7. The classical and exceptional Lie algebras * version 1.4 *

Transcription

1 7 The classical and exceptional Lie algebras * version 4 * Matthew Foster November 4, 06 Contents 7 su(n): A n 7 Example: su(4) = A sp(n): C n 5 73 so(n): D n 9 74 so(n + ): B n 3 75 Classical Lie algebras to rank 4; equivalent algebras 6 76 The exceptional algebras 6 In Sec 53, we demonstrated how the root system (adoint representation) for a generic Lie algebra can be constructed from the Cartan matrix or Dynkin diagram The latter encodes the inner products and norm ratios of the simple roots In this module, we assemble the Cartan matrix from the simple roots for the four classical Lie algebra families A n, B n, C n, and D n The needed data is extracted from the defining representation for each algebra We will also state (but not derive) the results for the five exceptional algebras G, F 4, and E 6,7,8 In the next module 8, we will show how the Cartan matrix can be used to obtain the full set of weights in any (finite dimensional) irreducible representation In addition, the weights of the defining representation for the orthogonal algebras B n [so(n + )] and D n [so(n)] turn out to give a useful alternative basis for H, since these prove to be orthonormal We will exploit this fact later in module 0 to construct spinor representations The discussion here largely follows chapter VIII of [] 7 su(n): A n Consider the space of all n n matrices with real or complex elements A basis for this space can be defined as follows: (Êab ) i = δ aiδ b Êab a b b b+ a a+ (7)

2 All unmarked entries are zero These elements satisfy the multiplication rule (Êab ) (Êcd ) Ê ab Ê cd = δ ai δ b δ c δ dk δ bc Ê ad, (7) i k so that [Ê ab, Ê cd ] = δ bc Ê ad δ da Ê cb (73) Note that [Êab, Êcc] = 0, (74) c which implies that the identity ˆ n = c Ê cc (75) generates a one-dimensional abelian ideal We want to construct root systems for generic semi-simple Lie algebras (no abelian ideals); we should therefore restrict ourselves to traceless n n matrices, ie the su(n)= A n Lie algebra In this case, an arbitrary element of the Cartan subalgebra can be written as Now Ĥ = λ i Ê ii, i= λ i = 0 (76) i= âdh (e ab ) [Ĥ, Ê ab ] = λ a Ê ab λ b Ê ab, (77) which implies that e ab is a root vector: [h, e ab ] = α ab (h)e ab, α ab ( i λ i e ii ) = λ a λ b (78) We identify the positive root vectors with set of upper triangular {Êab} For a generic Cartan subalgebra element h H defined via Eq (76), we claim that the set of simple roots Π can be associated to the first diagonal root vectors: root vector root α(h) e e α α (h) = λ λ e 3 e α α (h) = λ λ 3 e n,n e αn α n (h) = λ n λ n (79) Proof: A generic positive root is α i (h) = λ i λ, i < (70) This can be written as α i (h) = (λ i λ i+ ) + (λ i+ λ i+ ) + (λ i+ λ i+3 ) + + (λ λ ) = α l (7) Therefore every positive root can be expressed as a linear combination of the {α i } with non-negative integral coefficients Moreover, α i cannot be expressed as a linear combination of the other proposed simple roots To see this, assume the converse, ie that α i is positive, but not simple Then α i (h) = λ i λ i+ = i κ (i) α (h) = + κ (i) i (λ i λ i ) + κ (i) i+ (λ i+ λ i+ ) + (7) l=i Since all the κ (i) are positive, there is no way to get λ i+ with a negative coefficient (contradiction)

3 For the simple roots, we therefore have the associated Lie brackets [e αi, e α ] =δ,i+ e i,i+ δ,i e i,i+, [e αi, e α ] =δ i (e ii e i+,i+ ), [h, e αi ] =(λ i λ i+ )e αi, h defined as in Eq (76) (73a) (73b) (73c) Here the root vector associated to α is e α Ê+, Eq (73b) is consistent with the fact that simple roots are lowest weight states in mutually generated strings of the adoint representation [Eq (533)] It is useful to construct the {h αi } (i {,,, n }), ie the elements of the Cartan subalgebra H that represent the simple roots Consider the Killing form between generic elements of H We use the algorithm described in module 4 [Eq (4)] For we have the Killing form (h, h ) = since b λ b = 0 Therefore, since we conclude that Recall that [Eq (440)] a,b= a =b h i λ i e ii, ( ) [h, [h, e ab ]] = coeff of e ab h αi = e ii e i+,i+ n h a,b= λ e, (74) (λ a λ b )(λ a λ b) = n λ a λ a, (75) (h αi, h ) = α i (h ) = λ i λ i+, (76) Eqs (73b), (77) and (78) therefore give the Killing form of the simple root vectors, Therefore We can now compute the scalar products between simple roots, using = a= λ (α i ) e, λ (α i ) = n (δ i, δ i+, ) (77) h αi = [e α i, e αi ] (e αi, e αi ) (78) (e αi, e αi ) = n (79) [h αi, e α ] = α i, α e α = ( ) λ (α i ) λ (α i ) + eα = n (δ i, δ i+, δ i,+ + δ i+,+ )e α (70) α i, α = n (δ i, δ i+, δ i,+ ), simple root scalar products in A n (7) The Cartan matrix is defined as [Eqs (538) and (539)] A i = α i, α α, α = p α i;α (i ), (7) where p αi;α {0,,, 3} determines the number of states above α i in the root string generated by e α For A n, Eq (7) implies that A i = δ i δ i+, δ i,+ Â 0 0 0, Cartan matrix for A n (73)

4 The corresponding Dynkin diagram is shown in Fig 7 Because all roots have the same length, su(n) = A n is simply laced Figure 7: su(n) = A n 7 Example: su(4) = A 3 Using the root-building algorithm from Sec 53, we construct the roots for A 3 At k =, we have the simple roots [ 0 ] [ ] [ 0 ] (α ) (α ) (α 3) (74a) At k =, the sum ρ () α + α is a valid root since [Eq (7)] p α;α =, p α;α = (74b) The sum ρ () α + α 3 is a valid root since The Dynkin labels for these level roots are The non-zero depths are p α;α 3 =, p α3 ;α = [ ] [ ] ρ () ρ () (74c) (74d) m ρ () ;α = m ρ () ;α =, m ρ () ;α = m ρ () ;α3 = (74e) 3 At k = 3, we compute p =m () ρ = 0, ;α ρ () ;α p () ρ =m ;α () ρ = 0, ;α p () ρ =m ;α () ρ + =, ;α p () ρ =m ;α () ρ + =, ;α p () ρ =m ;α () ρ = 0, ;α p () ρ =m ;α () ρ = 0 ;α Thus ρ (3) ρ () + α 3 = ρ () + α = α + α + α 3 is the only k = 3 root, [ 0 ] ρ (3) (74f) (74g) The depths are m ρ (3) ;α =, m ρ (3) ;α = 0, m ρ (3) ;α 3 = (74h) 4

5 Figure 7: Root tree for su(4) = A 3 Roots are specified by their Dynkin labels The level k = simple roots appear at the top; linear combinations of these give the other positive roots The labels on diagonal lines indicate the simple root that gives the level (k + ) th root from the k th one 4 At k = 4, p ρ (3) ;α =m ρ (3) ;α = 0, p ρ (3) ;α =m ρ (3) ;α 0 = 0, p ρ (3) ;α 3 = m ρ (3) ;α 3 = 0 (74i) We conclude that there are no level k = 4 roots ρ (3) = α + α + α 3 θ is the highest root The results of the algorithm can be indicated with the branching diagram in Fig 7 Since the roots are non-degenerate [Proposition (VI), Sec 5] and each positive root ρ has a corresponding negative root ρ, we have a complete inventory for the set of roots for A 3 There are = roots, so the group dimension d = + 3 = 5, as expected for su(4) To get a geometrical picture of the algebra, we examine the Cartan matrix in Eq (73) The angle between simple roots {α, α } and between {α, α 3 } is 0, while α and α 3 are orthogonal Combined with the fact that the algebra is simply laced and the inventory of roots (Fig 7), one finds that the roots correspond to the twelve vertices of a certain quasiregular polyhedron (a cuboctahedron) See Fig 73 7 sp(n): C n Next we consider C n, the Lie algebra associated to symplectic Sp(n) transformations From module A, Eqs (A3) and (A33), the defining conditions for a symplectic transformation Û on a n-component complex vector are Û Û =ˆ n, Û T ˆɛ Û = ˆɛ, (7a) (7b) Figure 73: Root geometry for su(4) = A 3 The twelve roots correspond to the vertices of the green polyhedron (a cuboctahedron) The vertices of the embedded red (blue) tetrahedron correspond to the four weights of the defining 4 (conugate 4 ) representation, as demonstrated module 8 The box arrays on the right are the corresponding Young tableaux, also discussed in module 8 5

6 where the n n matrix ˆɛ is a block rank- Levi-Civita symbol, ˆɛ [ 0 ˆ n ˆ n 0 If we express Û as the exponentiation of an (antihermitian) n n matrix Ŷ Û = exp(ŷ ), ] (7) then the symplectic condition translates to As with ˆɛ, we decompose this into n n blocks, Ŷ T = ˆɛ Ŷ ˆɛ (73) ] [Ŷ Ŷ Ŷ = (74) Ŷ 3 Ŷ 4 Eq (73) then implies that Ŷ 4 = Ŷ T, Ŷ =Ŷ T, Ŷ 3 =Ŷ T 3 (75) Let, k n We can define the following basis elements for Ŷ, Ê k Êk Êk+n,+n, Ê k Ê,k+n + Êk,+n = Ê (k), Ê 3 k Ê +n,k + Ê k+n, = Ê 3 (k) (76a) (76b) (76c) The parentheses here denote symmetrization [Eq (4)] The total number of such elements is n + n(n + ) = n(n + ), which is the correct group dimension [Eq (A35)] Explicitly for n = 5 [sp(0)], we have (eg) Ê34, (77a) Ê , (77b) 6

7 Ê (77c) Using Eq (73), one can verify the following Lie brackets implied by the corresponding commutation relations: Exercise: Verify all of the Lie brackets in Eq (78) [e k, e lp ] = δ kl e p δ p e lk, (78a) [e k, e lp] = δ kl e p + δ pk e l, (78b) [e k, e 3 lp] = δ p e 3 kl δ l e 3 kp, (78c) [e k, e lp ] = 0, (78d) [e 3 k, e 3 lp] = 0, (78e) [e k, e 3 lp] = δ kl e p + δ p e kl + δ pk e l + δ l e kp (78f) A basis for the n-dimensional Cartan subalgebra H is given by the diagonal traceless elements {e }, n; a generic element h H is h = λ e (79) = Then [h, e k] =(λ λ k )e k, [h, e k ] =(λ + λ k )e k, [h, e 3 k] = (λ + λ k )e 3 k, (70a) (70b) (70c) so that the collection of {e,,3 k } excluding diagonal {e } is the set of root vectors The simple roots can be associated to the root vectors root vector root α(h) e e α α (h) = λ λ e 3 e α α (h) = λ λ 3 e n,n e α n α n (h) = λ n λ n e n,n e αn α n (h) = λ n (7) By definition, we take α > α > > α n (lexicographic ordering) Then positive roots are associated to root vectors {e k } with < k, and to {e k } Negative root vectors are {e k } ( > k) and {e3 k } The positive root associated to e k ( < k) is expressed in terms of simple roots identically as in Eq (7) For e i, the decomposition is (taking i WLoG) n λ i + λ = (λ i λ ) + (λ λ n ) + λ n = α l + α l + α n (7) l=i l= 7

8 The commutation relations between the simple root vectors are given by [e αi, e α ] = δ,i+ e i,i+ δ,i e i,i+, i, n, [e αi, e αn ] = δ i,n e n,n, i n, ( [e αi, e α ] = δ i e i,i e i+,i+), i, n, [e αi, e αn ] = 0, i n, [e αn, e αn ] = 4 e nn (73a) (73b) (73c) (73d) (73e) The Killing form between Cartan subalgebra elements h i λ i e ii, h λ e, (74) is given by Now since (h, h ) = = = p,q= p q p,q= ( [h, [ h, e ]] ) pq coeff of e pq + p q= p q= ( [h, [ h, e ]] ) pq coeff of e pq n (λ p λ q )(λ p λ q ) + (λ p + λ q )(λ p + λ q ) (λ p λ q )(λ p λ q) + p,q= =4(n + ) λ p λ p p= (λ p + λ q )(λ p + λ q) + 4 λ p λ p p,q= (h αi, h ) = α i (h ) = p= + { λ i λ i+, i n, λ n, i = n, p q= ( [h, [ h, e 3 ]] ) pq coeff of e 3 pq (75) (76) we conclude that h αi = h αn = 4(n + ) (n + ) e nn ( e ii e i+,i+) n = = λ (α i ) e, λ(α i ) = 4(n + ) (δ i, δ i+, ), i n, λ (αn) e, λ (αn) = (n + ) δ,n (77) Eqs (73c), (73e), (77) and (78) therefore give the Killing forms of the simple root vectors, (e αi, e αi ) =4(n + ), i n, (e αn, e αn ) =8(n + ) (78) We can now compute the scalar products between simple roots, using [h αi, e α ] = α i, α e α = ( λ (α i ) λ (α i ) +) eα = 4(n + ) (δ i, δ i+, δ i,+ ) e α, i, n, [h αn, e α ] = α n, α e α = ( λ (αn) λ +) (αn) eα = (n + ) δ +,n e α, n, (79) [h αn, e αn ] = α n, α n e αn = (n + ) e α n 8

9 Figure 74: sp(n) = C n Therefore α i, α = 4(n + ) (δ i, δ i+, δ i,+ ), i, n, α i, α n = (n + ) δ i,n, i n, α n, α n = (n + ), simple root scalar products in C n (70) The last equation implies that α n is a long root, with α n = α, n C n is not simply laced The corresponding Cartan matrix is given by [via Eq (7)] Â, Cartan matrix for C n (7) The Dynkin diagram is shown in Fig so(n): D n Next we consider D n, the Lie algebra associated to special orthogonal SO(n) transformations on even-dimensional vectors From Eq (A) in module A, the defining condition for an orthogonal transformation Û on a n-component vector is If we express Û as the exponentiation of an (antihermitian) n n matrix Ŷ then we have the antisymmetry condition Û T Û =ˆ n (73) Û = exp(ŷ ), Ŷ T = Ŷ (73) Since we would like to identify Cartan subalgebra elements with diagonal matrices, Eq (73) is inconvenient Instead, we make a unitary (but not orthogonal) basis transformation, Then Û ˆV Û ˆV, ˆV ˆV = ˆ n (733) Û T ˆMÛ = ˆM, ˆM ˆV T ˆV = ˆM T, ˆM ˆM = ˆ n (734) For Û = exp(ŷ ), ˆM Ŷ T ˆM = Ŷ (735) 9

10 We choose ˆV to have the block form ˆV = [ iˆ n ˆ n iˆ n ˆ n ], ˆM = ˆM = [ ] 0 ˆ n (736) ˆ n 0 As with ˆM, we decompose Ŷ into n n blocks, ] [Ŷ Ŷ Ŷ = (737) Ŷ 3 Ŷ 4 Eq (735) then implies that Ŷ 4 = Ŷ T, Ŷ = Ŷ T, (738) Ŷ 3 = Ŷ3 T This is almost the same as the symplectic case [Eq (75)], except the off-diagonal blocks are now antisymmetric Let, k n We can define the following basis elements for Ŷ, Êk Ê k Ê k+n,+n, Êk Ê,k+n Ê k,+n = Ê[k], Êk 3 Ê+n,k Êk+n, = Ê3 [k] (739a) (739b) (739c) The parentheses here denote antisymmetrization [Eq (4)] The total number of such elements is n + n(n ) = n(n ), which is the correct group dimension [Eq (73)] Relative to the n = 5 symplectic example in Eq (77), Ê34, (730a) Ê (730b) 0

11 Using Eq (73), the Lie brackets are given by [cf Eq (78)] Exercise: Verify the Lie brackets in Eq (73) A generic Cartan subalgebra H element is given by [e k, e lp] = δ kl e p δ p e lk, (73a) [e k, e lp] = δ kl e p δ pk e l, (73b) [e k, e 3 lp] = δ p e 3 kl δ l e 3 kp, (73c) [e k, e lp] = 0, (73d) [e 3 k, e3 lp ] = 0, (73e) [e k, e 3 lp] = δ kl e p + δ p e kl δ pk e l δ l e kp (73f) h = λ e (73) = Then The simple roots can be associated to the root vectors [h, e k] =(λ λ k )e k, [h, e k ] =(λ + λ k )e k, [h, e 3 k] = (λ + λ k )e 3 k root vector root α(h) e e α α (h) = λ λ e 3 e α α (h) = λ λ 3 e n,n e αn α n (h) = λ n λ n e n,n e α n α n (h) = λ n + λ n (733a) (733b) (733c) (734) By definition, we take α > α > > α n (lexicographic ordering) Then positive roots are associated to root vectors {e k } with < k, and to {e k } Negative root vectors are {e k } ( > k) and {e3 k } The positive root associated to e k ( < k) is expressed in terms of simple roots identically as in Eq (7) For e i, the decomposition is (taking > i WLoG) n λ i + λ = (λ i λ ) + (λ λ n ) + (λ n λ n ) + (λ n + λ n ) = α l + α l + α n + α n (735) The commutation relations between the simple root vectors are given by l=i l= [e αi, e α ] = δ,i+ e i,i+ δ,i e i,i+, i, n, [e αi, e αn ] = δ i,n e n,n, i n, ( [e αi, e α ] = δ i e i,i e i+,i+), i, n, [e αi, e αn ] = 0, i n, [e αn, e αn ] = e n,n + e n,n (736a) (736b) (736c) (736d) (736e) The Killing form between Cartan subalgebra elements h i λ i e ii, h λ e, (737) is given by (h, h ) = 4(n ) λ p λ p (738) p=

12 Exercise: Verify Eq (738) Now since we conclude that h αi = h αn = 4(n ) 4(n ) ( e ii e i+,i+) n (h αi, h ) = α i (h ) = = ( e n,n + e n,n) n { λ i λ i+, i n, λ n + λ n, i = n, λ (α i ) e, λ (α i ) = 4(n ) (δ i, δ i+, ), i n, = λ (αn) e, λ (αn) = 4(n ) (δ n, + δ n, ) (739) (730) Eqs (736c), (736e), (730) and (78) therefore give the Killing forms of the simple root vectors, Therefore We can now compute the scalar products between simple roots, using (e αi, e αi ) = 4(n ), i n (73) [h αi, e α ] = 4(n ) (δ i, δ i+, δ i,+ ) e α, i, n, [h αn, e α ] = 4(n ) δ,n e α, n, [h αn, e αn ] = 4(n ) e α n α i, α = 4(n ) (δ i, δ i+, δ i,+ ), i, n, α i, α n = 4(n ) δ i,n, i n, α n, α n = 4(n ), Since all roots have the same length, D n is simply laced The corresponding Cartan matrix is given by [via Eq (7)] (73) simple root scalar products in D n (733) Â, Cartan matrix for D n (734) The Dynkin diagram is shown in Fig 75 Figure 75: so(n) = D n

13 74 so(n + ): B n Finally we consider B n so(n + ) We can view the B n generators in the defining representation as equivalent to those for D n, but with an added zeroth row and zeroth column preceding the n n generators of so(n); ie a vector index i runs from 0 i n As in Sec 73, we begin by rotating the orthogonal condition Û T Û = ˆ n+ to a different basis Applying the transformation as in Eq (733), we choose ˆV = ˆ0,n ˆ0,n ˆ0,n ˆ0,n ˆ0 n, iˆ n iˆ n, ˆM = ˆV T ˆV = ˆ0 n, 0 ˆ n (74) ˆ0 n, ˆ n ˆ n ˆ0 n, ˆ n 0 Here ˆ0,n (ˆ0 n, ) denotes an n-fold row (column) of zeroes The (n+) (n+) generator Ŷ satisfies the condition as in Eq (735), which allows the block decomposition [cf Eqs (737) and (738)] b ĉ Ŷ,n ĉ,n = ˆd n, Ŷ Ŷ (74) Ŷ 3 Ŷ 4 Eq (735) then implies that ˆd n, Ŷ 4 = Ŷ T, Ŷ = Ŷ T, Ŷ 3 = Ŷ T 3, b =0, (743) ĉ,n = [ ˆd n, ] T, ĉ,n = [ ˆd n, ] T The basis elements for Ŷ include {Ê k, Ê k, Ê3 k } (, k n) from Eq (739), and Ê 4 Ê 0, Ê +n,0, Ê 5 Ê,0 Ê 0,+n, (744a) (744b) both with n The total number of such elements is n(n ) + n = n(n + ) Relative to the example in Eq (730), {Êk, Ê k, Ê3 k } are now block embedded in matrices (after one initial row and one initial column of zeroes), while (eg) Ê , (745a)

14 Ê3 5 In addition to the Lie brackets in Eq (73), we have Exercise: Verify the Lie brackets in Eq (746) A generic Cartan subalgebra H element is given by (745b) [e k, e 4 l ] = δ l e 4 k, (746a) [e k, e 5 l ] =δ kl e 5, (746b) [e k, e4 l ] =δ l e 5 k δ kl e 5, (746c) [e k, e 5 l ] =0, (746d) [e 3 k, e 4 l ] =0, (746e) [e 3 k, e 5 l ] =δ l e 4 k δ kl e 4, (746f) [e 4, e 4 k] = e 3 k, (746g) [e 5, e5 k ] = e k, (746h) [e 4, e 5 k] = e k (746i) h = λ e (747) = Then [h, e k] =(λ λ k )e k, [h, e k] =(λ + λ k )e k, [h, e 3 k ] = (λ + λ k )e 3 k, [h, e 4 ] = λ e 4, [h, e 5 ] =λ e 5 The simple roots can be associated to the root vectors root vector root α(h) e e α α (h) = λ λ e 3 e α α (h) = λ λ 3 e n,n e αn α n (h) = λ n λ n e 5 n e αn α n (h) = λ n (748a) (748b) (748c) (748d) (748e) (749) By definition, we take α > α > > α n (lexicographic ordering) The positive roots are associated to root vectors {e k } with < k, {e k }, and {e5 } Negative root vectors are {e k } ( > k), {e3 k }, and {e4 } The positive root associated to e k ( < k) is expressed in terms of simple roots identically as in Eq (7) For e i, the decomposition is (taking > i WLoG) n λ i + λ = (λ i λ ) + (λ λ n ) + λ n = α l + α l + α n (740) l=i l= 4

15 Eq (740) has almost the same form as the symplectic C n case, Eq (7); the difference is the prefactor of two for α n in Eq (740) In fact we will see that B n and C n are closely related (although not identical, except for B = C ) For e 5 i, the decomposition is The commutation relations between the simple root vectors are given by n λ i = (λ i λ n ) + λ n = α l + α n (74) l=i [e αi, e α ] = δ,i+ e i,i+ δ,i e i,i+, i, n, [e αi, e αn ] = δ i,n e 5 n, i n, ( [e αi, e α ] = δ i e i,i e i+,i+), i, n, [e αi, e αn ] = 0, i n, [e αn, e αn ] = e nn (74a) (74b) (74c) (74d) (74e) The Killing form between Cartan subalgebra elements h i λ i e ii, h λ e, (743) is given by ( (h, h ) =4 n ) n λ p λ p (744) p= Exercise: Verify Eq (744) Now since (h αi, h ) = α i (h ) = { λ i λ i+, i n, λ n, i = n, (745) we conclude that h αi = 4 ( ) ( ) n n e ii e i+,i+ = h αn = 4 ( )e n n,n = λ (α i ) e, λ (α i ) = λ (αn) e, λ (αn) = 4 ( n 4 ( n ) (δ i, δ i+, ), i n, )δ n, (746) Eqs (74c), (74e), (746) and (78) therefore give the Killing forms of the simple root vectors, ( (e αi, e αi ) = 4 n ), i n (747) We can now compute the scalar products between simple roots, using [h αi, e α ] = 4 ( ) n (δ i, δ i+, δ i,+ ) e α, i, n, [h αn, e α ] = 4 ( ) n δ,n e α, n, [h αn, e αn ] = 4 ( )e n αn (748) 5

16 Figure 76: so(n + ) = B n Therefore α i, α = 4 ( ) n (δ i, δ i+, δ i,+ ), i, n, α i, α n = 4 ( )δ n i,n, i n, α n, α n = 4 ( ) n simple root scalar products in B n (749) The last equation implies that α n is a short root, with α n = α /, n Like C n, B n is not simply laced The corresponding Cartan matrix is given by [via Eq (7)] Â, Cartan matrix for B n (740) The Dynkin diagram is shown in Fig 76 The Cartan matrices/dynkin diagrams for C n [Eq (7), Fig 74] and B n [Eq (740), Fig 76] are almost identical In both cases, the only non-vanishing simple root inner products involve nearest-neighbors α i and α i+, i n The first n simple roots share the same length, while α n is a long (short) root for C n (B n ) In fact, the root systems for B n and C n have the same geometry, except that long and short roots are interchanged On the other hand, we will see that D n and B n have more in common in terms of the n simplest (so called fundamental ) representations, described in module 8 In particular, both D n and B n possess fundamental spinor representations, which are qualitatively distinct and cannot be built by tensoring together defining vector indices By contrast, every representation of C n or A n can be built by tensoring together indices transforming in the the defining n-dimensional [(n + )-dimensional] representation Exercise: Find the inventory of roots and plot the 3D root geometry for B 3 and C 3, following the same approach used for A 3 in Sec 7 75 Classical Lie algebras to rank 4; equivalent algebras In Table, we show the classical Lie algebras through rank four Various accidental equivalences between different algebras of the same rank are indicated, and easily recognized by the identical Dynkin diagrams All four families are distinct for each rank above three Of course, equivalence of the Lie algebras does not imply equivalence of the corresponding symmetry groups 76 The exceptional algebras The four classical families studied above do not exhaust the possibilities for (semi-)simple Lie algebras It is perhaps surprising that there are only five additional possibilities corresponding to particular algebras of fixed rank This can be shown by a careful determination of allowable Dynkin diagrams The argument is explicated in chapter IX of Cahn [] and chapter 0 of Georgi []; So far we have used the terms defining and fundamental representations interchangably, as is common in physics However, for a given Lie algebra L, there exists of set of simplest representations (which always includes the defining one), and mathematicians refer to each member of this set as a fundamental representation We will define fundamental represenations precisely in module 8, using the fundamental weights 6

17 Table : Classical Lie algebras to rank 4 Rank Dynkin Cartan Compact # of positive roots A su() = sp() = so(3) A su(3) 3 B = C so(5) = sp(4) 4 D = A A so(4) = su() su() A 3 = D 3 su(4) = so(6) 6 3 B 3 so(7) 9 C 3 sp(6) 9 A 4 su(5) 0 B 4 so(9) 6 4 C 4 sp(8) 6 D 4 so(8) we won t belabor it here Instead, we simply state the Cartan matrices and Dynkin diagrams for the five exceptional algebras G, F 4, and E 6,7,8 Our labeling conventions are the same as chapter 3 of [3] Â G = Cartan matrices for the exceptional Lie algebras G, F 4, E 6, E 7, and E [ ] , Â E6 = , (76) Â F4 = 0 0, Â E7 = , (76) Â E8 =

18 Figure 77: The five exceptional algebras References [] Robert N Cahn, Semi-Simple Lie Algebras and Their Representations (Benamin/Cummings, Menlo Park, California, 984) [] Howard Georgi, Lie Algebras in Particle Physics, nd ed (Westview Press, Boulder, Colorado, 999) [3] Phillipe Di Francesco, Pierre Mathieu, David Sénéchal, Conformal Field Theory (Springer-Verlag, New York, 996) 8