Lie group and Lie algebra Guangyue Ji (棘 广 Ã)1 1International Center for Quantum Materials, Peking University, Beijing 100871, China 2 I. GROUP A. The definition of group A group is a set, G, together with an operation • (called the group law of G) that combines any two elements a and b to form another element, denoted a • b or ab. To qualify as a group, the set and operation, (G; •), must satisfy four requirements known as the group axioms: • Closure For all a, b in G, the result of the operation, a • b, is also in G. • Associativity For all a, b and c in G, (a • b) • c = a • (b • c). • Identity element There exists an element e in G such that, for every element a in G, the equation e • a = a • e = a holds. Such an element is unique, and thus one speaks of the identity element. • Inverse element For each a in G, there exists an element b in G, commonly denoted a−1 (or −a, if the operation is denoted "+"), such that a • b = b • a = e, where e is the identity element. B. Some familiar matrix groups 1. General linear group • General complex linear groups GL(n; C). Its definition is GL(n; C) = fAjA is a n × n complex matrix, and det A 6= 0g : (1) The complex matrix of GL(n; C) has 2n2 independent matrix elements. • General real linear groups GL(n; R). Its definition is GL(n; R) = fAjA is a n × n real matrix, and det A 6= 0g : (2) The real matrix of GL(n; R) has n2 independent matrix elements. 2. Special linear group • Special complex linear groups SL(n; C). Its definition is SL(n; C) = fAjA is a n × n complex matrix, and det A = 1g : (3) The group SL(n; C) has 2(n2 − 1) independent matrix elements. • Special real linear groups SL(n; R). Its definition is SL(n; R) = fAjA is a n × n real matrix, and det A = 1g : (4) The group SL(n; R) has (n2 − 1) independent matrix elements. 3 3. Unitary group and special unitary group • Unitary group U(n). Its definition is y y U(n) = UjU 2 GL(n; C); and U U = UU = en : (5) The unitary group U(n) has n2 independent matrix elements. • Special unitary group SU(n). definition SU(n) = fUjU 2 U(n); and det U = 1g : (6) The special unitary group SU(n) has n2 − 1 independent matrix elements. 4. Orthogonal group and special orthogonal group • Orthogonal group O(n). definition T O(n; C) = OjO 2 GL(n; C); and O O = en : (7) The orthogonal group O(n) has n(n − 1) independent matrix elements. • Special orthogonal group SO(n). definition SO(n) = fOjO 2 O(n); and det U = 1g : (8) 5. Symplectic group For 2n × 2n-dimensional matrix M and vector X and Y 0 1 0 1 x1 y1 B x2 C B y2 C B C B C B . C B . C B . C B . C B C B C B xn C B yn C X = B C ; Y=B C. (9) B xn+1 C B yn+1 C B C B C B xn+2 C B yn+2 C B C B C B . C B . C @ . A @ . A x2n y2n Under the transformation X ) X0 = MX, Y ) Y 0 = MY , if n X (xiyn+i − yixn+i) = const. (10) i=1 The sets of matrices M2n×2n consist of the symplectic group Sp(2n; C). The symplectic group Sp(2n; C) has 2n(2n+1) independent matrix elements. The sympletic group can also be defined by RT JR = J (11) with J 0 I J = n×n : (12) −In×n 0 4 C. 3-D harmonic oscillator Ref: Howard Georgi. Chapter 14. The Hamiltonian is 1 m!2 H^ = ~p2 + ~r2 (13) 2m 2 3 = ! ay a + ; (14) k k 2 where 1 ak = p (m!rk + ipk); (15) 2m! y 1 a = p (m!rk − ipk): (16) k 2m! If the ground state is j0i, satisfying ak j0i = 0 (17) then the energy eigenstates are ay ··· ay j0i (18) k1 kn with energy 3 !(n + ): (19) 2 The degeneracy of these states is interesting—it is the number of symmetric combinations of the n indices, k1 ··· kn, which is (n + 1)(n + 2) : (20) 2 Just the dimension of the (n; 0) representation of SU(3). This suggests that the model has a SU(3) symmetry. D. SU(2) ! SU(1; 1) Ref: PIERRE RAMOND. Chapter 5. The SU(2) Lie algebra has yet another interesting property: one can reverse the sign of two of its elements without changing the algebra. For example T 1 ! −T 1; (21) T 2 ! −T 2; (22) T 3 ! T 3: (23) This operation is called an involutive automorphism of the algebra. It is an operation which forms a finite group of two elements. Its eigenvalues are therefore ±1, and it splits the Lie algebra into even and odd subsets. One can use this to obtain a closely related algebra by multiplying all odd elements by i. This generates another Lie algebra with three elements LA, with L1 ≡ iT 1; (24) L2 ≡ iT 2; (25) L3 ≡ T 3: (26) 5 Note that L1;2 are no longer hermitian. The new algebra is L1;L2 = −iL3; (27) L2;L3 = iL1; (28) L3;L1 = iL2: (29) One can go further and absorb the i on the right-hand side of the commutators, and generate the algebra in terms of the real matrices 1 0 0 1 0 −1 ; ; : (30) 0 −1 1 0 1 0 Its representation theory is far more complicated than that of its compact coun- terpart. Its simplest unitary rep- resentations can be found in the infinite Hilbert space generated by one bosonic harmonic oscillator. To see this, consider the operators 1 L+ ≡ p ayay; (31) 2 2 1 L− ≡ p aa: (32) 2 2 The harmonic oscillator space splits into two representations of SO(2; 1) of even and occupation number states. E. 1-D translation group 1-D translation group T (a) is T (a) = x + a: (33) We choose the wavefunctions xm as basis of the spcae. For example, in the case of 2 base vectors, the result of transformation T (a) is 1 1 T (a) = : x x + a The entry in the column are base vectors. The matrix representation of T (a) is 1 1 0 1 1 T (a) = = : x a 1 x x + a 6 The finite dimension representation of T (a) is not unitary. If we continue to add the dimension of the Hilbert space, the representation of T (a) is 0 1 0 1 1 0 1 0 0 0 ::: 1 1 B x C a 1 0 0 ::: B x C B 2 C B C B 2 C T (a) B x C = B a2 2a 1 0 ::: C B x C B 3 C B C B 3 C x B 3 2 C x B C @ a 3a 3a 1 ::: A B C @ . A @ . A . ::::::::::::::: . 0 1 00 1 0 0 0 ::: 1 0 0 0 0 0 ::: 1 0 0 0 0 0 ::: 11 1 BB 0 1 0 0 ::: C B 1 0 0 0 ::: C B 0 0 0 0 ::: CC B x C BB C B C 2 B CC B 2 C = BB 0 0 1 0 ::: C + a B 0 2 0 0 ::: C + a B 1 0 0 0 ::: CC B x C BB C B C B CC B x3 C @@ 0 0 0 1 ::: A @ 0 0 3 0 ::: A @ 0 3 0 0 ::: AA B C @ . A ::::::::::::::: ::::::::::::::: ::::::::::::::: . 0 1 1 B x C a d B 2 C = e dx B x C B 3 C B x C @ . A . 0 1 1 B x + a C B 2 C = B (x + a) C : B 3 C B (x + a) C @ . A . F. Peter-Weyl theorem: • 1, the representations of a compact group are finite dimension and completely irreducible and are isomorphism to unitary representations. • 2, the representations of non-compact group are not completely irreducible. There are only two unitary repre- sentations, namely trival representation and infinite-dimension representation. II. LIE GROUP AND LIE TRANSFORMATION GROUP A. Structure function For a list of parameters (α1; α2; : : : ; αn), every list of parameters corresponds to an element of the group. For example, A(α1; : : : ; αn), B(β1; : : : ; βn), and C(γ1; : : : ; γn) = AB, where γ = '(α; β) is the function of (α; β) and satisfies: 0 0 • There is an identity element I(α1; : : : ; αn) = I(0;:::; 0) satisfying IA = AI = A; '(0; α) = '(α; 0) = α: (34) • There is an inverse element A−1 = A(¯α) for every A(α) satisfying A−1A = AA−1 = I; (35) '(¯α; α) = '(α; α¯) = 0: (36) • Associativity: 7 (AB)C = A(BC); (37) '('(α; β); γ) = '(α; '(β; γ)): (38) When A(α) acts on the space V , X0 = A(α)X, A(α) is called the Lie transformation group. Specially, if A(α) is V , A(α) is called the Lie group. B. The infinitesimal generator of Lie group Ref: Yuxin Liu. Chapter 1. For transformation A(α), its effect is X0 ) X = A(α)X0 = f(α; X0): (39) If V is multi-dimensional space, for every i, we have i i i X = f (α; X0) = f (0;X): (40) If we continue to change the group parameter by δα, we have i i i X + dX = f (α + dα; X0) (41) = f i(δα; X) (42) i i @f (β; X)) σ = f (0;X) + σ δα + ::: (43) @β β=0 i i σ = X + Uσ(X)δα : (44) Namely, i i σ dX = Uσ(X)δα : (45) In the parameter space, we have αµ + dαµ = 'µ(δα; α) (46) i µ @' (β; X)) σ = ' (0; α) + σ δα + ::: (47) @β β=0 µ µ σ = α + Vσ δα : (48) Namely, σ −1σ µ σ µ δα = V µ dα = Λµ(α)dα : (49) Finally, we have i i σ µ dX = Uσ(X)Λµ(α)dα : (50) i σ We call Uσ(X)Λµ(α) the infinitesimal operator.