Derivation Algebras And 2-Cocycles Of The Algebra Of (r,s)-Differential Operators

Let C[t,t^-1] be the algebra of Laurent polynomials over the complex numbers field, and (?) = d/dt the differential of C[t,t^-1]. Let q be a complex number which is not 0 and 1, similar to d, the q-differential ofC[t,t^-1] is defined as:Let r, s be two different complex numbers which are not 0 and 1, we define (r, s)-differential of C[t,t^-1] as follows:Let D be the associative algebra generated by {t,(?)}, and D_q be the algebra generated by {t,(?)_q}.There have been many papers making research on the derivations, 2-cocycles and automorphisms of V and D_q. Let D_r,s be the associative algebra generated by {t,(?)_r,s}, which is called (r,s)-differential operators algebra of C[t,t^-1]. In this paper, the derivation algebra of D_r,s and its Lie algebra D_r,s - will be described and all the non-trivial 2-cocycles will be determined under the condition of r^x s^y≠ 1(x,y∈ Z,x,y never be 0 at the same time).Let Z⁺ be the set of all nonnegative integers. And the condition r^x s^y ≠ 1(x, y ∈Z, x, y never be 0 at the same time) is always assumed through the paper.Define the automorphism ξ of C[t, t^-1] satisfying ξ(t) = st. Set D = (r -1)t(?)_r,s + ξ. For ξ = t(?)_r,s - r(?)_r,st, D ∈ D_r,s. Then we get a C-basis ofD_r,s, which is {t^m Dⁿ ξ^q| m∈Z,n,q∈ Z⁺}, that is to say,D_r,s = {∑ C(m,n,q)t^m Dⁿξ^q|C(m,n,q)∈ C}.Using the result above, in the second part of this paper, the derivation algebra of the associative algebra D_r,s is determined as foollows:Theorem 2.1: Der(D_r,s) = ad (D_r,s) (?) ∑_i=1³ C_σi , where σ_i (i = 1,2,3) are outer derivations of D_r,s, satisfying σ₁(t) = t, σ₁(D) = σ₁(ξ) = 0, σ₂(D) = D,σ₂ (t) = σ₂(ξ) = 0, σ₃(ξ) = ξ,σ₃(D) = σ₃(t) = 0.Upon theorem 2.1, the derivation algebra of the Lie algebra D_r,s - is discribed, and the result isTheorem 3.1: Der(D_r,s -)= ad (D_r,s(?)=∑_i=1³ Cσ_i(?)∑_j∈Z\{0}Cζ_j where ζ_i(i∈ Z \ { 0}) are the outer derivations of the Lie algebra D_r,s -, satisfyingIn the last part of this paper, all the nontrial 2-cocycles are determined as follows:Theorem4.1: dimH²(D(r,s) -,C) = ∞. Every 2-cocycle on D_r,s - is equivalent to one of the following 2-cocycles: where a_m,m1 are arbitrary constants with a_m,m1 = -a_m1,m.