A symplectic representation of E 7

Tevian Dray; Corinne A. Manogue; Robert A. Wilson

Commentationes Mathematicae Universitatis Carolinae (2014)

  • Volume: 55, Issue: 3, page 387-399
  • ISSN: 0010-2628

Abstract

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We explicitly construct a particular real form of the Lie algebra 𝔢 7 in terms of symplectic matrices over the octonions, thus justifying the identifications 𝔢 7 ≅ 𝔰𝔭 ( 6 , 𝕆 ) and, at the group level, E 7 ≅ Sp ( 6 , 𝕆 ) . Along the way, we provide a geometric description of the minimal representation of 𝔢 7 in terms of rank 3 objects called cubies.

How to cite

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Dray, Tevian, Manogue, Corinne A., and Wilson, Robert A.. "A symplectic representation of $E_7$." Commentationes Mathematicae Universitatis Carolinae 55.3 (2014): 387-399. <http://eudml.org/doc/261873>.

@article{Dray2014,
abstract = {We explicitly construct a particular real form of the Lie algebra $\mathfrak \{e\}_7$ in terms of symplectic matrices over the octonions, thus justifying the identifications $\mathfrak \{e\}_7\cong \mathfrak \{sp\}(6,\mathbb \{O\})$ and, at the group level, $E_7\cong \text\{Sp\}(6,\mathbb \{O\})$. Along the way, we provide a geometric description of the minimal representation of $\mathfrak \{e\}_7$ in terms of rank 3 objects called cubies.},
author = {Dray, Tevian, Manogue, Corinne A., Wilson, Robert A.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {exceptional Lie algebras; octonions; $E_7$; exceptional algebra; Freudenthal-Tits magic square; symplectic matrices; octonions; },
language = {eng},
number = {3},
pages = {387-399},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A symplectic representation of $E_7$},
url = {http://eudml.org/doc/261873},
volume = {55},
year = {2014},
}

TY - JOUR
AU - Dray, Tevian
AU - Manogue, Corinne A.
AU - Wilson, Robert A.
TI - A symplectic representation of $E_7$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2014
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 55
IS - 3
SP - 387
EP - 399
AB - We explicitly construct a particular real form of the Lie algebra $\mathfrak {e}_7$ in terms of symplectic matrices over the octonions, thus justifying the identifications $\mathfrak {e}_7\cong \mathfrak {sp}(6,\mathbb {O})$ and, at the group level, $E_7\cong \text{Sp}(6,\mathbb {O})$. Along the way, we provide a geometric description of the minimal representation of $\mathfrak {e}_7$ in terms of rank 3 objects called cubies.
LA - eng
KW - exceptional Lie algebras; octonions; $E_7$; exceptional algebra; Freudenthal-Tits magic square; symplectic matrices; octonions;
UR - http://eudml.org/doc/261873
ER -

References

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  9. Wilson R.A., 10.1090/S0002-9939-2013-11838-1, Proc. Amer. Math. Soc. 142 (2014), 867–880. MR3148521DOI10.1090/S0002-9939-2013-11838-1
  10. Manogue C.A., Schray J., 10.1063/1.530056, J. Math. Phys. 34 (1993), 3746–3767. Zbl0797.53075MR1230549DOI10.1063/1.530056
  11. Kincaid J., Dray T., Division algebra representations of S O ( 4 , 2 ) , arXiv: 1312.7391. 
  12. Kincaid J.J., Division algebra representations of S O ( 4 , 2 ) , Master's thesis, Oregon State University, 2012, available at http://ir.library.oregonstate.edu/xmlui/handle/1957/30682. 
  13. Dray T., Huerta J., Kincaid J., The 2 × 2 Lie group magic square, in preparation. 
  14. Röhrle G., 10.1090/conm/153/01310, Contemp. Math. 153 (1993), 143–155. Zbl0832.20071MR1247502DOI10.1090/conm/153/01310

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