ATOMISTIC SUBSEMIRINGS OF THE LATTICE OF SUBSPACES OF AN ALGEBRA

作者：Sage Daniel S^*

来源：Communications in Algebra, 2013, 41(10): 3652-3667.

DOI：10.1080/00927872.2012.674588

• Atomistic semiring
• G-algebra
• Hypergroup
• Subrepresentation semiring
• Primary 16Y60
• 20N20
• Secondary 16W22

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.

• 出版日期2013-10-3