Realization Of The Quantum Symmetric Space Of Type B As A Module Algebra Of U_q(so(2n+1))

The discussion in the present paper arises from exploring intrinsically the struc-tual nature of the quantum symmetric space of type B. The rings of q-differential operators on quantum planes of type B are isomorphic to the rings of classical differential operators. We construct decomposition of the ring of q-differential operators into tensor products of the rings of q-differential operators with less variables. After the recurrence of calculation we get that Diff(R_q²ⁿ⁺¹)≌ (Diff_q²(1))^?(2n+1). Then we define a subalgebra U_q^N whose elements can commute with the length operators and the Laplace operators. Then the quantum symmetric space is a module algebra of the U_q^N . U_q^N has a Hopf algebra structure which is isomophic to U_q ,(so(2n + 1)) as Hopf algebras. Therefore we realise the quantum symmetric space of type B as a module algebra of U_q (so{2n + 1)).