A Quantum Deformation of the Second Weyl Algebra: Its derivations and poisson derivations

Since the introduction of quantum algebras in the 1980's, many have introduced quantum deformations of the Weyl algebras. Two such examples are the quantum Weyl algebras and the Generalized Weyl algebras. In this thesis, we use a different approach to find deformations of (a quadratic extension of) the second Weyl algebra A_2(C), and compare some properties of these deformations to those of A_2(C).

Let n be a nilpotent Lie algebra, U(n) the enveloping algebra of n and Q a primitive ideal of U(n). Dixmier [12] proved that the factor algebra U(n)/Q is isomorphic to an n^{th} Weyl algebra A_n(C), where n ∈ N_{≥1}. This isomorphism gives a route to construct potential deformations for any Weyl algebra. Let g = g^ − ⊕ h ⊕ g^ + represent a simple Lie algebra. Now, Dixmier's result holds for n = g^+. Since U_ q^+ (g) is a q-deformation of U(g^+), it is natural to consider U_q^+(g)/P, where P is a primitive ideal of U_q^+ (g), as a potential deformation of the Weyl algebras.

This thesis focuses on the case where g = G_2. We find a family of primitive ideals (P_{α,β})_{(α,β)∈C^2\(0,0)} of U _ q^ (G_2) whose corresponding quotients A_{α,β} := U_q^+ (G_2)/P_{α,β} are simple noetherian domains of Gelfand-Kirillov dimension 4. In view of Dixmier's result, we consider A_{α,β} as a q-deformation of (a quadratic extension of) A_2(C). The derivations of the Weyl algebras are all known to be inner derivations [5]. Motivated by this, we also study the derivations of A_{α,β} and compare them to those of the Weyl algebras. The final part of the thesis studies a Poisson derivation of a semiclassical limit \mathcal{A}_{α,β} of A_{α,β}. Interestingly, the Poisson derivations of \mathcal{A}_{α,β} and the derivations of A_{α,β} are congruent.