THE HALL ALGEBRA OF CYCLIC SERIAL ALGEBRA

In this paper, orders < 1 and < 2 on ((Z)+)nm are introduced and also regarded as orders on the isomorphism classes of finite modules of finite cyclic algebra R with n simple modules and all the indeocomposable projective modules have length m through a one-to-one correspondence between ((Z+)nm and the isomorphism classes of finite R modules. Using this we prow that the Hall algebra of a cyclic serial algebra is identified with its Loewy subalgebra, and its rational extension has a basis of BPW type, and is a (((Z)+)nm, <2) filtered ring with the associated graded ring as an iterated skew polynomial ring. These results are also generalized to the Hall algebra of a tube over a finite field.