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By Chen G.

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14) are actually in the Lie algebra of the Euclidean group. A simple computation shows that for n≥1  n   y1  ..   Yn Y n−1 y   Y .  ,   =    yn  0 ··· 0 0 ··· 0 where y is the column vector with entries y1 , · · · , yn . 14), then etX is of the form   ∗  ..   etY .  etX =  .  ∗  0 ··· 0 1 40 3. LIE ALGEBRAS AND THE EXPONENTIAL MAPPING Now, we have already established that etY is in O(n) for all t if and only if Y = −Y . 14) with Y satisfying Y tr = −Y . A similar argument shows that the Lie algebra of P(n; 1) is the space of all (n + 2) × (n + 2) real matrices of the form   y1  ..

In our new notation, we may say Ad = ad By the defining property of Ad, we have the following identity: For all X ∈ g, Ad(eX ) = eadX . 17) are linear operators on the Lie algebra g. This is an important relation, which can also be verified directly, by expanding out both sides. 1. Structure Constants. Let g be a finite-dimensional real or complex Lie algebra, and let X1 , · · · , Xn be a basis for g (as a vector space). Then for each i, j, [Xi , Xj ] can be written uniquely in the form n [Xi , Xj ] = cijk Xk .

Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 8. Proof. ) Cm ≤ const. m2 . Since e e domain of the logarithm for all sufficiently large m. 7 Cm ≤ const. the logarithm gives X X Y + + Cm m m X m + Y m + Cm 2 ≤ const. m2 . Exponentiating Y X + + Cm + Em m m Y e m e m = exp and X Y m emem = exp (X + Y + mCm + mEm ) . Since both Cm and Em are of order exponential) lim m→∞ X Y em em 1 m2 , we have (using the continuity of the m = exp (X + Y ) which is the Lie product formula. 10. Let X be an n × n real or complex matrix.

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A Characterization of Alternating Groups by the Set of Orders of Maximal Abelian Subgroups by Chen G.


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