An alternative to the Chevalley description of \(U[sl(n+1)]\) and \(U_q[sl(n+1)]\) (Q1386515)

The special linear Lie algebra \(sl(n+1)\) and its universal enveloping algebra \(U[sl(n+1)]\) are usually described by means of their Chevalley generators (\(h_j\), \(e_j\) and \(f_j\)) and the corresponding relations (the Cartan relations and the Serre relations). Here, these algebras are described via creation and annihilation generators, with the appropriate relations. This provides an alternative definition which is closely related to quantum statistics. Next, the authors consider the \(q\)-deformed or quantized algebra \(U_q[sl(n+1)]\), and also define it in terms of deformed creation and annihilation generators. The emphasis is on finding the relations between the new generators; deriving the comultiplication or other Hopf algebra properties is more difficult.