Twisted generalized Weyl algebras (TGWAs) are defined as the quotient of a
certain graded algebra by the maximal graded ideal I with trivial zero
component, analogous to how Kac-Moody algebras can be defined. In this paper we
introduce the class of locally finite TGWAs, and show that one can associate to
such an algebra a polynomial Cartan matrix (a notion extending the usual
generalized Cartan matrices appearing in Kac-Moody algebra theory) and that the
corresponding generalized Serre relations hold in the TGWA. We also give an
explicit construction of a family of locally finite TGWAs depending on a
symmetric generalized Cartan matrix C and some scalars. The polynomial Cartan
matrix of an algebra in this family may be regarded as a deformation of the
original matrix C and gives rise to quantum Serre relations in the TGWA. We
conjecture that these relations generate the graded ideal I for these algebras,
and prove it in type A_2.