New PDF release: 3-Sasakian Geometry, Nilpotent Orbits, and Exceptional

By Boyer Ch. P.

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Additional info for 3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients

Example text

A web W(n + 1, n, r) is called a Bol web Bal am+l (n + 1, n, r) if it is a Bol web B a s ( n + l , n , r ) forallas, l < ~ a s < ~ n + l ; s = l . . . m + l ; l < < . m < < . n + l . I f m = n , a web Bal an+l (n + 1, n, r) is said to be a Moufang web and denoted by M(n + 1, n, r). It is easy to prove that for a Moufang web M(n + 1, n, r), and only for it, all its threesubwebs are Moufang webs M(3, 2, r). Consider two different leaves Fn~+l and Fn~+l of the foliation ~n+l of a web W(n + 1, .....

N)pnJl'"Jpl'"kl'"kpn ji JPl "" Ukln "" U1 . . A. V. Goldberg 38 and the coefficients ( 1 )Pl X . . i (n)Pn jl ... k! (S--pl p,=0 (Pl Pl ..... 0-P2... 91) provided that 0-~2. . . +0-nPn)S--(0-~2) s . . . (O'nPn)S -- 1 ~ O. 92) Note that if u~9 -- ... 92) agree with those indicated above for a binary quasigroup. 89) are called the canonical expansions, and the variables occurring in them are called the canonical parameters. 14) are satisfied. Thus F : X1 • • Xn ~ Xn+l is an n-loop. 91) we have F i (u jl , 0"2uj2 .

An) of an (n + 1)-web W(n + 1, n, r) find an algebraic construction similar to local Akivis algebras for a threeweb W(3, 2, r). 31 Differential geometry of webs This problem was solved by Smith in [467] (see also [204]). In this subsection we present Smith's results. We will consider a local analytic n-loop L (al . . . an) over R given by an r-tuple F = (F 1, . . 53) where F i, i = 1 , . . 54) (~,3,y) satisfying the conditions i Fi (O . . . O, uaJa , 0 . . . 0) -- ua. 53) is the n-loop F m ( F i , .

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3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients by Boyer Ch. P.


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