Cayley algebras. | 3 | Exceptional Lie Algebras | N. Jacobson

ABSTRACT

Let https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg7_23.tif"/> be a (generalized) Cayley algebra over an arbitrary field https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg7_24.tif"/> (of characteristic ≠ 2). We have the usual involution π: a → ā in https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg7_25.tif"/> and we write n(a)1 = aā = āa, a+ā = t(a)1. Then n(a) is a quadratic form which permits composition: n(ab) = n(a) n(b). Let https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg7_26.tif"/> denote the space of elements of trace 0 (t(a) = 0) in https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg7_27.tif"/> and write a_R and a_L for the right and left multiplications determined by https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg7_28.tif"/> . The algebra https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg7_29.tif"/> is alternative and hence determines a Jordan algebra https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg7_30.tif"/> relative to the composition a ⋅ b = 1 2 ( ab + ba ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/eq23.tif"/> and we write R a = 1 2 ( a R + a L ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/eq24.tif"/> . If https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg7_31.tif"/> then x² = -n(x) and 8if a = αl + x then n(a) = α² + n(x). It follows that https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg8_1.tif"/> is the Jordan algebra of the bilinear form ( x , y ) = − 1 2 [ n ( x + y ) − n ( x ) − n ( y ) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/eq25.tif"/> in the space https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg8_2.tif"/> . The results of the last section now imply that the mappings R_x, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg8_3.tif"/> , are skew in https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg8_4.tif"/> relative to n ( a , b ) = 1 2 [ n ( a + b ) − n ( a ) − n ( b ) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/eq26.tif"/> and the Lie algebra https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg8_5.tif"/> of all skew linear mappings in https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg8_6.tif"/> is the set of mappings of the form R x + Σ [ R x i R z i ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/eq27.tif"/> where x, x_i, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg8_7.tif"/> We shall now show that the mappings x_R, x_L, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg8_8.tif"/> , are in https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg8_9.tif"/> .