In the Hopf algebra $SL\_q(N)$, it can be shown, using direct calculations, that $S(u^1\_i)u^j\_1 = q^{-1}u^j\_1S(u^1\_i)$. Can anyone see a more elegant way of establishing this?


Moreover, does anyone know of a similar relation in the more general case of $S(u^1\_r)u^i\_j$?


Edit (referneces): By $SL\_q(N)$ I mean the quantized coordinate algebra (not the quantized enveloping algebra). I am using the conventions of Klimyk and Schmudgen, Chpt 4 for the N=2 case, or Chpt 9 for the general case. 



The relation $S(u^1_i)u^j_1 = q^{-1}u^j_1S(u^1_i)$
This is a reasonably known result. That $S(u^1\_i)u^j\_1 = q^{-1}u^j\_1S(u^1\_i)$, was originally proven (to the best of my knowdledge) in FRT's '89 paper "Quantum Groups and Lie Algebras" - the paper is in Russian though. The only English write up of the proof that I known is in Theorem 1 of Vainermann and Podkolzin's '99 [paper](http://docs.google.com/viewer?a=v&q=cache%3AH1PD0VCip6MJ%3Aciteseerx.ist.psu.edu/viewdoc/download%253Fdoi%253D10.1.1.145.4800%2526rep%253Drep1%2526type%253Dpdf+quantum+Stieffel+manifold&hl=ga&pid=bl&srcid=ADGEESj18ZxTGm6y0t_mQ4GrrzMrIRnK_9B_Oynw9gDehj-Z6NEiQHuWDEhtgQxsfu8rEmlZnp1b-O6GGfNhD5mEJQK15EUlHs9WiQGsbIZJp61PjBfrTgJtrEnovntBZZBh_OmYcjXk&sig=AHIEtbRxdwbZXWeC01h1RfkqMawEnDG86w) on Quantum Stiefel Manifolds. It gives a general comm rel for $[S(u^i\_j),u^r\_s]$, for the general $N$ case, using just the $R$-matrix construction of the $SU\_q(N)$. I am sure there are other versions around somewhere though.


