On the Hall Algebra of Semigroup Representations over F1
نویسنده
چکیده
Let A be a finitely generated semigroup with 0. An A–module over F1 (also called an A–set), is a pointed set (M, ∗) together with an action of A. We define and study the Hall algebraHA of the category CA of finite A–modules. HA is shown to be the universal enveloping algebra of a Lie algebra nA, called the Hall Lie algebra of CA. In the case of the 〈t〉 the free monoid on one generator 〈t〉, the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent 〈t〉-modules) is isomorphic to Kreimer’s Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when A is a quotient of 〈t〉 by a congruence, and the monoid G ∪ {0} for a finite group G.
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