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Generalized down-up algebras were introduced by Cassidy and Shelton (J Alg 2004) as a generalization of down-up algebras, originally defined by Benkart and Roby (J Alg 1998). The latter were motivated by combinatorial operators on a (q, r)-differential poset. Those generalized down-up algebras which are Noetherian are part of a larger class of algebras known as generalized Weyl algebras (GWA). When the underlying base ring of a GWA is a polynomial ring in one variable, the automorphism group of such a GWA was studied in detail by Bavula and Jordan (Trans Amer Math Soc 2001). A Noetherian generalized down-up algebra is a GWA whose base ring is a polynomial algebra in two variables, and it has Gelfand-Kirillov dimension 3. In joint work with P. Carvalho (Comm Alg 2009) we determined the automorphism groups of these algebras, under certain restrictions. In this talk, we will introduce these algebras and study their automorphisms. Time allowing, we shall also refer ongoing related joint work with P. Carvalho, S. Launois and C. Lomp concerning the notion of a non-commutative Noetherian unique factorization ring (Chatters and Jordan 1986).
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