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

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.

## $𝔤={G}_{2}$

The irreducible finite dimensional representation of ${U}_{q}\left({G}_{2}\right)$ are parametrized by h.w. $\lambda ={\lambda }_{1}{\omega }_{1}+{\lambda }_{2}{\omega }_{2},$ where ${\omega }_{1}$ and ${\omega }_{2}$ are the fundamental weights (see [Bou1981]) and ${\lambda }_{i}\in {ℤ}_{+}\text{.}$ If ${e}_{\lambda }\in {V}^{\lambda }$ the h.w. vector, ${H}_{1}{e}_{\lambda }={\lambda }_{1}{e}_{\lambda },$ ${H}_{2}{e}_{\lambda }=3{\lambda }_{2}{e}_{\lambda }\text{.}$

The basic representation of ${U}_{q}\left({G}_{2}\right)$ have h.w. $\lambda ={\omega }_{1}\text{.}$ Let us consider the basics in ${V}^{{\omega }_{1}}$ formed by weight vectors,  ${e}_{1}={e}_{{\omega }_{1}},$ ${e}_{2}={e}_{{\alpha }_{1}+{\alpha }_{2}},$ ${e}_{3}={e}_{{\alpha }_{1}},$ ${e}_{4}={e}_{0},$ ${e}_{5}={e}_{1},$ ${e}_{6}=e-{\alpha }_{1}-{\alpha }_{2},$ ${e}_{7}=e-{\omega }_{1}\text{.}$ The roots are shown on the figure 1. ${\alpha }_{1} {\alpha }_{2} {\omega }_{1} {\omega }_{2} Figure 1.$ In this basis the elements ${H}_{i},{X}_{i}^{±}$ 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 $b=\sqrt{\frac{\text{sh}\left(\frac{h}{2}\right)}{\text{sh}\left(\frac{3h}{2}\right)}},$ $c=\sqrt{\frac{\text{sh}\left(h\right)}{\text{sh}\left(\frac{3h}{2}\right)}}\text{.}$

In this normalization the commutation relations between the generators ${H}_{i},{X}_{i}^{±}$ 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 ${V}^{{\omega }_{1}}$ is selfdual: ${V}^{{\omega }_{1}^{*}}={V}^{{\omega }_{1}}\text{.}$

A tensorial product of the two basic representations of ${U}_{q}\left({G}_{2}\right)$ has the following decomposition: $Vω1⊗Vω1= V2ω1⊕Vω2 ⊕Vω1⊕V0. (7.3)$ In this decomposition $\stackrel{‾}{2{\omega }_{1}}=0,$ $\stackrel{‾}{{\omega }_{2}}=\stackrel{‾}{{\omega }_{1}}=1,$ $\stackrel{‾}{0}=0\text{.}$ It is not difficult to calculate the matrix ${K}_{{\omega }_{1}}^{{\omega }_{1}{\omega }_{1}}\text{:}$ $e1 = g ( cq34e2⊗ e3-bq32 e1⊗e4+b q-32e4 ⊗e1-cq-14 e3⊗e2 ) , e2 = g ( bq12e2⊗ e4-cq54 e1⊗e5+c q-54e5 ⊗e1-bq-12 e4⊗e2 ) , e3 = g ( bq12e3⊗e4- cq54e1⊗e6+ cq-54e6⊗ e1-bq-12 e4⊗e3 ) , e4 = gb ( -qe1⊗e7- q32e2⊗ e6+e3⊗ e5+ (q12-q-12) e4⊗e4-e5⊗e3 +q-32e6⊗ e2+q-1e7 ⊗e1 ) , e5 = g ( bq12e4⊗ e5-cq54e2 ⊗e7+cq-54 e7⊗e2-b q-32e5⊗e4 ) , e6 = g ( bq12e4⊗e6 -cq54e3⊗ e7+cq-54 e7⊗e3-b q-12e6⊗ e4 ) , e7 = g ( cq34e5⊗ e6-bq32 e4⊗e7+b q-32e7 ⊗e4-cq-34 e6⊗e5 ) , g = 1q2q-2, (7.4)$ So the matrix ${\left({K}_{{\omega }_{1}}^{{\omega }_{1}{\omega }_{1}}\right)}_{i}^{jk}$ $ei=∑j,k=17 (Kω1ω1ω1)ijk ej⊗ek (7.5)$ is given explicitly.

Theorem 7.1. The matrix ${R}^{{\omega }_{1}{\omega }_{1}}$ satisfies the following relations: $-q =(q-1) { -α +β } (7.6) -q =(q-1) { -α +β } . (7.7)$ Here $\alpha =q+{q}^{-1},$ $\beta =\frac{{q}^{-3}\left({q}^{3}-1\right)\left({q}^{4}+1\right)}{q-1}\text{.}$

 Proof. From the theorem 1.5 we have the spectral decomposition of ${R}^{{\omega }_{1}{\omega }_{1}}\text{:}$ ${\omega }_{1} {\omega }_{1} =q {\omega }_{1} {\omega }_{1} 2{\omega }_{1} {\omega }_{1} {\omega }_{1} - {\omega }_{1} {\omega }_{1} {\omega }_{2} {\omega }_{1} {\omega }_{1} -q-3 {\omega }_{1} {\omega }_{1} {\omega }_{1} {\omega }_{1} {\omega }_{1} +q-6 {\omega }_{1} {\omega }_{1} 0 {\omega }_{1} {\omega }_{1} . (7.8)$ One-dimensional projector is defined in section 1 (1.47): ${\omega }_{1} {\omega }_{1} 0 {\omega }_{1} {\omega }_{1} = 1trVω1(q-ρ) =q2+1q q-1 (q7-1)(q-6+1) . (7.9)$ Substituting (7.8), (7.9) in the l.h.s. of 7.6 and using the decomposition of units in ${V}^{{\omega }_{1}\otimes 2}\text{:}$ $= 2{\omega }_{1} + {\omega }_{2} + {\omega }_{1} + 0$ we obtain (7.6). The equality (7.7) is obtained by rotating the pictures (7.6) by ${90}^{\circ }$ (making crossing-transformation). $\square$

Considering the equalities (7.6) and (7.7) as a system of linear equations for ${R}^{{\omega }_{1}{\omega }_{1}}$ and ${\left({R}^{{\omega }_{1}{\omega }_{1}}\right)}^{-1}$ we obtain an expression for these matrices in terms of the matrix ${K}_{{\omega }_{1}}^{{\omega }_{1}{\omega }_{1}}$ (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 ${G}_{2}$ to the $𝔰𝔩\left(2\right)$ triple formed by ${X}_{1}^{±},{H}_{1}\text{.}$ The representation ${V}^{{\omega }_{1}}$ is the sum of three $𝔰𝔩\left(2\right)\text{-irreducible}$ components. $Vω1 |𝔰𝔩(2) =Vω1⊗V2 ⊗V∼1. (7.12)$ Here $\text{dim} {V}^{1}=\text{dim} {\stackrel{\sim }{V}}^{1}=2,$ $\text{dim} {V}^{2}=3\text{.}$ The spaces ${V}^{1},{V}^{2}$ and ${\stackrel{\sim }{V}}^{1}$ are formed by the vectors $\left({e}_{1},{e}_{2}\right),$ $\left({e}_{3},{e}_{4},{e}_{5}\right)$ and $\left({e}_{6},{e}_{7}\right)$ correspondingly.

For $𝔰𝔩\left(2\right)$ CGC we have: $V2⊂V1⊗V1$ $e2 = e1⊗e1, e0 = 1q1/2+q-1/2 ( q14e-1⊗ e1+q-14 e1⊗e-1 ) , e-2 = e-1⊗e-1, (7.12)$ where ${e}^{±1}$ are the bases in $2\text{-dimensional}$ representation of ${U}_{q}\left(𝔰𝔩\left(2\right)\right)$ and ${e}^{2},{e}^{0},{e}^{-2}$ are the bases of a $3\text{-dimensional}$ one. $V2⊂V2⊗V2$ $e2 = 1q+q-1 ( q12e2⊗ e0-q-12 e0⊗e2 ) , e0 = 1q+q-1 ( e2⊗e-2+ (q12-q-12) e0⊗e0- e-2⊗e2 ) , e-2 = 1q+q-1 ( q12e0⊗ e-2-q-12 e-2⊗e0 ) . (7.13)$ 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 $2\text{-dimensional}$ representation. An index $2$ corresponds to a $3\text{-dimensional}$ representation.

From (7.4) and (7.12)-(7.14) we obtain the $𝔰𝔩\left(2\right)\text{-block}$ form of the matrix ${K}_{{\omega }_{1}}^{{\omega }_{1}{\omega }_{1}}\text{:}$ $Kω1ω1ω1= =1q2+q-2$ $0 -q 1 2 2 0 q-1 1 2 1 0 0 0 0 0 0 xq-54 2 1 1 0 y 2 2 2 0 -xq54 2 1 1 0 0 0 0 0 0 q 1 2 1 0 -q-1 1 1 2 (7.15)$ 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 ${P}_{{\omega }_{1}}^{{\omega }_{1}{\omega }_{1}}\text{:}$ ${\omega }_{1} {\omega }_{1} {\omega }_{1} {\omega }_{1} {\omega }_{1} =1q2-q-2 0 0 0 0 q2 1 2 1 1 2 0 0 0 x2q-32 1 1 2 1 1 0 0 0 - 1 1 2 1 1 0 0 0 xyq-54 1 1 2 2 1 0 0 0 0 0 0 0 -x2 1 1 2 1 1 0 0 0 - 2 1 1 1 2 0 0 0 xyq-54 2 2 2 1 1 0 0 0 q-2 2 1 1 2 1 0 0 0 y2 2 2 2 2 2 0 0 0 q2 2 1 1 2 1 0 0 0 -yxq54 2 2 2 1 1 0 0 0 - 2 1 1 1 2 0 0 0 -x2 1 1 2 1 1 0 0 0 0 0 0 0 -yxq54 1 1 2 2 2 0 0 0 - 1 2 1 2 1 0 0 0 x2q52 1 1 2 1 1 0 0 0 q-4 1 2 1 1 2 0 0 0 0 (7.16)$ and the similar expression for crossing-conjugate matrix. For one dimensional projector there is a similar representation: ${\omega }_{1} {\omega }_{1} {\omega }_{1} {\omega }_{1} = 0 0 0 0 0 0 0 0 q92 1 1 0 0 0 0 0 0 0 q94 1 2 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 q94 2 1 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 2 1 q-94 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 q-94 1 2 0 0 0 0 0 0 0 1 1 q-92 0 0 0 0 0 0 0 0 (7.17)$ where $\begin{array}{c} 1 -1 1 \end{array}={q}^{-\frac{1}{4}},$ $\begin{array}{c} 1 1 -1 \end{array}={q}^{\frac{1}{4}},$ $\begin{array}{c} 2 -2 2 \end{array}=q,$ $\begin{array}{c} 2 0 0 \end{array}=1,$ $\begin{array}{c} 2 2 -2 \end{array}={q}^{-1}\text{.}$

Substituting (7.15)-(7.17) in (7.10) and (7.11) we obtain an explicit expression for ${R}^{{\omega }_{1}{\omega }_{1}}$ though $𝔰𝔩\left(2\right)\text{-CGC.}$