Narrow Results

A reality-based algebra (RBA) is a finite-dimensional associative algebra with involution over $ℂ$ whose distinguished basis $B$ contains $1$ and is closed under pseudo-inverse. An integral RBA is one whose structure constants in its distinguished basis are integers. If the algebra has a one-dimensional representation taking positive values on $B$, then we say that the RBA has a positive degree map. These RBAs have a standard feasible trace, and the multiplicities of the irreducible characters in the standard feasible trace are the multiplicities of the RBA. In this paper, we show that for integral RBAs with positive degree map whose multiplicities are rational, any finite subgroup of torsion units whose elements are all of degree $1$ and have algebraic integer coefficients must have order dividing a certain positive integer determined by the degree map and the multiplicities. The paper concludes with a thorough investigation of the properties of RBAs that force multiplicities to be rational.

Table Algebras (A,B) with distinct non-real basis elements a, b∈ B satisfying , (m, n∈ ℝ⁺) are studied in this paper. The two cases where either a is real and b is non-real or a, b are both real were completely classified by Arad and Blau. Here, we study the remaining case where a and b are non-real.

Let (A,B,f) be a C-algebra. For any subset N of B, its kernel ker_fN is a subset of . The kernel ker_f has been used to obtain important results for C-algebras (see [2] and [4]). In this paper, we study further properties of ker_f. In particular, for a transitional C-algebra (A,B,f) and quotient subsets M and N such that N ⊆ M and M is also a C-subset, we prove that . In order to show this result, exact sequences in C-algebras are studied. Finally, we present a few examples and applications.

Table Algebras (A, B) with distinct non-trivial basis elements a, b ∈ B satisfying either or for some m, n ∈ ℝ⁺ are studied in this paper. The case was partially classified by Arad and Blau, and the case was partially classified by Arad and Cohen.

In this paper we prove the Krull-Schmidt type theorems for direct products of association schemes and tensor products of table algebras.