By Ashok Das, Susumu Okubo

The booklet is meant for graduate scholars of theoretical physics (with a heritage in quantum mechanics) in addition to researchers attracted to purposes of Lie team conception and Lie algebras in physics. The emphasis is at the inter-relations of illustration theories of Lie teams and the corresponding Lie algebras.

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**Example text**

Let U (a) denote a matrix representation of a group G of transformations. The matrix U (a) may or may not be unitary. ) If the matrices are unitary, then the representation of the group is known as a unitary representation. 83) the representation is known as a unitary representation. Let us give a simple 2 × 2 matrix example to illustrate this. Let us consider the matrix 1 U=√ 2 1−i i 0 . 85) and, similarly U †U = 1 2 3 1+i 1−i 2 = E2 . 86) On the other hand, let us consider a nonsingular matrix S= 1 −1 0 1 , S −1 = 1 1 .

124) is unitary in this space of functions. However, in order to calculate explicitly the unitary matrix, we need to change the basis functions. Let us consider the normalized infinite dimensional basis functions 1 2 1 √ n π2 n! 125) where Hn (x) denotes the nth order Hermite polynomial and ∞ dx φ∗n (x)φm (x) = δnm . 127) m=0 where Umn (a) denote the matrix elements of the infinite dimensional matrix U (a) providing a unitary representation for T1 . 129) satisfying [α, α† ] = ✶, [α, α] = 0 = [α† , α† ].

2) then, the set of all such n × n matrices {U (a)} forms a group and provides a (n × n matrix) representation of the group G. 2) the product on the left hand side is the natural associative matrix product which defines the product rule for such a matrix group. Indeed it is straightforward to check that such a set of matrices forms a group in the following manner. 3) is well defined with the matrix product rule and is in the set of matrices {U (a)}. 1), namely, for any two elements a, b ∈ G of the group, the group multiplication leads to ab = c ∈ G.