Notes on Lie Algebras

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 E-mail: iavramid@nmt.edu

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

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

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

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

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

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

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

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

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

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 2 1 0... 0 ( αi, α j ) m i j = 2( ) = α j, α j 1 2 1... 0 0 1 2... 0. (5.74)........... 0 0 0... 2 roots.tex; October 26, 2010; 15:19; p. 10