Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links, I

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 2 June 2014

Notes and References

This is a typed copy of the LOMI preprint Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links, I by N.Yu. Reshetikhin.

Recommended for publication by the Scientific Council of Steklov Mathematical Institute, Leningrad Department (LOMI) 6, June, 1987.


The basic representation of Vq(𝔰𝔩(n)) have the h.w. ω1 and Vω1n. The matrix elements of the generators do not depend on q in this representation. π(Xi+)= Eii+1, π(Xi-)= Ei+1i, π(Hi)= Eii-Ei+1,i+1 (4.1) where (Eij)αβ=δiαδjβ are the basic matrices in n.

Tho Tensorial square of Vω1 is decomposed into the sum of two irreducible components (Vω1)2= V2ω1Vω2. (4.2)

The KGC of this decomposition are determined by the relations: eij2ω1= 11+q-1 (eiej+q-12ejei), i<j, eii2ω1= eiei (4.3) eijω2= 11+q-1 (eiej-q-1/2ejei), i<j (4.4) where eij2ω1 and eijω2 are the bases in V2ω1 and in Vω2 respectively.

From the theorem 1.5 we find the basic R-matrix for Uq(𝔤𝔩(n)): Rω1ω1= P2ω1ω1ω1 q1/2- Pω2ω1ω1 q-1/2 (4.5) or using (4.3), (4.4) Rω1ω1= q1/2i=1n EiiEii+ ijn EijEji- (q-1/2-q1/2) j>iEjj Eii. (4.6) In the form (4.6) the matrix Rω1ω1 was found by Jimbo [Jim1985].

For 𝔤=𝔤𝔩(n) the decomposition of VλVω1 is well known: VλVω1= k=1n Vλ(k) (4.7) where λ(k)=(λ1,,λk+1,λk+1,,λn) if λ=(λ1,,λn), (λi+,λiλi+1) and the sum is taken only over k satisfying the condition λk>λk+1. Using the rule (4.7) and the theorem 3.1 one can calculate the elements Pλ(q,a). The representations λ=kω1 and λ=ωk are contained in (Vω1)k with multiplicity equal to one. It is not difficult to calculate the corresponding projectors Pλ: Pkω1 = qk(k-1)4[k]! wSk q(w)2 πw, (4.8) Pωk = q-k(k-1)4[k]! wSk (-q)-(w)2 πw, (4.9) where the sum taken is over the element of symmetric group Sk, πsi=gi if Si is the elementary transposition in Sk, the elements gi are defined by (3.2) and πw=πs1πsk if w=s1sk is the representation of w in the nonreducible product of the transpositions, [k]!=[k][1], [k]=(qk/2-q-k/2)/(q1/2-q-1/2). The formulae (4.8), (4.9) were obtained in [Jim1985] as the q-analogs of the Young sym (antisym) metrisators. The following theorem describes the connection between the algebra CNω1(𝔤𝔩(n)) and Hecke algebra N(q). The last one is the assotiative algebra with units generated by the elements 1,gˆi, i=1,,N-1 satisfying the following relations: gˆi gˆi+1 gˆi = gˆi+1 gˆi gˆi+1 gˆi gˆj = gˆj gˆi, |i-j|>1 gˆi2 = (q-1) gˆi+q. (4.10)

Theorem 4.1. The algebra CNω1(𝔤𝔩(n)) is the factor of Hecke algebra N(q) over the ideal JN formed by the elements wSn+1 (-q)-(w)2 πw=0 where πw are the same as in (4.8) and (4.9). If n+1>N, CNω1(𝔤𝔩(n))N(q). The generators gi of CNω1(𝔤𝔩(n)) correspond to the generators gˆi of Hecke algebra gˆi=q1/2gi.

This theorem in equivalent to the fact that in the space (Vω1)N the representation of Hecke algebra α(gˆi)=q1/2gi (this fact was noted by Jimbo [Jim1985]) is defined. The theorem 4.1 means that the irreducible components of this representation of N(q) have the Young diagrams λ with no more than n rows.

An explicit expression for matrix elements of gi in the basics (3.4) of the representation Wλ can be found from (3.9). Omitting the details, we give the answer: w(λ,λ+ek,λ+ek+e,λ+e) = (qd+12-q-d+12) (qd-12-q-d-12) (q12+q-12) (qd2-q-d2) , (4.11) w(λ,λ+ek,λ+ek+e,λ+ek) = { q12-q-12 qd-1 , k q-12, k= -q12, =k+1,λk=λ (4.12) where d=λk-k-λ+. This expression was given in [Wen1985] and [Ker1987] following a different argumentation.


*) From this point we will use the normalization and the notation of books [Bou1981] for the highest weights and the roots of simple Lie algebras.

page history