Automorphism Groups of Lie Algebras from Quantum Tori

摘要

The q-quantum torus C_q := C_q[t_1~(±1), t_2~(±1),…, t_n~±1] is the unit al associative algebra over the complex number field C generated by t_1~(±1) ,t_2~(±1),…, t_n~(±1) and subject to the defining relations t_it_j = q_(ij)t_jt_i and t_it_i~(-1)=t_i~(-1)t_i=1, where the matrix q = (q_(ij)) ∈ M_(n×n)(C) with q_(ii) = 1 and q_(ij)~(-1) . Let D be a subspace of D_n=＋_(i=1)~n Cθ_i. where θ_i is the degree derivation on C_q sending t_1~(k_1)t_2~(K_2)…t_n~(k_n) to k_it_1~(k_1)t_2~(k_2)…t_n~(k_n).In this paper, under a mild condition, we determine the automorphism groups of Lie algebras C_q/C,L = C_q/C ＋ D_n, L= C_q/C ＋ ad c_q ＋ D_n and the Weyl-type algebra C_q[D].