‘Lie algebra’ means Lie algebra over R, and ‘representation’ means finite-dimensional representation over C. 1. Define the exponential eX of a square matrix X, and show that eX is invertible. Determine eX in each of the following cases: X= X= 0 0 1 0 1 , X= 0 1 , X= 1 0 1 1 1 1 , X= 0 1 , X= 2 0 −1 , 1 0 −1 0 . 0 −1 Which of these 6 matrices X are themselves expressible in the form X = eY , (a) with a real matrix Y , (b) with a complex matrix Y ? (Justify your answers in all cases.) 2. Define…
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