References about an action $U_q(n) \times \mathbb{C}_q[N] \to \mathbb{C}_q[N]$.

66 Views Asked by At

Let $N$ be the unipotent subgroup of a Lie group $G$ consisting of all upper triangular matrices and $n$ the Lie algebra of $N$. I found in some paper that there is an action $U_q(n) \times \mathbb{C}_q[N] \to \mathbb{C}_q[N]$ which is given by $$ E_i.x(E) = x(EE_i), \\ K_i.x=K_{i,\lambda}.x=q^{\lambda(\alpha_i)-(\gamma, \alpha_i)}x, \\ F_i.x=\frac{v_i^{l_i}q^{-(\gamma, \alpha_i)}xx_i - v^{-l_i}x_ix}{v_i-v_i^{-1}}. $$ What are the references in which this action is first defined? If we let $q \to 1$, do this action becomes the action of the Lie algebra $n$ on $\mathbb{C}[N]$ which is given by $a(f)(x) = \frac{d}{dt}|_{t=0} f(e^{-at}x)$? Thank you very much.