l~
(2.9)
E~be defined by (1.18), let (1.19) holdfor k~
(2.10)
Then (1.11) holdsfork=p+sandl=p: [x~±
i,
x~]= [x~~±
5,x~± ~]
±~(a1,
a1)(x~~±~x~ +x~x~0±1)
Proof Note that (1. 19) is of the form [~k,xfl=±(a,,a/)x~/+k±
~
d’~rX,t7±r,
(2.11)
Osr~k—2
set
=
h10, and define inductively ~ O~r~k—2
d~rI~ir, k=l,2,...,s.
Then
~ and
x~]= ±(a1, aj)x,±k±/,k=0, 1,..., s , le~~ ,
(2.12)
S.Z. Levendorskill On generators and defining relations of Yangians
/~ik=
h,~+polynomial in h,0, h11,
7
..., h,,kl,
hlk=,c’k+ (another) polynomial in h10, h,1,
...,
h1,k!
.
(2.13)
Using (2.12) and (2.10), we obtain x~]=
±(a,,
a,) ‘[ ~
~]±(a1,
=
[~, [x~~±1,x~]] 1 [~, x~2(±~ a [x~~±5,x~+1]±(a1, a1) 1))]
=
[x~~±5,x~±1] ±~(a1, a,)(x~~±5x~ +x~x~~±5)
=
[x~~±5,~
x~~±1], xfl a1)
This proves the lemma. Due to (2.2), (2.7) and (2.8), the conditions oflemma 2.1 hold forsi~l,p=O. Hence, the conclusion of lemma 2.1 gives [x~,x~]=
±~(a~, a1)(x,~x~ +x~x,~)
(2.14)
.
Takethe + sign and commutate with x~: [h12,x~]+[x~, h10]=~(a1,a,)(h,1x~+x~h11+x~h10+h,0x~) =
[h11,x~] [h10,x~] + ~(a1, a1) (h11x~+x~5h,1) —
Hence, [h12,x~]= [h11,x~]+~(a1, a,)(h~1x~5 +x~h11),
(2.15)
and analogously, [h12,x~]=[h11,x~]—~(a1,a,)(h,1x~
+x~h11)
.
(2.16)
Since [h1,h12]=0, we deduce from (2.15), (2.16): [h12,x,~J]=[h11,x,~1±1 ]±~(a1, a,)(h11x~+x~’h11); in addition, we see that (2.ll)—(2.13) hold for kz~2. Lemma 2.2. Formula(1.11)holdsfork, le7L±,j~i. Proof Set ~ ~
+x~x~),
and note that (2.12) with k= 1, 2 gives the following implications:
(2.17)
S. Z. Levendorskil/ On generators anddefining relations of Yangians
8
X~(i,p,l)=0 (i, p+ 1, 1) +X~(i,
1+ 1) =0
X~(i,p+2,l)+2X~—(i,p+1,l+l)+X~(i,p,l+2)=0; (2.18) X~(i,p,l)=0
~.
X~(i,p+2,l)+X~(i,p,l+2)=0.
(2.19)
Hence, X~(i,p,l)=0
X~(i,p+1,l+1)=0.
(2.20)
Now (2.8), (2.14) and (2.20) give X’~(i,k,l)=0 for0~l
(2.21)
Suppose, (2.21) holds for 0~
k+l~
(2.22)
(2.23) [h
±
1k, Xit
± =
i.k—!
, X,~+
±~(aI,aI)(hIklx,~+x~hIkI),k~>l,l~>0,k+l~
Since [h,~,h,1] =0 for l~p,we deduce from (2.24)~: [h10,x~,]=[h11,_1,x~m±i] ± ~(a1, a,)(h1,~_1x~~ +x~,h,~1) formcZ~hence,(2.11)—(2.l3)holdwithk=pand/el~,and 0= [h,~,h,~]=[h,1,,~]= =
—
[x~,x~],
~]
(a,, a1){ [x~0, x~]— [x~, x~]}.
(2.25)
Further, by commutating ~ with both sides of (2.23) with s=2p— 1, we obtain [x~0, x~]— [x~,
, x~]
(2.26)
By comparing (2.25) and (2.26), we conclude that (2.23) holds for s= r+ 1 =2p and that for q-
S. Z. LevendorskillOn generatorsand defining relations of Yangians
[hi,rq, hjq±1] = [X~~q, _r —
x~], ~,q±
± —1 LX,,r+ 1, X, 0 j
—
9
~]
r + — LXi,r_q, X,q+ I
— —
Hence, (2.22) holds for s=r+ 1. Finally, by commutating E~with (2.24) ±with s=r and taking into account that (2.22) holds for s~
+
—1
j
1~X~0 =
1 —
—
+
LX,.2p, X11
xzj I — [x~~1, x~] =
~
~
(2.27)
.
Due to (2.23), [h,~,h,1] =0 for l’~
h1~]= [h,,~+~, h,p] = [x~~+ =
—
)~,
[see (2.11)—
q~X~], h,~]
i
(a1, aj)([x~p+i...q, x~]
—
[X~p+i...q, X~p±q]) .
(2.28)
On the other hand, using (2.24) ±several times with k+ l~2p gives 11.
1,
1...FFL,
L”i,p± 1, ‘tlp J
—
L L “l,p+ 1,
=
[h,0, ~ ~
±
1
Xj,p
ii,
—1J..F
Xi1 i
‘
+
—
FL,
LX1,p_ I ,
L”i,p±I X11
x,j] (I,
L O”kj±p
~
~
± 1,p—j
~
i,p±j— I
—
1, l,p+j— 1
i,p—jJ,
it
[h1o,x~~±2]] ~
(h1,~1x~1
O~j±p =
(a1, a~)(~ ~
+
xj ]
—
~
i,
x~~+2])
{ [h1,~_1x~±1..1+xt,±1_ih1,~..1,xj]
O~j~kp
+~
h1~1x~.±1 +x~.÷1h1,~1]}.
(2.29)
In Y(g), the left-hand side in (2.29) is equal to zero and so is the first term on the right-hand side; therefore the sum over j is zero as well. But this sum can be represented as a sum of monomials with the property that the sum of the second indices is not greater than 2p; since by the induction hypothesis the sets of such vanishing sums in Y(g) and Y(g) coincide, we see that in ?(g), (2.29) may be rewritten as
S. Z. Levendorskit / On generators and defining relations of Yangians
10
By comparing this equality with (2.28), we find [h1~±1,h,~]= 0, and (2.23) holds for s=2p+ 1 =r+ 1. Now, the proof of (2.22) and (2.24) for s=r+ 1 is concluded just as in the case r odd above. Thus, we have proved all the relations (l.7)-.-( 1.1 1) forj= i; below, we prove (l.7)—(l.l2) for i~tj. First we prove (1.11). For k=l=0 it is just (1.4). If (a1, a~)=0, then by commutating successively with fi.~we obtain (1.11) for all k>_~0 and 1=0; then, by commutating with i~1,weobtain (1.11) for all k, leL. If (a,, a1) 0, we commutate X~(i,j; k, 1) := [x~±1 , x~] — [xi,
x~±~ I ~b,1(x~x1~ +x~xj)
with /~and independently with /~•~ to obtain: ifX~(i,j; k, l)=0, then (X~(i,j; k+ 1, 1), X~(i,j; k, 1+ 1)) is a solution to the homogeneous system of two equations with the determinant (a1,a1) —(a1,a1) (ct~,a,) —(a1,a1•)
~o
(230)
Hence, X~(i,j; k, 1) =0
X~(i,j; k+ 1,!) =0,
X~(i,j; k, 1+1) =0
SinceX~(i,j; 0,0) =0, we deduce (1.11) for all k, 1. Relation (1.9) is proved similarly: if (a1, a~)=0then we commutate [x~,x~]=0 with i~and obtain [x~ , x~]= 0 for kEZ±;after that, we commutate with /~and obtain (1.9) for all k, le7L~. If (a,, a1)0, we commutate [x~, x~]= 0 with ~ and independently with i~to obtain [x~~1, x~]=0, [x,~.,xj~± 1=0. Hence, the equality [x~, x10]=0 (it]) yields (1.9) for all k, 1. Now, we prove (1.10) for the + sign; the proof for the — sign is similar: N,
±1
‘ti,k±1~ X11 J
11
±
—1
±lrr
+
=
I L~i,k+ I , X10 j, X11 j — [ ~
=
It. ± 1 I “1k~Xjt± ii
1,11.
U11
±
~ft,kX1/
±1 i, X~j j,
—
X10
+1,
X11 ~
To prove (1.7) we first note that the h1k satisfy (1. 19) not only for i j (as was shown before) but for i j as well. Indeed, if we substitute commutation 1hlk for in hlkthe and ((a relations involving (i, i)~ ((a1, a1) / (a,, a1) )k± 1, a1)/ (a1, a1)) ~ for x~,we will get the commutation relations involving i =1. But 1hlk• hence, this substitution, due to (1.17), replaces h•k with ((a,, a~)/ (a,, a1) )k±
S.Z. Levendorskit / On generators and defining relations of Yangians
11
(1.19) with i=jgives (1.19) with iitj. By using (1.18), (1.19), we can construct /c,1,k (k=0, 1,...) such that (2.31)
[~J,k,x~]=±(ai,aj)x~k±t, 7,j,k =h,k
+ polynomial in h,
0,
1
...,
h,,kj
(2.32)
(cf. proof of lemma 2.2). Using (l.2)~and (2.1), we get
[h10,h1~]= [h,0, [x1~, xj
I] =
(a1, a1)’ (h11—h11) =0;
therefore (1.7) is proved for k=0 and all lEZ~.Suppose (1.7) holds for k=r. We want to show that it holds for k=r+ 1. Due to (2.31) and the induction hypothesis we may substitute ~ij,k for h1k; after that (2.32) allows us to prove (1.7) like we did in the case k=0. Now all that is left to do is to prove (1.12). If (a,, a~)=0,then (1.5) is [x~, x~]=0, and by commutating successively with /~and E~,we obtain [xi, x~] =0, for all k, lE7~. This means that (1.12) holds. Now, let (a1, a1) ~0. For lE~~and a non-increasing tuple k= (k1, k2, k,~), where k~e7L± and m= 1—a,1, denote by X~(k; 1) the left-hand side of relation ...,
(1.12).
First we show thatX~(0; 1) =0 for all lc7L~,i.e., (1.12) holds for k= (0, ..., 0) and all le7L~.For 1=0, it is just (1.4). Assume X~(0; /)=0 for ls~rand commutate with Ji~.1and with We get a system of two homogeneous linear equations with the unknowns X~((1, 0, ..., 0); r), X~(0; r+ 1) and the determinant (2.30). Hence, X~(0; 1) =0 for ~ Finally, we use induction on the number s of the first zero entry k, of k. For s=0, we have X~(k; 1) =X~(0; 1) =0. Assume X-~(k; 1) =0 for all k with s=s(k) ~ r and all leZ~.By commutating with h,~and using the induction hypothesis, we get X~(k’; 1) =0 for all land k’=k’(k, p) which are obtained from (k1, kr, p, 0, ..., 0) by reordering. Hence, X~(k; 1) =0 for all k with s=s(k)~r+1 and all leZ~,and (1.12) is proved. )~.
...,
The author expresses his gratitude to V.G. Drinfeld for discussion ofthe paper. References [I] [2] [3] [4]
[5]
V. Chari and A.N. Pressley, Yangians and R-matrices, L’Enseignement Mathematique 36 (1990) 267—302. V. Chari and AN. Pressley, Fundamental representations of Yangians, Tata preprint (1990). V.G. Drinfeld, Hopf algebras and the quantum Yang—Baxter equation, DokI. Akad. Nauk SSSR 283 (1985) 254—258 [Soy.Math. DokI. 32 (1985)]. V.G. Drinfeld, A new realization of Yangians and quantized affine algebras, DokI. Akad. Nauk SSSR296 (1987) 212—216 [Soy. Math. DokI. 36(1988)]. V.G. Drinfeld, Quantum groups, in: Proc. Intern. Congr. Math. (Berkeley, 1988), Vol. 1, pp. 798—820.