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.


All formulae in thin caae are very similar to those in part one of the previous section.

The h.w. of finite dimensional representation of are parametrized by the numbers λ=(λ1,,λn), λ1λn, λi+. If eλ is the h.w. vector, Hieλ=(λi-λi+1)eλ, i=1,,n-1, Hneλ=λneλ, Xi+eλ=0.

The basic representation of Uq(𝔰𝔭(2n)) have the h.w. ω1=(1,0,,0), Vω12n π(Xi+) = Eii+1- EN-i,N-i+1, i=1,,n-1 π(Xn-) = Enn+1, π(Xi-)= π(Xi+)t π(Hi) = Eii- Ei+1,i+1- EN-i+1,N-i+1+ EN-i,N-i π(Hn) = 2Enn-2 En+1,n+1. (6.1) The tensor square of the basic representation is the sum of three irreducible components: Vω1Vω1= V2ω1 Vω2 V(0) (6.2) and we have the following spectral decomposition of Rω1ω1: Rω1ω1= q12P2ω1ω1ω1 -q-12 Pω2ω1ω1- q-2n+12 P0ω1ω1. (6.3)

As in the cases of 𝔤=𝔰𝔬(2n+1) and 𝔰𝔬(2n) one can calculate the matrices K2ω1ω1ω1, Kω2ω1ω1, and K0ω1ω1. Substituting these matrices into (6.3) we obtain the matrix Rω1ω1: Rω1ω1 = q12iEii Eii+ij,j EjiEij +q-12ii EiiEii+ (q12-q-12) i>jEii Ejj -(q12-q-12) i>jEiEj qj-i2 EijEij (6.4) where i=2n+1-i; ϵi=1, i=1,,n, ϵj=-1, i=n+1,,2n; i=i-12, in, i=i+12, in+1. This matrix can also be extracted from [Jim1986-2].

The matrices Rλμ and Pλ for any h.w. λ,μ can be found from the theorems 2. - 3. and from the ramification rule: VλVω1= k=12n Vλ(k) (6.5) where λ(k)=(λ1,,λk+1,,λn), λ(n+k)=(λ1,,λk-1,,λk), k=1,,n.

As in the case 𝔤=𝔰𝔬(2n+1),𝔰𝔬(2n) we have the following propositions.

Proposition 6.1. The algebra CNω1(𝔰𝔭(2n)) is the factor of B.-W. algebra with m=(i(q1/2-q-1/2)), =iq-2n+12 over the ideal formed by the elements J=Pωn+1 (q)1n+1 (6.6) acting nontrivially only in the multipliers of (Vω1)N with numbers in+1. The elements Pk(q) are defined by (5.11) with the matrix Q= ( 111 q q-k qk-2-2n q2 q-2k q2(k-2-2n) ) (6.7) and CMω1(𝔰𝔭(2n))BWM for n+1>M.

If we consider the block basis in Vω1 connected with the embedding Uq(𝔰𝔭(2n))Uq(𝔰𝔩(n)), Vω1=Vω1Vω1*, dimVω1=dimVω1*=n (6.8) we obtain the representation of Rω1ω1 in the block form with the blocks constructed from Uq(𝔰𝔩(n)) R-matrices.

Proposition 6.3. The matrices (Rω1ω1)±1 have the following block structure in the basis (6.9): = 0 0 0 0 a ( +q-n+12 ) 0 0 0 0 0 0 0 (6.9) = 0 0 0 0 0 0 0 -a ( +qn+12 ) 0 0 0 0 (6.10) a=q12-q-12 where we use the notations of sections 2 and 5.

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