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By Ji L.

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6. Properties of the Lie Algebra We will now establish various basic properties of the Lie algebra of a matrix Lie group. The reader is invited to verify by direct calculation that these general properties hold for the examples computed in the previous section. 14. Let G be a matrix Lie group, and X an element of its Lie algebra. Then eX is an element of the identity component of G. Proof. By definition of the Lie algebra, etX lies in G for all real t. But as t varies from 0 to 1, etX is a continuous path connecting the identity to eX .

If C is invertible, then eCXC = CeX C −1. 6. eX ≤ e X . It is not true in general that eX+Y = eX eY , although by 4) it is true if X and Y commute. This is a crucial point, which we will consider in detail later. ) Proof. Point 1) is obvious. Points 2) and 3) are special cases of point 4). To verify point 4), we simply multiply power series term by term. ) Thus eX eY = I +X+ X2 + ··· 2! I +Y + Y2 +··· 2! 2. 5) eX eY = ∞ m m=0 k=0 ∞ 1 X k Y m−k = k! (m − k)! m=0 m! m k=0 m! X k Y m−k . (m − k)!

EX = C  C . λn 0 e D 30 3. LIE ALGEBRAS AND THE EXPONENTIAL MAPPING Thus if you can explicitly diagonalize X, you can explicitly compute eX . 1) is real. For example, take −a 0 0 a X= i 1 1 and i respectively. Thus the invertible matrix Then the eigenvectors of X are 1 i C= . , with eigenvalues −ia and ia, i 1 0 1 to the eigenvectors of X, and so (check) and 1 0 C −1XC is a diagonal matrix D. Thus X = CDC −1 : maps the basis vectors eX = 1 i i 1 = cos a sin a e−ia 0 − sin a cos a 0 eia 1/2 −i/2 −i/2 1/2 .

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2-idempotent 3-quasigroups with a conjugate invariant subgroup consisting of a single cycle of length four by Ji L.


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