Abstract:

This is the first one of two papers studying the quantum group of sl ^∞_∞ type via the Ringel-Hall algebra of A^∞_∞ type. For this we need to deal with finite-dimensional representations of the infinite-dimensional path algebra kA^∞_∞ over any field k. In the present paper, we first study the category of the finite-dimensional representations of kA^∞_∞ by determining all its indecomposable objects and their extensions explicitly ; then we investigate the Ringel-Hall algebra H(kA^∞_∞) for a finite field k. The main viewpoint of this investigation is to regard H(kA^∞_∞) as the direct limit of the Ringel-Hall algebra H(kA_∞) and H(kA_∞) as the direct limit of the Ringel-Hall algebra H(kA_n). In particular, we get
a PBW-basis of H(kA^∞_∞) and show that H(kA_^∞∞) coincides with its composition subalgebra.

Key words: Path algebra, Ringel-Hall algebra, Quantum group