By Francesco Iachello

This ebook, designed for complex graduate scholars and post-graduate researchers, introduces Lie algebras and a few in their purposes to the spectroscopy of molecules, atoms, nuclei and hadrons. The ebook includes many examples that support to explain the summary algebraic definitions. It offers a precis of many formulation of useful curiosity, equivalent to the eigenvalues of Casimir operators and the scale of the representations of all classical Lie algebras.

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This booklet, designed for complex graduate scholars and post-graduate researchers, introduces Lie algebras and a few in their purposes to the spectroscopy of molecules, atoms, nuclei and hadrons. The e-book comprises many examples that aid to explain the summary algebraic definitions. It presents a precis of many formulation of functional curiosity, corresponding to the eigenvalues of Casimir operators and the size of the representations of all classical Lie algebras.

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**Example text**

0] . [1, 1, 1, . . 3 Orthogonal Algebras, n = Odd There are ν = (n−1)/2 fundamental representations, one of which is a spinor representation spin(n), n = odd [1, 0, . . , 0] [1, 1, . . , 0] . 30) ... [1, 1, . . , 1, 0] 1 1 1 1 2, 2, . . 4 Orthogonal Algebras, n = Even There are ν = n/2 fundamental representations, two of which are spinor representations spin(n), n = even [1, 0, . . , 0] [1, 1, . . , 0] ... 31) [1, 1, . . 1, 0, 0] 1 1 1 1 2, 2, . . , 2, 2 1 1 1 1 2, 2, . . 5 Symplectic Algebras There are ν = n/2 fundamental representations.

3. 5 Examples of Groups of Transformations 35 u = a11 u + a12 v v = a21 u + a22 v . 39) This is a 8 parameter group. 41) The corresponding group, U (2), is a four parameter group. 42) one obtains the three parameter group SU (2). Example 4. 43) a11 a∗11 + a12 a∗12 = 1 . 5 Relation Between SO(3) and SU (2) Both SO(3) and SU (2) are three parameter groups. It is of importance to ﬁnd their relationship. Consider the following combination of the complex spinor u, v x1 = u2 ; x2 = uv ; x3 = v 2 . 48) 2 2 −i (a11 a12 − a∗11 a∗12 )z z = − (a∗11 a12 + a11 a∗12 )x + i (a∗11 a12 − a11 a∗12 )y x = +(a11 a∗11 − a12 a∗12 )z This is a real orthogonal transformation in three dimensions, satisfying x 2 + y 2 + z 2 = x2 + y 2 + z 2 .

N), and their derivatives, ∂θ∂ i . The elements are Eij = θi ∂ . 23) The commutation relations can be obtained from the basic commutation relations ∂ ∂ ∂ , θj ≡ θj + θj = δ ij . 24) ∂θi ∂θ ∂θ i i + These realizations will not be discussed in these notes. A detailed account is given by Berezin. A. Berezin, An Introduction to Superanalysis, D. 26) called Fermi commutation relations. These realizations will be discussed in Chap. 8. 27) with i, j = 1, . . n. 23). They deﬁne again the Lie algebra u(n).