Notes on Lie Algebras

Transcription

1 NEW MEXICO TECH (October 23, 2010) DRAFT Notes on Lie Algebras Ivan G. Avramidi Department of Mathematics New Mexico Institute of Mining and Technology Socorro, NM 87801, USA

2 1 1 Lie Algebras A (real or complex) Lie algebra L is an m-dimensional (real or complex) vector space with a antisymmetric bilinear binary operation called the Lie bracket, satisfying the Jacobi identity: for any X, Y, Z L [, ] : L L L, (1.1) [X, [Y, Z] + [Y, [Z, X] + [Z, [X, Y] = 0. (1.2) The algebra is called Abelian if [L, L] = 0. A subspace A L of the Lie algebra L is a subalgebra of L if [A, A] A. A subspace J L is an ideal of L if [J, L] L. Every ideal is a subalgebra. The maximal ideal N satisfying the condition [N, L] = 0 is called the center. The center is always Abelian. Every real Lie algebra L naturally defines a complex Lie algebra L C = L il = {Z Z = X + iy, X, Y L} (1.3) called the complexification of L with the Lie bracket extended by linearity. Then the original real Lie algebra L is called the real form of the complex Lie algebra L C. Every complex Lie algebra L allows a realification (real restriction) to a real Lie algebra L R if one represents the complex vector space as a real vector space of double dimension. Let T a, a = 1,..., m, form a basis in the Lie algebra L. That is, every element X of the Lie algebra g can be represented in the form X = X a T a, (1.4) Then T a are called the generators of the algebra L. The generators must satisfy the commutation relations [T a, T b ] = C c abt c, (1.5) where C a bc are some real constants called the structure constants of the algebra. The structure constants are anti-symmetric in the last two indices and must satisfy the conditions the Jacobi identity C e [a d C d bc] = 0. (1.6) Here the brackets denote the anti-symmetrization over all indices inlcuded in the brackets and the vertical lines indicate the indices excluded from the anti-symmetrization. This identity puts severe restriction on the structure of the Lie algebra and allows the complete classification of all Lie algebras. roots.tex; October 26, 2010; 15:19; p. 1

3 2 Notice that under the rescaling T a λt a the structure constants rescale accordingly C c ab λc c ab. That is why, the structure constants are defined up to a uniform factor. Moreover, under a linear transformation of the basis they change as tensors in R m. A map ρ : L M (1.7) of a Lie algebra L into a Lie algebra M is a homomorphism if it is linear and ρ([x, Y]) = [ρ(x), ρ(y)]. (1.8) A representation of the Lie algebra L in a vector space V is a homomorphism ρ : L End(V). (1.9) so that the elements of the Lie algebra X are represented by the endomorphisms ρ(x) of the vector space V and the Lie bracket is represented by the commutator of these matrices, ρ([x, Y]) = [ρ(x), ρ(y)]. (1.10) The adjoint representation of the Lie algebra is a homomorphism into the endomorphisms of the Lie algebra, ad : L End(L), (1.11) such that for any X L, ad X End(L) is an endomorphism defined by 2 Compact Semisimple Lie Algebras ad X (Y) = [X, Y]. (1.12) Let C a be m m real matrices defined by (C a ) b c = C b ac. Then the Jacobi identity can be written in the matrix form as the same commutation relations [C a, C b ] = C c abc c. (2.13) This simply means that ad X is represented by a real m m matrix in the sense C(X) = X a C a, (2.14) (ad X Y) a = [C(X)] a by b. (2.15) The Killing form on a Lie algebra is a symmetric bilinear form defined by It has the property where This means that B(X, Y) = tr ad X ad Y = tr C(X)C(Y). (2.16) B([X, Y], Z) + B(Y, [X, Z]) = 0. (2.17) B(X, Y) = B ab X a Y b, (2.18) B ab = tr C a C b = C c adc d bc. (2.19) roots.tex; October 26, 2010; 15:19; p. 2

4 3 A Lie algebra with no Abelian ideals is called semi-simple. A Lie algebra is semi-simple if and only if the Killing form is non-degenerate. A Lie algebra which has no nontrivial ideals is called simple. A semi-simple Lie algebra is the sum of simple ideals. A Lie algebra is compact if it has a positive-definite quadratic form (, ) such that ([X, Y], Z) + (Y, [X, Z]) = 0. (2.20) Every complex Lie algebra is non-compact since it cannot have a positive definite quadratic form. A Lie algebra of a compact Lie group is compact. Lie algebra is compact semi-simple if and only if the Killing form negativedefinite. For compact semi-simple algebras in particular, tr C(X) = 0, (2.21) tr C a = C b ab = 0. (2.22) We restrict ourselves to compact simple algebras. That is why, the matrix defines a positive definite inner product on the Lie algebra There is a basis in which this metric is Euclidean γ ab = B ab (2.23) (X, Y) = B(X, Y) = γ ab X a Y b. (2.24) γ ab = c 2 δ ab, (2.25) where c is a normalization constant. We will use it to raise and lower group indices. In particular, by defining C cad = γ cb C b ad (2.26) we see that it is antisymmetric in all indices. Every compact algebra is the direct sum of two ideals, the center, and a semisimple algebra. That is, every compact Lie algebra L is the sum of ideals where N is the center and S i are simple algebras. L = N S 1 S n, (2.27) roots.tex; October 26, 2010; 15:19; p. 3

5 4 3 Simple Lie Algebras 3.1 Root System Let V be a vector space. Let T End(V) be a set of linear transformations of the vector space V. A subspace W is invariant under T if TW W. The set T is called semi-simple if the complement of every invariant subspace is also invariant. A subalgebra H of a semi-simple algebra L is called the Cartan subalgebra if it is the maximal Abelian subalgebra of L, and for any X H the endomorphism ad X is semi-simple. The dimension r of the Cartan subalgebra is called the rank of the Lie algebra L. A linear functional α : H C on the Cartan subalgebra H is called the root if for any X H, the value α(x) is a non-zero eigenvalue of the adjoint endomorphism ad X. In other words, let L α = {Y L ad X Y = α(x)y, X H}. (3.28) be the corresponding eigenspace. Then α is the root if L α 0. Then L α is called the root vector. The roots are the non-zero eigenvalues of the matrix C(X) for X H. The set of all roots is denoted by. For a semi-simple complex Lie algebra L the roots have the following properties: 1. There is the decomposition of the Lie algebra L = H E (3.29) where E = L α ( = L α L α ). (3.30) α α + 2. For any α, dim L α = 1, that is, all eigenvalues are non-degenerate. 3. If α + β 0, then B(L α, L β ) = 0. (3.31) 4. For any root α there is a unique element of the Cartan subalgebra H α H such that for any X H B(H α, X) = α(x). (3.32) roots.tex; October 26, 2010; 15:19; p. 4

6 5 5. For any root α, α is also a root, that is, α. 6. For any root α, α(h α ) = B(H α, H α ) 0. (3.33) 7. For any root α, if X α L α, and X α L α, then 8. For any two roots α, β such that α + β 0, [X α, X α ] = B(X α, X α )H α. (3.34) [L α, L β ] = L α+β. (3.35) Thus, the Cartan subalgebra H and the root vectors L α give a basis for the Lie algebra L. There is a one-to-one correspondence between the roots α and the elements H α of the Cartan subalgebra. Therefore, there is a root vector E α L α such that for any X H [X, E α ] = B(X, H α )E α. (3.36) It is easy to see that B (α, β) = B(H α, H β ). (3.37) Further, for a complex semi-simple algebra the root system has the following properties. 1. If α, then α is also a root, α, but for any k ±1, kα is not a root, kα. 2. For any two roots α, β such that α ±β there are integers p βα and q βα such that for any integer k, β k = β + kα are roots, β k, if p βα k q βα, and not roots otherwise. Then B(β, α) 2 B(α, α) = (p βα + q βα ). (3.38) 3. For any two roots α, β (p β,α + q β,α ) B(β, α) = 2 γ (p γ,α + q γ,α ). (3.39) 4. Killing form defines a real positive definite metric on the real span of the Cartan subalgebra, H = X X = r α H α, r α R. (3.40) α Moreover, the whole complex Cartan subalgebra is H = H ih. (3.41) roots.tex; October 26, 2010; 15:19; p. 5

7 6 Now, one can choose as a basis in the Lie algebra a basis in the Cartan subalgebra and the corresponding root vectors. Then the commutation relations for the complex semi-simple Lie algebra takes especially simple form. Namely, for every root one can choose a root vector E α L α such that for all α, β, Here the constants N αβ satisfy and are determined by [H i, H j ] = 0 for all H i, H j H, (3.42) [H i, E α ] = α(h i )E α, for all H i H, (3.43) [E α, E β ] = 0, if α + β 0 and α + β, (3.44) [E α, E α ] = H α, (3.45) [E α, E β ] = N αβ E α+β if α + β. (3.46) N αβ = N α, β, (3.47) Nαβ 2 = q βα(1 p βα ) B(α, α), (3.48) 2 where p and q are determined from the sequence β + kα as defined above. 3.2 Positive Roots, Simple Roots, The root vectors are linearly dependent. We need to find a basis in the root space. Let H be the subalgebra of the Cartan subalgebra defined above. Let X 1,... X r be a basis in H. Then X H is called positive if the first non-zero components is positive. The set of positive roots is denoted by +. A positive root α + is called simple if it cannot be respresented as the sum of two positive roots. The set of all simple roots is denoted by Π. The system of simple roots of a semi-simple algebra has the following properties. 1. If α, β Π then β α Π. 2. If α, β Π and α β, then B(β, α) m α (β) = 2 B(α, α) (3.49) is a nonpositive integer. roots.tex; October 26, 2010; 15:19; p. 6

8 7 3. The system of symple roots Π is linearly independent and gives a basis in H. Every positive root γ + can be represented in the form where k i are non-negative integers. γ = l k i α i, (3.50) i=1 4. If a positive root γ + is not simple, then it is the sum of a simple root and a positive root, that is, there is positive root β + such that α = γ β Π is a simple root, or, γ = α + β. The Cartan matrix is the r r matrix ( αi, α j ) m i j = 2( ), (3.51) α j, α j where α i, i = 1,..., r, are simple roots and (, ) is the usual inner product in R r. Clearly, the diagonal elements of the matrix are all equal to two. The matrix is not necessarily symmetric, but if m i j = 0, then m ji = 0. The only possible values for the off- diagonal matrix elements are 0, ±1, ±2, and ±3. The Dynkin diagram of a semi-simple Lie algebra is constructed as follows. For every simple root, place a dot. For a simple Lie algebra, the simple roots are at most of two sizes. Darken the dots corresponding to the smaller roots. Connect the i-th and j-th dots by a number of straight lines equal to m i j m ji. For a semi-simple algebra which is not simple, the diagram will have disjoint pieces, each of which corresponds to a simple algebra. 4 Representations of Compact Semisimple Lie Algebras Let L be a complex simple Lie algebra and ρ : L End(V) be a representation of the Lie algebra L in an N-dimensional complex vector space V. Every representation decomposes into a direct sum of irreducible representations. We consider irreducible representations below. A linear functional ν : H C on the Cartan subalgebra H is called the weight of the representation ρ if for any X H, the value ν(x) is a non-zero eigenvalue of the endomorphism ρ(x). In other words, let V ν = {Y L ρ(x)y = ν(x)y, X H}. (4.52) be the corresponding eigenspace. Then ν is the weight of the representation ρ if V ν 0. Then V ν is called the weight space and its elements weight vectors. If α is a root of the Lie algebra L, then the image of the endomorphism ρ(e α ) is in the eigenspace V ν+α, ρ(e α )V V ν+α. roots.tex; October 26, 2010; 15:19; p. 7

9 8 A weight µ is called the highest weight if for any positive root α of the algebra, µ + α is not a weight and in the weight space V µ there is a vector u such that it generates the whole space V under the action of the endomorphisms ρ(x), that is V = {ρ(x)u X L}. (4.53) We say that a weight ν is positive if it can be represented as a linear combination of simple roots with positive integer coefficients, ν = k α α, (4.54) α Π where k α are positive integers. This obviously defines a partial ordering of weights. Then the highest weight µ of the representation is the largest weight. A weight ν is called simple if the corresponding weight space V ν is one-dimensional, dim V ν = 1. The highest weight of any representation is simple. Every weight ν can be represented as a difference of the highest weight and a positive weight, that is, ν = µ λ, (4.55) where with positive coefficients k α. λ = k α α (4.56) α Π Every representation has a unique highest weight. For any simple root α, is a positive integer. B(α, µ) 2 B(α, α) (4.57) Every functional µ satisfying this condition determines a finite-dimensional representation. Thus, the highest weight characterizes the representation uniquely. The set W of all weights of a representation is called the weight diagram of the representation. The dimension of an irreducible finite-dimensional representation ρ with the highest weight µ is given by the Weyl formula B(α, µ + σ) dim V =, (4.58) B(α, σ) α + where is one-half of the sum of all positive roots of the algebra. σ = 1 α (4.59) 2 α + roots.tex; October 26, 2010; 15:19; p. 8

10 9 A basis in the space of all weights is given by the simple roots of the algebra α Π. For a given basis α i Π of simple roots we define the dual basis β j by 2 B(β j, α i ) B(β j, β j ) = δ i j. (4.60) The elements of the dual basis are called fundamental weights. Every fundamental weight β determines an irreducible representation ρ β called the fundamental representation with the highest weight β. The number of inequivalent fundamental representations is equal to the rank of the algebra. An irreducible representation ρ is determined by its highest weight µ or by the positive integers B(α, µ) m α = 2 B(α, α), α Π, (4.61) which are equal to the coefficients of the expansion of the highest weight in terms of the simple roots µ = m α α. (4.62) One can show that for an irreducible N-dimensional representation ρ the matrix, called the Casimir operator, γ ab ρ(t a )ρ(t b ), (4.63) where γ ab is the inverse of the matrix γ ab, commutes with all generators and, therefore, is proportional to the identity α Π γ ab ρ(t a )ρ(t b ) = C 2 (ρ)i, (4.64) where C 2 (ρ) is a number characterizing the representation. Therefore, tr ρ(t a )ρ(t b ) = N m C 2(ρ)γ ab. (4.65) This means that tr ρ(x)ρ(y) = N m C 2(ρ)(X, Y). (4.66) Therefore, for an element X = X a C a we have similarly, for any ρ(x) = X a ρ(t a ) X a = m N X a = γ ab tr (XC b ), (4.67) 1 C 2 (ρ) γab tr [ρ(x)ρ(t b )]. (4.68) roots.tex; October 26, 2010; 15:19; p. 9

11 10 5 S U(N) The algebra su(n) is formed by N N complex traceless anti-hermitian matrices. Its dimension is equal to m = N 2 1. The Cartan subalgebra is generated by N N traceless diagonal purely imaginary matrices; its dimension (the rank of the algebra) is r = (N 1). The complexification of the algebra su(n) is sl(n, C) which contains any traceless N N matrices. The roots are described as follows. Let e i, i = 1,..., N + 1, be an orthonormal basis in the Euclidean space R N+1. Let N+1 v = e i. (5.69) i=1 Let H be a hyperplane at the end of the vector v, which is orthogonal to the vector v. Let Z N+1 be the lattice in R N+1 with integer coordinates Z N+1 = u N+1 RN+1 u = k i e i, i=1 k i Z. (5.70) The set Z N+1 H is the set of all points on the hyperplane H with integer coefficients. Let be the set of vectors in Z N+1 H of length 2. Then (su(n)) = { α i j = e i e j i, j = 1,..., N; i j }. (5.71) This is nothing but the root system of S U(N). It contains N(N 1) vectors. The set of positive roots is + (su(n)) = { α i j = e i e j i, j = 1,..., N; i < j }. (5.72) The system of simple roots is given by Π(su(N)) = { α i = e i e i+1 i = 1,..., N 1 } (5.73) It contains N 1 vectors. Its Cartan matrix is ( αi, α j ) m i j = 2( ) = α j, α j (5.74) roots.tex; October 26, 2010; 15:19; p. 10