The nonstandard quantum plane

Let k be a field of char(k) 6= 2, and suppose that q ∈ k is not a root of unity. The associated quantum plane [5, §IV.1], denoted by kq[x, y], is defined to be the free k-algebra k{x, y} generated by x and y, modulo the relation yx = qxy. The set of monomials {xy}i, j≥0 is a basis for the underlying k-vector space, and for every pair (i, j) of nonnegative integers, we have yx = qxy . There is a natural action on the quantum plane by the quantum group Uq, which is defined to be the k-algebra generated by the four variables E, F, K, K−1, modulo the relations: KK−1 = K−1K = 1 , KEK−1 = qE , KFK−1 = q−2F , (1)