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.

𝔤=𝔰𝔬(2n+1) and 𝔤=𝔰𝔬(2n)

The h.w. of those algebras can be integer or halfinteger vectors. For 𝔤=𝔰𝔬(2n+1), λ=(λ1,,λn), λ1λ2λn, λi12+, on the h.w. vector eλ:Hieλ=(λi-?), i=1,,n-1, Hneλ=λneλ. For 𝔤=𝔰𝔬(2n), λ=(λ1,,λn-1,λn), λ1λn-1λn, λi12+, i=1,,n-1, λn12, Hieλ=(λi-λi+1)eλ, i=1,,n-1, Hneλ=(λn-1+λn). The representation with integers h.w. λ form a category of tensoriol representations of 𝔤. The basic representation in this category is the representation with h.w. ω1=(1,0,,0). In the category of finite dimensional representations basic representation of 𝔰𝔬(2n+1) have h.w. ωn=(12,,12). For 𝔰𝔬(2n) there are two basic representations in this category. These representations are ω1 and ωn or ω1 and ωn-1 or ωn and ωn-1. All three possibilities are equivalent. Let us start to study the algebras CNω(𝔤) and R-matrices connected with algebras Uq(𝔰𝔬(2n)) and Uq(𝔰𝔬(2n+1)) from the category of a tensorial representation.

Tensorial representations of Uq(𝔰𝔬(2n)) and Uq(𝔰𝔬(2n+1))

The vectorial representation Vω1 of Uq(𝔰𝔬(2n+1)) and Uq(𝔰𝔬(2n)) do not depend on q: Uq(𝔰𝔬(2n+1)) π(Xj+) = Ejj+1- EN-j,N+1-j, j=1,, π(Xn+) = 12- (En+1,n+2+Enn+1) π(Hj) = Ejj- Ej+1,j+1- EN+1-j,N+1-j +EN-j,N-j, j=1,,n-1 π(Hn) = Enn -En+2,n+2 π(Xj-) = π(Xj+)t (5.1) Uq(𝔰𝔬(2n)) π(Xj+) = Ejj+1- EN-j,N+1-j, j=1,,n-1 π(Xn+) = Enn-1- Enn+2, π(Hj) = Ejj- Ej+1j+1- EN+1-j+ EN-jN-j, j=1,,n-1 π(Hn) = Enn +En-1,n-1- En+1,n+1- En+2,n+2, π(Xj-) = π(Xj+)t. (5.2) Here Eij are the basic matrices in Vω1N, N=2n+1, for Uq(𝔰𝔬(2n)) and N=2n for Uq(𝔰𝔬(2n)), t is the transposition. The vector representations of Uq(𝔰𝔬(2n)) and Uq(𝔰𝔬(2n+1)) are self-adjoint: ω1*=ω1. The tensor product of two vectorial representations is decomposed into the sum of three irreducible components (Vω1)2= V2ω1 Vω2V0. (5.3)

After some calculations one can find the matrices K2ω1ω1ω1, Kω2ω1ω1, K0ω1ω1. They are determined by the following formulae ij,i<j, eij2ω1= 11+q {eiej+q1/2ejei}, ii,eii2ω1 =eiei, eii2ω1=ni { q1/2·eiei +q-1/2ei ei- (q-1/2-q1/2) q-1/2-qN-12 ( qN-12k>i qi-k2 ekek+ q-1/2ki qi-k2 ekek ) } (5.4) eijω2= eiej-q1/2ejei1+q ,i<j eiiω2= ni { eiei- eiei+ (q-12-q12) q12+qN-12 ( qN2k>i qi-k2 ekek+q ki qi-k2 ekek ) } . (5.5) eij0= (q12-q-12) (q-N4-q-N4) (qN-24+q-N-24) δijk qi-k2 ekek (5.6) where ni are normalization constants, j=N+1-j, eij2ω1, eijω2, eij0 are the bases in V2ω2, Vω2, V0 respectively, i= { i+12, in i, i=n+1 i-12, i>n+1 forUq(𝔰𝔬(2n+1)), i= { i+12, in i-12, i>n forUq(𝔰𝔬(2n)). From the theorem 1.5 we obtain the matrix Rω1ω1: Rω1ω1= P2ω1ω1ω1 q-1/2- Pω2ω1ω1 q1/2+ P0ω1ω1 q-N-12. (5.7) Substituting the expressions (5.4)-(5.6) in this formula we obtain the representation of Rω1ω1 as N2×N2 matrix: Rω1ω1 = q1/2 iN-12 EiiEii+ EN+12,N+12 EN+12,N+12 +ij,j EjiEij +q-1/2 ii Eii Eii -(q-1/2-q1/2) i>jEjj Eii+(q-1/2-q1/2) i>j qj-i2 EijEij, (5.8) where Eij are the basic matrices in N and the second term is present only for Uq(𝔰𝔬(2n+1)).

The matrix (5.8) can be extracted from the work [Jim1986-2]: Rω1ω1=limxx-1/2Ř(x), where Ř(x) is the solution of Yang-Baxter equation corresponding to the series Bn(1) and Dn(1).

The following decompositions are well known: Vω1Vλ= k=12n+1 Vλ(k), 𝔤=𝔰𝔬(2n+1), (5.9) Vω1Vλ= k=12n Vλ(k), 𝔤=𝔰𝔬(2n). (5.10) Here λ(k)=(λ1,,λk+1,,λn), λ(n+1+k)=(λ1,,λk-1,,λn), λ(k+1)=λ in (5.9) and λ(n+k)=(λ1,,λk-1,,λn) in (5.10), 1kn.

Using the theorem 3.1 and the decompositions (5.9) (5.10) we obtain the formulae for projectors Pλ(q,a). For Pωk(q) (mult(ωk(ω1)k)=1) we have the following representation Pωk(q)= n1,,nk=0c i=1kqni,1 ω ω ω ω α(n1,,nk) (5.11) where Q=(qi,j)i=0,1,2j=0,1,2= ( 111 qq-kq-N+k q2q-2kq-2N+2k ) -1 (5.12) and α(n1,,nk) is defined by (3.19).

The algebras CMω1(𝔰𝔬(2n-1)) and CMω1(𝔰𝔬(2n+1)) are connected with the Birman-Wenzl algebra. The last one is an associative algebra formed by the generators 1, Gi, Ei, i=1,,M-1 with the following relations GiGj = GjGi, |i-j|2 GiGi+1Gi = Gi+1GiGi+1, Gi+Gi-1 = m(1+Ei), EiEi+1Ei = Ei=EiEi-1 Ei Gi+1GiEi+1 = EiGi+1Gi= EiEi+1 Gi-1GiEi-1 = EiGi-1Gi= EiEi-1 Gi+1EiGi+1 = Gi-1Ei+1 Gi-1 Gi-1EiGi-1 = Gi-1Ei-1 Gi-1 Gi+1EiEi+1 = Gi-1Ei+1 Gi-1EiEi-1 = Gi-1Ei-1 Ei+1EiGi+1 = Ei+1Gi-1 Ei-1EiGi+1 = Ei-1Gi-1 GiEi = EiGi= -1Ei EiGi+1Ei = EiGi-1Ei= Ei. (5.13)

Theorem 5.1. The algebras CMω1(𝔰𝔬(2n)) and CMω1(𝔰𝔬(2n+1)) are factors of Birman-Wenzl algebra with m=i(q1/2-q-1/2), =-iqN-12 over the ideals formed by the elements J=Pωn+1 (q)1,,n+1 acting nontrivially only in the multipliers of (Vω1)M with numbers in+1. In other multipliers J acts as a unit matrix. Here Pωn+1(q) is defined by (5.11), (5.12). The generators Gi of B.W. algebra are connected with the generators of CMω1(𝔤) by the identification Gj=igj, the generators Ej are determined by (5.13). For n+1>M, CMω1BWM.

Consequence. The representation α of B.W. algebra with m=i(q1/2-q-1/2), =-iqN-12, α(G)=i(1Rω1ω11) defined in the space (Vω1)M.

The structure of algebras CMω1(𝔰𝔬(2n)), CMω1(𝔰𝔬(2n+1)) and CMω1(𝔰𝔩(n)) is in agreement with the natural embeddings 𝔰𝔩(n)𝔰𝔬(2n+1), 𝔰𝔩(n)𝔰𝔬(2n).

Theorem 5.2. The algebras Uq(𝔰𝔬(2n)) and Uq(𝔰𝔬(2n+1)) contain Uq(𝔰𝔩(n)) as a Hopf subalgebra.

This theorem follows from the commutation relations (1.1)-(1.3) and from the formulae (1.4) (1.5) for coproduct in Uq(𝔤).

Let us consider the restriction of the vector representation of Uq(𝔰𝔬(2n)) and Uq(𝔰𝔬(2n+1)) on the subalgebra Uq(𝔰𝔩(n)). We have Vω1 = Vω1 Vω1* V0forUq (𝔰𝔬(2n+1)) Vω1 = Vω1 Vω1* forUq(𝔰𝔬(2n)). Here Vω1 and Vω1* are vector and conjugated to vector (Vω1*=Vωn-1) representations of Uq(𝔰𝔩(n)). The Uq(𝔰𝔩(n)) R-matrices Rω1ω1, Rω1ω1*, Rω1*ω1 and Rω1*ω1* in accordance with the notations of the section 2 we represent by the graphs: , , , . From the symmetries of R-matrices we have Rω1ω1=Rω1*ω1*, Rω1*ω1=Rω1*ω1T.

Theorem 5.3. The basic R-matrices of Uq(𝔰𝔬(2n)) and Uq(𝔰𝔬(2n+1)) have the following block structures at the restriction of Uq(𝔰𝔬(2n)) and Uq(𝔰𝔬(2n+1)) on Uq(𝔰𝔩(n)) for Uq(𝔰𝔬(2n+1)): = n×n n n×n n 1 n n×n n n×n n×n 0 0 0 0 0 0 0 0 n 0 a 0 0 0 0 0 0 n×n 0 0 a ( -q-n-32 ) 0 aq-n-34 0 0 0 n 0 0 0 0 0 0 0 0 1 0 0 aq-n-34 0 1 0 0 0 0 n 0 0 0 0 0 a 0 0 n×n 0 0 0 0 0 0 0 0 n 0 0 0 0 0 0 0 0 n×n 0 0 0 0 0 0 0 0 a=(q-12-q12) (5.16) = n×n n n×n n 1 n n×n n n×n n×n 0 0 0 0 0 0 0 0 n 0 0 0 0 0 0 0 0 n×n 0 0 0 0 0 0 0 0 n 0 0 -a 0 0 0 0 0 1 0 0 0 0 1 0 -aqn-34 0 0 n 0 0 0 0 0 0 0 0 n×n 0 0 0 0 -aqn-34 0 -a ( -qn-32 ) 0 0 n 0 0 0 0 0 0 -a 0 n×n 0 0 0 0 0 0 0 0 (5.17) for Uq(𝔰𝔬(2n)): = n×n n×n n×n n×n n×n 0 0 0 n×n 0 a ( -q-n-12 ) 0 n×n 0 0 0 n×n 0 0 0 (5.18) = 0 0 0 0 0 0 0 -a ( -q-n-12 ) 0 0 0 0 (5.19)

This theorem follows from the formulae (5.8) and (4.6).

Corollary. There exist the following embeddings of the algebras CNω1: CMω1 (𝔰𝔬(2n+1)) CMω1 (𝔰𝔩(n))× CMω1(𝔰𝔩(n)).

Using the theorem 5.3 and the action (4.11), (4.12) of CMω1(𝔰𝔩(n)) in Kλ-basis in Wλ one can find the action of CMω1(𝔰𝔬(2n+1)) and CMω1(𝔰𝔬(2n)) in Kλ-basis in Wλ. Explicit formulae will be given in a separate publication.

The spinor representations

The spinor representations Vωn and Vωn-1 of Uq(𝔰𝔬(2n)) each have a dimension 2n-1. For Uq(𝔰𝔬(2n+1)) spinor representation Vωn has dimension 2n. The bases in spinor representations are parametrized by the weights ε=(ε1,,εn), εi=±12. For Uq(𝔰𝔬(2n)) there are auxiliary restrictions on ε: i=1nεi=2n for Vωn and i=1nεi=-2-n for Vωn-1. The spinor representations of Uq(𝔰𝔬(2n+1)) and Uq(𝔰𝔬(2n)) are defined by the action of the generators on the basic vectors eε: Uq(𝔰𝔬(2n+1)): Xi+eεiεi+1 = -eεi+1,εi-1,, Xieεn = 12ch(h4) eεn+1, Hieε = (εi-εi+1) eε,i=1,,n-1 Hneε = εneε Xi- = (Xi+)t. (5.20) Uq(𝔰𝔬(2n)): Xi+eεiεi+1 = -eεi+1,εi+1-1,, Xn+eεn-1εn = -eεn-1+1,εn+1,, Hieε = (εi-εi+1) eε, Hneε = (εn-1+εn) eε, Xi- = (Xi+)t. (5.21) Here r.h.s. =0 if the weight ε of eε does not belong to the set of weights of the space Vωn (or Vωn-1). The tensor product of spinor representations has the following spectral decomposition Uq(𝔰𝔬(2n)): Vωn-1 Vωn-1 VωnVωn = { (s=0n2-1Vω2s) V2ωn, neven (s=1n-12Vω2s-1) V2ωn, nodd (5.22) VωnVωn-1 = { (s=1n2-1Vω2s-1) Vωn+ωn-1, neven (s=0n2-2Vω2s) Vωn+ωn-1, nodd (5.23) Vω1Vωn = { Vωn-1 Vωn+ω1, neven Vωn Vωn+ω, nodd (5.24) Vω1Vωn-1 = { Vωn Vωn-1+ω1, neven Vωn-1 Vωn-1+ω1, nodd (5.25) Uq(𝔰𝔬(2n)): VωnVωn = (s=0n-1Vωs) V2ωn Vω1Vωn = VωnVω1+ωn. (5.26)

Let us describe the KFC of these decompositions. As in the previous part of this section we shall use the graphical representation of the matrices Rλμ and Kνλμ given in section 2.

It is convenient to consider the space Vs=VωnVωn-1 for Uq(𝔰𝔬(2n)) and the matrix Kssω1= ( Kωnωnω1 Kωn-1ωnω1 Kωnωn-1ω1 Kωn-1ωn-1ω1 ) For Uq(𝔰𝔬(2n+1)) we shall write Vs=Vωn.

Theorem 5.5. The matrices Kω1 for Uq(𝔰𝔬(2n)) and for Uq(𝔰𝔬(2n+1)) satisfy the following relations: s s s ω1 ω1 =q-1/2 s s s ω1 ω1 +(1+q-N-22) s s ω1 ω1 (5.27)

The proof of this theorem follows from theorem 3.1.

It is interesting that the relations () between the matrix elements of Kωnωnω1 and Kssω1 permit us to introduce q-analog of Clifford algebra. If we write (γi)αβ= s s ω1 α β i (5.28) the relations (5.27) give the q-anticommutation relations for the matrices γi: γiγj = -q-i/2 γjγi, i<j,ij γi2 = 0 γiγi = -γiγi- (q-1-1) k>i qi-k2 γkγk+ (1+qN-22) qi2 (5.29) with normalization condition iγiγi= χω1. Using explicit form of the spinorial and vector representations the matrices Kssω1 can be calculated exactly. But we will not present here these calculations because they take up too much space and are not very instructive for out purposes.

The matrices Kω1ss are the crossing transformations of the matrices Kssω1 (see (2.20)). Using the formulae of the section 5.1 and (5.27) we obtain an expression for the projector Pωkss: s s* ωk s* s =Nks s s s* s* ω1 ω1 ω1 ω1 ω1 ω1 ωk (5.30) where Nks are some normalisation constants and projector P:(Vω1)kVωk was given by (5.11).

So, we have the expression for spinor-spinorial R-matrices through the matrices Kω1ss and Rω1ω1: ωn ωn = s=0n-1 q14((2n+1)k-k2-n2-n2) ωn ωn ωk ωn ωn +qn8 ωn ωn 2ωk ωn ωn (5.31) ω(n-1n) ω(n-1n) = { s=0n2-1 (-1) q(ns-s2-n24+n8) ω(n-1n) ω(n-1n) ω2s ω(n-1n) ω(n-1n) +qn8 ω(n-1n) ω(n-1n) 2ω(n-1n) ω(n-1n) ω(n-1n) , neven s=1n-12 (-1) q(ns-s2-(n+1)24+n8) ωn-1 ωn-1 ω2s-1 ωn-1 ωn-1 +qn8 ωn-1 ωn-1 2ωn-1 ωn-1 ωn-1 , nodd (5.32) ωn ωn-1 = { s=1n2-1 (-1) q(ns-s2-(n+1)24+n8) ωn ωn-1 ω2s-1 ωn-1 ωn +qn8-14 ωn ωn-1 ωn+ωn-1 ωn-1 ωn , neven s=0n2-2 (-1)q(ns-s2-n24+n8) ωn ωn-1 ω2s ωn-1 ωn +qn8-14 ωn ωn-1 ωn+ωn-1 ωn-1 ωn nodd (5.33) ω1 ω(n-1n) = { q ω1 ω(n-1n) ω1+ω(n-1n) ω(n-1n) ω1 -q-2n-14 ω1 ω(n-1n) ω(nn-1) ω(n-1n) ω1 , neven q ] ω1 ω(n-1n) ω1+ω(n-1n) ω(n-1n) ω1 -q-2n-14 ω1 ω(n-1n) ω(n-1n) ω(n-1n) ω1 , nodd. (5.34) Here Pωkωsωs are given by (5.30).

Using the commutation relations (5.27) it is easy to prove the following identities: ω1 ω1 ω1 ω1 ωs 2n = (qq12)n (1+qn-12)(1+q12) k1=12n-1 k2k1,k1 kik1,k1,,ki-1,ki-1 (-q)12(k1++kn) ω1 ω1 ω1 ω1 [k1,,kn] 2n ki = maxj { j(1,,2n)\ (k1,k1,,ki-1,ki-1) } . (5.35) Here [k1kn] is the joint in which the line numbered (the numeration being from left to right) is connected with the ki-line in such a way that the ki-line goes above kj-line if i<j.

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