CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP

Year 2017, Volume: 21 Issue: 21, 121 - 126, 17.01.2017

Allen Herman Gurmail Singh

https://doi.org/10.24330/ieja.296156

Cited By: 3

Abstract

A reality-based algebra (RBA) is a finite-dimensional associative
algebra that has a distinguished basis B containing 1A, where 1A is the identity
element of A, that is closed under a pseudo-inverse condition. If the RBA
has a one-dimensional representation taking positive values on B, then we
say that the RBA has a positive degree map. When the structure constants
relative to a standardized basis of an RBA with positive degree map are all
integers, we say that the RBA is integral. Group algebras of finite groups are
examples of integral RBAs with a positive degree map, and so it is natural to
ask if properties known to hold for group algebras also hold for integral RBAs
with positive degree map. In this article we show that every central torsion
unit of an integral RBA with algebraic integer coefficients is a trivial unit of
the form ζb, for some ζ is a root of unit in C and b is an element of degree 1 in B. 

Keywords

Reality based algebra, table algebra, association schemes

References

• [1] H. I. Blau, Table algebras, European J. Combin., 30(6) (2009), 1426-1455.
• [2] A. Herman and G. Singh, On the torsion units of integral adjacency algebras of
• finite association schemes, Algebra, 2014 (2014), Article ID 842378, 5 pages.
• [3] D. G. Higman, Coherent algebras, Linear Algebra Appl., 93 (1987), 209-239.
• [4] G. Singh, Torsion Units of Integral Group Rings and Scheme Rings, Ph.D.
• Thesis, University of Regina, 2015.
• [5] M. Takesaki, Theory of Operator Algebras I, Encyclopaedia of Mathematical
• Sciences, 124, Springer-Verlag, Berlin, 1979.
• [6] B. Xu, On isomorphisms between integral table algebras and applications to
• finite groups and association schemes, Comm. Algebra, 42(12) (2014), 5249-5263.