Atomistic subsemirings of the lattice of subspaces of an algebra

TL;DR

This paper introduces a new algebraic structure called the condensation monoid derived from atomistic subsemirings of an algebra's subspace lattice, providing novel invariants for certain representations.

Contribution

It defines the condensation monoid and focal subalgebra, and demonstrates their properties and applications to G-algebras and representation theory.

Findings

The condensation monoid can be a group under certain conditions.

The focal subalgebra is associated with the identity atom.

New invariants for finite-dimensional irreducible projective representations are introduced.

Abstract

Let A be an associative algebra with identity over a field k. An atomistic subsemiring R of the lattice of subspaces of A, endowed with the natural product, is a subsemiring which is a closed atomistic sublattice. When R has no zero divisors, the set of atoms of R is endowed with a multivalued product. We introduce an equivalence relation on the set of atoms such that the quotient set with the induced product is a monoid, called the condensation monoid. Under suitable hypotheses on R, we show that this monoid is a group and the class of k1_A is the set of atoms of a subalgebra of A called the focal subalgebra. This construction can be iterated to obtain higher condensation groups and focal subalgebras. We apply these results to G-algebras for G a group; in particular, we use them to define new invariants for finite-dimensional irreducible projective representations.