### Abstract

Starting with a representation φ{symbol} of a Euclidean Jordan algebra V by selfadjoint operators on a real Euclidean vector space E, we introduce a clan structure in V_{E}: = E ⊕ V. By the adjunction of a unit element to VE, we obtain a clan V ^{0}_{E} with unit element. By computing the determinant of the right multiplication operators of V ^{0} _{E}, we get an explicit expression of the basic relative invariants of V ^{0}_{E} in terms of the Jordan algebra principal minors of V and the quadratic map associated with φ{symbol}. For the dual clan of V ^{0} _{E}, we also obtain an explicit expression of the basic relative invariants in a parallel way.

Original language | English |
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Pages (from-to) | 163-202 |

Number of pages | 40 |

Journal | Kyushu Journal of Mathematics |

Volume | 67 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 25 2013 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Kyushu Journal of Mathematics*,

*67*(1), 163-202. https://doi.org/10.2206/kyushujm.67.163

**Clans defined by representations of euclidean Jordan algebras and the associated basic relative invariants.** / Nakashima, Hideto; Nomura, Takaaki.

Research output: Contribution to journal › Article

*Kyushu Journal of Mathematics*, vol. 67, no. 1, pp. 163-202. https://doi.org/10.2206/kyushujm.67.163

}

TY - JOUR

T1 - Clans defined by representations of euclidean Jordan algebras and the associated basic relative invariants

AU - Nakashima, Hideto

AU - Nomura, Takaaki

PY - 2013/7/25

Y1 - 2013/7/25

N2 - Starting with a representation φ{symbol} of a Euclidean Jordan algebra V by selfadjoint operators on a real Euclidean vector space E, we introduce a clan structure in VE: = E ⊕ V. By the adjunction of a unit element to VE, we obtain a clan V 0E with unit element. By computing the determinant of the right multiplication operators of V 0 E, we get an explicit expression of the basic relative invariants of V 0E in terms of the Jordan algebra principal minors of V and the quadratic map associated with φ{symbol}. For the dual clan of V 0 E, we also obtain an explicit expression of the basic relative invariants in a parallel way.

AB - Starting with a representation φ{symbol} of a Euclidean Jordan algebra V by selfadjoint operators on a real Euclidean vector space E, we introduce a clan structure in VE: = E ⊕ V. By the adjunction of a unit element to VE, we obtain a clan V 0E with unit element. By computing the determinant of the right multiplication operators of V 0 E, we get an explicit expression of the basic relative invariants of V 0E in terms of the Jordan algebra principal minors of V and the quadratic map associated with φ{symbol}. For the dual clan of V 0 E, we also obtain an explicit expression of the basic relative invariants in a parallel way.

UR - http://www.scopus.com/inward/record.url?scp=84880812420&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880812420&partnerID=8YFLogxK

U2 - 10.2206/kyushujm.67.163

DO - 10.2206/kyushujm.67.163

M3 - Article

VL - 67

SP - 163

EP - 202

JO - Kyushu Journal of Mathematics

JF - Kyushu Journal of Mathematics

SN - 1340-6116

IS - 1

ER -