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
aram@unimelb.edu.au

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.

𝔤=G2

The irreducible finite dimensional representation of
Uq(G2) are parametrized by h.w. λ=λ1ω1+λ2ω2, where ω1 and ω2 are the fundamental weights (see [Bou1981]) and λi+. If eλVλ the h.w. vector, H1eλ=λ1eλ, H2eλ=3λ2eλ.

The basic representation of Uq(G2) have h.w. λ=ω1. Let us consider the basics in Vω1 formed by weight vectors,
 e1=eω1, e2=eα1+α2, e3=eα1, e4=e0, e5=e1, e6=e-α1-α2, e7=e-ω1. The roots are shown on the figure 1. α1 α2 ω1 ω2 Figure 1. In this basis the elements Hi,Xi± are represented by the matrices: π(X1+)= ( 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 ) ,π(X2+)= ( 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) (7.1) π(H1)= ( 1 -1 0 2 0 -2 0 1 -1 ) ,π(H2)= ( 0 1 0 -1 0 1 0 -1 0 ) (7.2) π(Xi-)= π(Xi+)t. Here b=sh(h2)sh(3h2), c=sh(h)sh(3h2).

In this normalization the commutation relations between the generators Hi,Xi± have the following form: [H1,X1±]= ±2X1±, [H2,X1±]= 3X1± [H1,X2±]= 3X2±, [H2,X2±]= ±6X2± [Xi+,Xj-]= sh(h2Hi) sh(3h2) ,q=eh.

The representation Vω1 is selfdual: Vω1*=Vω1.

A tensorial product of the two basic representations of Uq(G2) has the following decomposition: Vω1Vω1= V2ω1Vω2 Vω1V0. (7.3) In this decomposition 2ω1=0, ω2=ω1=1, 0=0. It is not difficult to calculate the matrix Kω1ω1ω1: e1 = g ( cq34e2 e3-bq32 e1e4+b q-32e4 e1-cq-14 e3e2 ) , e2 = g ( bq12e2 e4-cq54 e1e5+c q-54e5 e1-bq-12 e4e2 ) , e3 = g ( bq12e3e4- cq54e1e6+ cq-54e6 e1-bq-12 e4e3 ) , e4 = gb ( -qe1e7- q32e2 e6+e3 e5+ (q12-q-12) e4e4-e5e3 +q-32e6 e2+q-1e7 e1 ) , e5 = g ( bq12e4 e5-cq54e2 e7+cq-54 e7e2-b q-32e5e4 ) , e6 = g ( bq12e4e6 -cq54e3 e7+cq-54 e7e3-b q-12e6 e4 ) , e7 = g ( cq34e5 e6-bq32 e4e7+b q-32e7 e4-cq-34 e6e5 ) , g = 1q2q-2, (7.4) So the matrix (Kω1ω1ω1)ijk ei=j,k=17 (Kω1ω1ω1)ijk ejek (7.5) is given explicitly.

Theorem 7.1. The matrix Rω1ω1 satisfies the following relations: -q =(q-1) { -α +β } (7.6) -q =(q-1) { -α +β } . (7.7) Here α=q+q-1, β=q-3(q3-1)(q4+1)q-1.

Proof.

From the theorem 1.5 we have the spectral decomposition of Rω1ω1: ω1 ω1 =q ω1 ω1 2ω1 ω1 ω1 - ω1 ω1 ω2 ω1 ω1 -q-3 ω1 ω1 ω1 ω1 ω1 +q-6 ω1 ω1 0 ω1 ω1 . (7.8) One-dimensional projector is defined in section 1 (1.47): ω1 ω1 0 ω1 ω1 = 1trVω1(q-ρ) =q2+1q q-1 (q7-1)(q-6+1) . (7.9) Substituting (7.8), (7.9) in the l.h.s. of 7.6 and using the decomposition of units in Vω12: = 2ω1 + ω2 + ω1 + 0 we obtain (7.6).

The equality (7.7) is obtained by rotating the pictures (7.6) by 90 (making crossing-transformation).

Considering the equalities (7.6) and (7.7) as a system of linear equations for Rω1ω1 and (Rω1ω1)-1 we obtain an expression for these matrices in terms of the matrix Kω1ω1ω1 (7.4):

Proposition 7.1. =1q+1 [ q2 +q-3 (q3-1)(q4+1) (q-1) ( +q ) +q-1 ] (7.10) =1q+1 [ q2 +q-3 (q3-1)(q4+1) (q-1) ( q + ) +q-1 ] . (7.11)

Let us consider now the restriction of G2 to the 𝔰𝔩(2) triple formed by X1±,H1. The representation Vω1 is the sum of three 𝔰𝔩(2)-irreducible components. Vω1 |𝔰𝔩(2) =Vω1V2 V1. (7.12) Here dimV1=dimV1=2, dimV2=3. The spaces V1,V2 and V1 are formed by the vectors (e1,e2), (e3,e4,e5) and (e6,e7) correspondingly.

For 𝔰𝔩(2) CGC we have: V2V1V1 e2 = e1e1, e0 = 1q1/2+q-1/2 ( q14e-1 e1+q-14 e1e-1 ) , e-2 = e-1e-1, (7.12) where e±1 are the bases in 2-dimensional representation of Uq(𝔰𝔩(2)) and e2,e0,e-2 are the bases of a 3-dimensional one. V2V2V2 e2 = 1q+q-1 ( q12e2 e0-q-12 e0e2 ) , e0 = 1q+q-1 ( e2e-2+ (q12-q-12) e0e0- e-2e2 ) , e-2 = 1q+q-1 ( q12e0 e-2-q-12 e-2e0 ) . (7.13) In accordance with the graphical representation of CGC given in section 2 we can associate tho following picture with the formulae (7.12) and (7.13) 1 1 2 (7.12), 2 2 2 (7.13). (7.14) An index 1 corresponds to a 2-dimensional representation. An index 2 corresponds to a 3-dimensional representation.

From (7.4) and (7.12)-(7.14) we obtain the 𝔰𝔩(2)-block form of the matrix Kω1ω1ω1: Kω1ω1ω1= =1q2+q-2 0 -q 1 2 2 0 q-1 1 2 1 0 0 0 0 0 0 xq-54 2 1 1 0 y 2 2 2 0 -xq54 2 1 1 0 0 0 0 0 0 q 1 2 1 0 -q-1 1 1 2 (7.15) where x=(1-q21-q3)12 q14,y= ((1-q)(1+q2)(1-q3))12.

From (7.15) we obtain the following expression for projector Pω1ω1ω1: ω1 ω1 ω1 ω1 ω1 =1q2-q-2 0 0 0 0 q2 1 2 1 1 2 0 0 0 x2q-32 1 1 2 1 1 0 0 0 - 1 1 2 1 1 0 0 0 xyq-54 1 1 2 2 1 0 0 0 0 0 0 0 -x2 1 1 2 1 1 0 0 0 - 2 1 1 1 2 0 0 0 xyq-54 2 2 2 1 1 0 0 0 q-2 2 1 1 2 1 0 0 0 y2 2 2 2 2 2 0 0 0 q2 2 1 1 2 1 0 0 0 -yxq54 2 2 2 1 1 0 0 0 - 2 1 1 1 2 0 0 0 -x2 1 1 2 1 1 0 0 0 0 0 0 0 -yxq54 1 1 2 2 2 0 0 0 - 1 2 1 2 1 0 0 0 x2q52 1 1 2 1 1 0 0 0 q-4 1 2 1 1 2 0 0 0 0 (7.16) and the similar expression for crossing-conjugate matrix. For one dimensional projector there is a similar representation: ω1 ω1 ω1 ω1 = 0 0 0 0 0 0 0 0 q92 1 1 0 0 0 0 0 0 0 q94 1 2 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 q94 2 1 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 2 1 q-94 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 q-94 1 2 0 0 0 0 0 0 0 1 1 q-92 0 0 0 0 0 0 0 0 (7.17) where 1 -1 1 =q-14, 1 1 -1 =q14, 2 -2 2 =q, 2 0 0 =1, 2 2 -2 =q-1.

Substituting (7.15)-(7.17) in (7.10) and (7.11) we obtain an explicit expression for Rω1ω1 though 𝔰𝔩(2)-CGC.

page history