The upper triangular algebra loop of degree 4

Kenneth Walter Johnson; M. Munywoki; Jonathan D. H. Smith

Commentationes Mathematicae Universitatis Carolinae (2014)

  • Volume: 55, Issue: 4, page 457-470
  • ISSN: 0010-2628

Abstract

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A natural loop structure is defined on the set U 4 of unimodular upper-triangular matrices over a given field. Inner mappings of the loop are computed. It is shown that the loop is non-associative and nilpotent, of class 3. A detailed listing of the loop conjugacy classes is presented. In particular, one of the loop conjugacy classes is shown to be properly contained in a superclass of the corresponding algebra group.

How to cite

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Johnson, Kenneth Walter, Munywoki, M., and Smith, Jonathan D. H.. "The upper triangular algebra loop of degree $4$." Commentationes Mathematicae Universitatis Carolinae 55.4 (2014): 457-470. <http://eudml.org/doc/262023>.

@article{Johnson2014,
abstract = {A natural loop structure is defined on the set $U_4$ of unimodular upper-triangular matrices over a given field. Inner mappings of the loop are computed. It is shown that the loop is non-associative and nilpotent, of class 3. A detailed listing of the loop conjugacy classes is presented. In particular, one of the loop conjugacy classes is shown to be properly contained in a superclass of the corresponding algebra group.},
author = {Johnson, Kenneth Walter, Munywoki, M., Smith, Jonathan D. H.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {algebra group; quasigroup; loop; supercharacter; fusion scheme; algebra groups; quasigroups; loops; supercharacters; fusion schemes; conjugacy classes},
language = {eng},
number = {4},
pages = {457-470},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The upper triangular algebra loop of degree $4$},
url = {http://eudml.org/doc/262023},
volume = {55},
year = {2014},
}

TY - JOUR
AU - Johnson, Kenneth Walter
AU - Munywoki, M.
AU - Smith, Jonathan D. H.
TI - The upper triangular algebra loop of degree $4$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2014
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 55
IS - 4
SP - 457
EP - 470
AB - A natural loop structure is defined on the set $U_4$ of unimodular upper-triangular matrices over a given field. Inner mappings of the loop are computed. It is shown that the loop is non-associative and nilpotent, of class 3. A detailed listing of the loop conjugacy classes is presented. In particular, one of the loop conjugacy classes is shown to be properly contained in a superclass of the corresponding algebra group.
LA - eng
KW - algebra group; quasigroup; loop; supercharacter; fusion scheme; algebra groups; quasigroups; loops; supercharacters; fusion schemes; conjugacy classes
UR - http://eudml.org/doc/262023
ER -

References

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  2. André C.A.M., 10.1006/jabr.1995.1187, Algebra 175 (1995), 287–319. Zbl1064.20005MR1338979DOI10.1006/jabr.1995.1187
  3. Bruck R.H., 10.1090/S0002-9947-1946-0017288-3, Trans. Amer. Math. Soc. 60 (1946), 245–354. Zbl0061.02201MR0017288DOI10.1090/S0002-9947-1946-0017288-3
  4. Bruck R.H., A Survey of Binary Systems, Springer, Berlin, 1958. Zbl0141.01401MR0093552
  5. Diaconis P., Isaacs I.M., 10.1090/S0002-9947-07-04365-6, Trans. Amer. Math. Soc. 360 (2008), 2359–2392. Zbl1137.20008MR2373317DOI10.1090/S0002-9947-07-04365-6
  6. Johnson K.W., Smith J.D.H., 10.1016/S0195-6698(89)80032-0, Eur. J. Comb. 10 (1989), 47–56. Zbl0667.20053MR0977179DOI10.1016/S0195-6698(89)80032-0
  7. Smith J.D.H., An Introduction to Quasigroups and Their Representations, Chapman and Hall/CRC, Boca Raton, FL, 2007. Zbl1122.20035MR2268350
  8. Smith J.D.H., Romanowska A.B., Post-Modern Algebra, Wiley, New York, NY, 1999. Zbl0946.00001MR1673047

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