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

Year 2017, , 121 - 126, 17.01.2017
https://doi.org/10.24330/ieja.296156

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. 

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.
Year 2017, , 121 - 126, 17.01.2017
https://doi.org/10.24330/ieja.296156

Abstract

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.
There are 10 citations in total.

Details

Subjects Mathematical Sciences
Journal Section Articles
Authors

Allen Herman This is me

Gurmail Singh This is me

Publication Date January 17, 2017
Published in Issue Year 2017

Cite

APA Herman, A., & Singh, G. (2017). CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP. International Electronic Journal of Algebra, 21(21), 121-126. https://doi.org/10.24330/ieja.296156
AMA Herman A, Singh G. CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP. IEJA. January 2017;21(21):121-126. doi:10.24330/ieja.296156
Chicago Herman, Allen, and Gurmail Singh. “CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP”. International Electronic Journal of Algebra 21, no. 21 (January 2017): 121-26. https://doi.org/10.24330/ieja.296156.
EndNote Herman A, Singh G (January 1, 2017) CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP. International Electronic Journal of Algebra 21 21 121–126.
IEEE A. Herman and G. Singh, “CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP”, IEJA, vol. 21, no. 21, pp. 121–126, 2017, doi: 10.24330/ieja.296156.
ISNAD Herman, Allen - Singh, Gurmail. “CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP”. International Electronic Journal of Algebra 21/21 (January 2017), 121-126. https://doi.org/10.24330/ieja.296156.
JAMA Herman A, Singh G. CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP. IEJA. 2017;21:121–126.
MLA Herman, Allen and Gurmail Singh. “CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP”. International Electronic Journal of Algebra, vol. 21, no. 21, 2017, pp. 121-6, doi:10.24330/ieja.296156.
Vancouver Herman A, Singh G. CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP. IEJA. 2017;21(21):121-6.