nLab Bruck loop

Definition

Definition

A left Bruck loop is a left Bol loop $(K,\cdot)$ satisfying the automorphic inverse property

$$(x y)^{-1} = x^{-1} y^{-1} \mathrlap{\,.}$$

Remark

(Alternative terminologies) Some authors say (left) K-loop or left gyrocommutative gyrogroup?.

Literature

• H. Kiechle: Theory of K-loops, Lecture Notes in Mathematics 1778, Springer (2002)

• Sejong Kim, Jimmie Lawson: Smooth Bruck loops, symmetric spaces, and nonassociative vector spaces, Demonstratio Mathematica 44 4 (2012) 755–779 [doi:10.1515/dema-2013-0343]

• Florian Girelli: Snyder space-time: K-loop and Lie triple system, SIGMA 6 074 (2010) [sigma:2010/074, pdf, arxiv:1009.4762]