Yıl 2017, Cilt 21, Sayı 21, Sayfalar 121 - 126 2017-01-17
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## CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP

#### Allen Herman [1] , Gurmail Singh [2]

##### 209 161

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.

Reality based algebra, table algebra, association schemes
• [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.
Konular Matematik Makaleler Yazar: Allen Herman Yazar: Gurmail Singh
 Bibtex @araştırma makalesi { ieja296156, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {Prof. Dr. Abdullah HARMANCI}, year = {2017}, volume = {21}, pages = {121 - 126}, doi = {10.24330/ieja.296156}, title = {CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP}, key = {cite}, author = {Singh, Gurmail and Herman, Allen} } 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. DOI: 10.24330/ieja.296156 MLA Herman, A , Singh, G . "CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP". International Electronic Journal of Algebra 21 (2017): 121-126 Chicago Herman, A , Singh, G . "CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP". International Electronic Journal of Algebra 21 (2017): 121-126 RIS TY - JOUR T1 - CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP AU - Allen Herman , Gurmail Singh Y1 - 2017 PY - 2017 N1 - doi: 10.24330/ieja.296156 DO - 10.24330/ieja.296156 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 121 EP - 126 VL - 21 IS - 21 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.296156 UR - http://dx.doi.org/10.24330/ieja.296156 Y2 - 2016 ER - EndNote %0 International Electronic Journal of Algebra CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP %A Allen Herman , Gurmail Singh %T CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP %D 2017 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 21 %N 21 %R doi: 10.24330/ieja.296156 %U 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 (Ocak 2017): 121-126. http://dx.doi.org/10.24330/ieja.296156