Split Casimir operator for simple Lie algebras, solutions of Yang-Baxter equations and Vogel parameters
aa r X i v : . [ m a t h - ph ] F e b Split Casimir operator for simple Liealgebras, solutions of Yang-Baxterequations and Vogel parameters
A.P.Isaev a,b,c , S.O.Krivonos a a Bogoliubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, 141980 Dubna, Russia b St.Petersburg Department of Steklov Mathematical Institute of RAS,Fontanka 27, 191023 St. Petersburg, Russia c Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia [email protected], [email protected]
Abstract
We construct characteristic identities for the split (polarized) Casimir operators ofthe simple Lie algebras in defining (minimal fundamental) and adjoint representa-tions. By means of these characteristic identities, for all simple Lie algebras wederive explicit formulae for invariant projectors onto irreducible subrepresentationsin T ⊗ in two cases, when T is the defining and the adjoint representation. In thecase when T is the defining representation, these projectors and the split Casimiroperator are used to explicitly write down invariant solutions of the Yang-Baxterequations. In the case when T is the adjoint representation, these projectors andcharacteristic identities are considered from the viewpoint of the universal descrip-tion of the simple Lie algebras in terms of the Vogel parameters. ontents b C for Lie algebra sℓ ( N ) . . . . . . . . . . . . . . . . 73.1.1 Operator b C for sℓ ( N ) in the defining representation. . . . . . . . . 73.1.2 Operator b C for sℓ ( N ) in the adjoint representation. . . . . . . . . . 93.2 Split Casimir operator b C for Lie algebras so ( N ) and sp (2 n ) . . . . . . . . . 173.2.1 Operator b C for so ( N ) and sp (2 n ) in the defining representation . . 173.2.2 Operator b C for so ( N ) and sp (2 n ) in the adjoint representation . . . 193.2.3 The algebra so (8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Universal characteristic identities for operator b C + in the case of Lie algebrasof classical series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 b C for exceptional Lie algebras 25 R -matrices in the fundamental rep-resentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1.2 Split Casimir operator and R -matrix for the algebra g . . . . . . . 264.1.3 Split Casimir operator and R -matrix for the algebra f . . . . . . . 284.1.4 Split Casimir operator and R -matrix for the algebra e . . . . . . . 314.1.5 Split Casimir operator and R -matrix for the algebra e . . . . . . . 324.1.6 Algebra e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Characteristic identities for operator b C and invariant projectors for excep-tional Lie algebras in the adjoint representations. . . . . . . . . . . . . . . 334.2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.2 Algebra g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.3 Algebra f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.4 Algebra e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2.5 Algebra e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2.6 Algebra e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Universal characteristic identities and general comments . . . . . . . . . . 38 b C for simple Lie algebrasin the adjoint representation and Vogel parameters 40 It is known that the special invariant operator, the split (or polarized) Casimir operator b C (see definition in Section ), plays an important role both in the description of the Lie1lgebras g themselves and in the studies of their representation theory. On the other hand,the split Casimir operator b C is the building block (see e.g. [1], [2] and references therein)for constructing g -invariant solutions r and R of semiclassical and quantum Yang-Baxterequations [ r ( u ) , r ( u + v )] + [ r ( u + v ) , r ( v )] + [ r ( u ) , r ( v )] = 0 , (1.0.1) R ( u ) R ( u + v ) R ( v ) = R ( v ) R ( u + v ) R ( u ) . (1.0.2)We use here the standard matrix notation which will be explained below. Recall [3] that g -invariant rational solutions of the Yang-Baxter equations (1.0.2) allow one to define theYangians Y ( g ) within the so-called RT T -realization.In this paper, we demonstrate the usefulness of the g -invariant split Casimir operator b C in the representation theory of Lie algebras. Namely, for all simple Lie algebras g ,explicit formulas are found for invariant projectors onto irreducible representations thatappear in the expansion of the tensor product T ⊗ T of two representations T . Theseprojectors are constructed in terms of the operator b C for two cases, when T is the defining(minimal fundamental) and when T ≡ ad is the adjoint representation of g .It is natural to find such invariant projectors in terms of g -invariant operators, whichin turn are images of special elements of the so-called centralizer algebra. The idea of thisapproach is not new. For example, invariant projectors acting in tensor representationsof sℓ ( N ) algebras are called Young symmetrizers and constructed as images of specialelements (idempotents) of group algebra C [ S r ] of the symmetric group S r . The algebra C [ S r ] centralizes the action of the sℓ ( N ) in the space of tensors of rank r (in the spaces ofrepresentations T ⊗ r ). In this paper, we consider a very particular problem of constructinginvariant projectors in representation spaces of T ⊗ , where T is the defining, or adjointrepresentation but for all simple Lie algebras g . Our approach is closely related to theone outlined in [4]. In [4], such invariant projectors were obtained in terms of severalspecial invariant operators and the calculations were performed using a peculiar diagramtechnique. In our approach, we try to construct invariant projectors in the representationspace V ⊗ of T ⊗ by using only one g -invariant operator which is the split Casimir operator b C . It turns out that for all simple Lie algebras g in the defining representations allinvariant projectors in V ⊗ are constructed as polynomials in b C . It is not the case forthe adjoint representation, i.e. not for all algebras g the invariant projectors in V ⊗ areconstructed as polynomials of only one operator b C ad ≡ ad ⊗ b C . Namely, in the case of sℓ ( N ) and so (8) algebras there are additional g -invariant operators which are independentof b C ad and act, respectively, in the antisymmetrized and symmetrized parts of the space V ⊗ . We construct such additional operators explicitly in Sections and .Our study of the split Casimir operator b C was motivated by the works [13], [14]and [15], and by the idea of finding formulas for solutions of the Yang-Baxter equationexpressed in terms of only the operator b C . For defining (minimal fundamental) repre-sentations of the simple Lie algebras g (except for the algebra e ), such formulas werederived in this paper (see equations (3.1.16), (3.2.12), (4.1.29), (4.1.44), (4.1.51) and(4.1.58) below). Note that these formulas are obtained by using well-known [26], [27]spectral decompositions for rational R -matrices . For the adjoint representations of thesimple Lie algebras g , as it was argued in [33], there are no such formulas (we need to All these spectral decompositions can also be obtained from the spectral decompositions of trigono-metric R -matrices (see e.g. [2] (section 7.2.4), [28], [29]) in the special limit q → g ; see [34], [35]). However, in the caseof the adjoint representation of g , the knowledge of the characteristic identities for b C ad turns out to be a key point for understanding the so-called universal formulation of thesimple Lie algebras [11] (see also the historical notes in [4], section 21.2). Though somecharacteristic identities and formulas for certain g -invariant projectors can be found in adifferent form in [4], we believe that the methods we used and the results obtained canbe useful for future research, e.g. from the viewpoint of technical applications of the splitCasimir operator.In our paper, to simplify the notation, we everywhere write sℓ ( N ), so ( N ) and sp (2 n )instead of sℓ ( N, C ), so ( N, C ) and sp (2 n, C ). Let g be a simple Lie algebra with the basis X a and defining relations[ X a , X b ] = C dab X d , (2.1.1)where C dab are the structure constants. The Cartan-Killing metric is defined in the stan-dard way g ab ≡ C dac C cbd = Tr (ad( X a ) · ad( X b )) , (2.1.2)where ad denotes adjoint representation: ad( X a ) db = C dab . Recall that the structure con-stants C abc ≡ C dab g dc are antisymmetric under permutation of indices ( a, b, c ). We denotean enveloping algebra of the Lie algebra g as U ( g ). Let g df be the inverse matrix to theCartan-Killing metric (2.1.2). We use this matrix and construct the operator b C = g ab X a ⊗ X b ∈ g ⊗ g ⊂ U ( g ) ⊗ U ( g ) , (2.1.3)which is called the split (or polarized) Casimir operator of the Lie algebra g . This operatoris related to the usual quadratic Casimir operator C (2) = g ab X a · X b ∈ U ( g ) , (2.1.4)by means of the formula ∆( C (2) ) = C (2) ⊗ I + I ⊗ C (2) + 2 b C , (2.1.5)where ∆ is the standard comultiplication for enveloping algebras U ( g ):∆( X a ) = ( X a ⊗ I + I ⊗ X a ) . (2.1.6)The following statement holds (see, for example, [7], [9]). Proposition 2.1.1
The operator b C , given in (2.1.3), does not depend on the choice ofthe basis in g and satisfies the condition (which is called ad -invariance or g -invariance): [∆( A ) , b C ] = [( A ⊗ I + I ⊗ A ) , b C ] = 0 , ∀ A ∈ g , (2.1.7)3 here ∆ is comultiplication (2.1.6). In addition, the operator b C obeys the equations [ b C , b C + b C ] = 0 ⇒ [ b C , b C ] = 12 [ b C , b C − b C ] , (2.1.8) which use the standard notation b C = g ab X a ⊗ X b ⊗ I , b C = g ab X a ⊗ I ⊗ X b , b C = g ab I ⊗ X a ⊗ X b . (2.1.9) Here I is the unit element in U ( g ) and b C ij ∈ U ( g ) ⊗ U ( g ) ⊗ U ( g ) . Relations (2.1.8) indicate that the split Casimir operator (2.1.3) realizes the Kono-Drinfeld Lie algebra and can be used as a building block for constructing solutions tothe quasi-classical (1.0.1) and quantum (1.0.2) Yang-Baxter equations. In particular, thesolution for the quasi-classical Yang-Baxter equation (1.0.1) is the operator r ( u ) = b C/u (see e.g. [2]).
Remark.
Let the normalization of the generators H i , E α in the Cartan-Weyl basis of thealgebra g be chosen so that we have for (2.1.4) and (2.1.3): C (2) = g ij ( H i H j ) + X α ( E α E − α ) ⇒ b C = g ij ( H i ⊗ H j ) + X α ( E α ⊗ E − α ) , (2.1.10)where the sum goes over all roots α and g ij is the inverse matrix to the metric in the rootspace g ij = X α α i α j . (2.1.11)Then the split Casimir operator is decomposed into the sum b C = ( r + + r − ) of two solutions r = r + and r = r − of a constant (i.e. independent of the spectral parameter) semiclassicalYang-Baxter equation [ r , r ] + [ r , r ] + [ r , r ] = 0. These solutions are written inthe form (see e.g. [1],[2]) r + = 12 g ij ( H i ⊗ H j ) + X α> ( E α ⊗ E − α ) , r − = 12 g ij ( H i ⊗ H j ) + X α> ( E − α ⊗ E α ) , where the sum goes over all positive roots α > g . The generators X a of a simple Lie algebra g satisfy the defining relations (2.1.1) and, inthe adjoint representation, X a are implemented as matrices ad( X a ) db = C dab . In this casethe split Casimir operator (2.1.3) is written as( b C ad ) a a b b ≡ (ad ⊗ ad) a a b b ( b C ) = C a hb C a fb g hf . (2.2.1)By definition this operator satisfies identities (2.1.8). Below we need one more ad-invariantoperator ( K ) a a b b = g a a g b b . (2.2.2)4he operators (2.2.1) and (2.2.2) act in the tensor product V ad ⊗ V ad of two spaces V ad = g of the adjoint representation and have the symmetry properties ( b C ad ) a a b b = ( b C ad ) a a b b and K a a b b = K a a b b , which are conveniently written in the form( b C ad ) = P ( b C ad ) P = ( b C ad ) , K = P K P = K , where 1 , V ad in the product ( V ad ⊗ V ad ) and P is permutationmatrix in ( V ad ⊗ V ad ): P ( X a ⊗ X a ) = ( X a ⊗ X a ) = ( X b ⊗ X b ) P b b a a , P b b a a = δ b a δ b a . (2.2.3)Here ( X a ⊗ X b ) is the basis in the space ( V ad ⊗ V ad ). Define the symmetrized and anti-symmetrized parts of the operator b C ad ( b C ± ) a a b b = 12 (( b C ad ) a a b b ± ( b C ad ) a a b b ) , b C ± = P ( ad ) ± b C ad = b C ad P ( ad ) ± , (2.2.4)where P ( ad ) ± = ( I ± P ) and I is the unit operator in ( V ad ) ⊗ . Proposition 2.2.1
The operators b C ad , b C ± and K , given in (2.2.1), (2.2.2) and (2.2.4),satisfy the identities b C − = − b C − , (2.2.5) b C − K = 0 = K b C − , b C ad K = K b C ad = − K , (2.2.6) b C + K = K b C + = − K . (2.2.7) Proof.
To prove equality (2.2.5), we note that b C − has a useful expression followed fromthe Jacobi identity C dab C rdc + cycle( a, b, c ) = 0 (see e.g. [4]):( b C − ) a a b b = − C a a d C db b , C a a d ≡ C a d b g b a . (2.2.8)Using this expression and identities C db b C b b a = δ da ⇔ ad( C (2) ) fr = g ab C fa d C db r = δ fr , (2.2.9)which are equivalent to the definition (2.1.2) of the Cartan-Killing metric, we calculate b C − and obtain (2.2.5). The first equality in (2.2.6) follows from the evident relations( I − P ) K = 0 = K ( I − P ). The second equality in (2.2.6) is proved with the help ofidentities (2.2.9) and complete antisymmetry of the constants C abc = C dab g dc . Relations(2.2.7) are derived from (2.2.6).Now we take into account definitions (2.2.1), (2.2.2), (2.2.3) and relations (2.2.5),(2.2.8), (2.2.9), and C aba = 0, which is valid for all simple Lie algebras, and obtain generalformulas for the traces Tr ( b C ad ) = 0 , Tr ( b C ± ) = ± dim g , Tr ( b C ) = dim g , Tr ( b C − ) = − Tr ( b C − ) = dim g , Tr ( b C ) = Tr ( b C − b C − ) = dim g , Tr ( K ) = dim g , Tr ( I ) = (dim g ) , Tr ( P ) = dim g . (2.2.10)5here Tr ≡ Tr Tr is the trace in the space V ad ⊗ V ad (indices 1 and 2 are attributed tofactors in the product V ad ⊗ V ad ). These formulas will be used in what follows.Using the characteristic identity (2.2.5) for the operator b C − , one can construct twomutually orthogonal projectorsP = − b C − , P = 2 b C − + P (ad) − ⇒ P i P k = P i δ ik , (2.2.11)which decompose the antisymmetrized part P ( ad ) − (ad ⊗ ad) of the representation (ad ⊗ ad)into two subrepresentations X , = P , (ad ⊗ ad). Dimensions of these subrepresentationsare equal to the traces of corresponding projectors (2.2.11)dim X = Tr (P ) = dim g , dim X = Tr (P ) = 12 dim g (dim g − , (2.2.12)where we use the general formulae (2.2.10). Since the constants C db b play the role ofthe Clebsch-Gordan coefficients for the fusion ad ⊗ → ad, we see from the explicit form(2.2.8) of the operator b C − that the projector P , given in (2.2.11), extracts the adjointrepresentation X = ad in P ( ad ) − (ad ⊗ ). Thus, the adjoint representation is always con-tained in the antisymmetrized part P ( ad ) − (ad ⊗ ). The first formula in (2.2.12) confirms theequivalence of X and ad. Note also that X is not necessarily irreducible representationfor all simple Lie algebras. As we will see below (see Remark after Proposition ),the representation X is reducible for algebras of the series A n = sℓ ( n + 1). The invariant metric for a simple Lie algebra g is uniquely determined up to a normaliza-tion constant (see e.g. [7]), i.e. such metric is always proportional to the Cartan-Killingmetric (2.1.2). Therefore, for any irreducible representation T of the simple Lie algebra g we have Tr (cid:0) T ( X a ) · T ( X b ) (cid:1) = d ( T ) g ab , (2.3.1)where the coefficient d ( T ) characterizes the representation T . Indeed, d ( T ) is expressedin terms of values c ( T )2 of the quadratic Casimir (2.1.4) in the irreducible representationof T using the well-known relation c ( T )2 dim( T ) = d ( T ) dim( g ) , which is obtained from (2.3.1) by contraction with the inverse metric g ab .Let T λ and T λ be two irreducible representations with the highest weights λ and λ acting in the spaces V λ and V λ . Let the representation T λ ⊗ T λ be decomposed intoirreducible representations T λ with the highest weights λ as follows: T λ ⊗ T λ = P λ n λ T λ ,where n λ is the multiplicity of occurrence of T λ in the expansion of T λ ⊗ T λ . Denotethe space of the representation T λ as V λ . Then, from (2.1.5) and expansion V λ ⊗ V λ = P λ n λ V λ , we obtain T ( λ × λ ) ( b C ) · ( V λ ⊗ V λ ) = P λ n λ ( c ( λ )2 − c ( λ )2 − c ( λ )2 ) V λ ⇔ T ( λ × λ ) ( b C ) · V λ = ( c ( λ )2 − c ( λ )2 − c ( λ )2 ) V λ , (2.3.2)6here we use the concise notation T ( λ × λ ) := ( T λ ⊗ T λ ). Here c ( λ )2 is the value of thequadratic Casimir operator C (2) , defined in (2.1.10), in the representation with the highestweight λ c ( λ )2 = ( λ, λ + 2 δ ) , δ := r X f =1 λ ( f ) = 12 X α> α , (2.3.3) λ ( f ) are the fundamental weights of the rank r Lie algebra g , α are the roots of g andsummation is over positive roots ( α > T ( b C ) is diagonalizablefor simple Lie algebras and in general its spectrum is degenerate. Therefore, formula(2.3.2) implies the characteristic identity Y λ ′ (cid:16) T ( λ × λ ) ( b C ) − ˆ c λλ ,λ (cid:17) = 0 , ˆ c λλ ,λ := 12 ( c ( λ )2 − c ( λ )2 − c ( λ )2 ) , (2.3.4)where the prime in Q ′ λ means that the product does not run over all weights λ thatparticipate in the expansion: T λ ⊗ T λ = P λ n λ T λ but only those λ that correspond tounequal eigenvalues ˆ c λλ ,λ .In the next sections, we obtain explicit expressions for the split Casimir operator T ( λ × λ ) ( b C ) ≡ ( T λ ⊗ T λ )( b C ) for all simple Lie algebras in the case when both representa-tions T λ and T λ are either defining or adjoint. In the case when T λ and T λ are adjointrepresentations λ = λ = λ ad , the characteristic identity (2.3.4) takes the form Y λ ′ (cid:16) ad ⊗ ( b C ) −
12 ( c ( λ )2 − c ( λ ad )2 ) (cid:17) ≡ Y λ ′ (cid:16) ad ⊗ ( b C ) − ˆ c ( λ )2 (cid:17) = 0 , (2.3.5)ˆ c ( λ )2 := 12 ( c ( λ )2 − c ( λ ad )2 ) = 12 c ( λ )2 − , (2.3.6)where λ ad is the highest weight of the adjoint representation of the algebra g , which isequal to the highest root θ of g . In the definition (2.3.6) of values ˆ c ( λ )2 of the operatorad ⊗ ( b C ) we use the condition (see (2.2.9)) c ( λ ad )2 ≡ c ( θ )2 = 1 . (2.3.7)Note that this formula is consistent with (2.3.3) only if the metric in the root space ofthe algebra g is given by (2.1.11), which corresponds to the condition( θ, θ ) = t − , (2.3.8)where t is the dual coxeter number of the algebra g . In the next sections, we demonstratethis fact explicitly and also find an explicit form for the characteristic identities (2.3.5)for all finite-dimensional simple Lie algebras. b C for Lie algebra sℓ ( N ) b C for sℓ ( N ) in the defining representation. In this Subsection, to fix the notation, we give the standard definitions of the Lie algebra sℓ ( N ) := sℓ ( N, C ) and the corresponding operator b C in the defining representation. Onecan find these definitions in many monographs and textbooks (see e.g. [4], [7]).7e denote the space C N of the defining representation T of the algebra sℓ ( N ) as V N .Choose the basis in sℓ ( N ) consisting of traceless matrices T ij = e ij − δ ij I N /N ⇒ T ij = e ij ( i = j ) , T ii = e ii − I N /N , (3.1.1)where e ij are N × N matrix units, and I N is the unit N × N matrix. The elements (3.1.1)satisfy the defining relations[ T ij , T km ] = δ jk T im − δ im T kj ≡ C rsij,km T rs , (3.1.2)where for structure constants we have the explicit expression C rsij,km = δ jk δ ri δ sm − δ im δ rk δ sj . (3.1.3)Using this expression and definition (2.1.2), we find the Cartan-Killing metric for sℓ ( N ): g ij,kℓ = C rsij,mn C mnkℓ,rs = 2( N δ jk δ iℓ − δ ij δ kℓ ) = g kℓ,ij , (3.1.4)and, for the basis (3.1.1) in the defining representation, equality (2.3.1) givesTr( T ij T kℓ ) = 12 N g ij,kℓ ⇒ d ( T ) = 12 N . (3.1.5)The inverse to (3.1.4) metric g ij,kℓ is defined by the traceless conditions g ii,kℓ = 0 = g ij,kk and relations g mn,ij g ij,kℓ = ¯ I kℓmn , ¯ I kℓmn ≡ δ km δ ℓn − N δ mn δ kℓ , (3.1.6)where the projector ¯ I kℓmn plays the role of the identity operator in the adjoint representationspace V ad ; we identify V ad with the subspace of traceless tensors in V N ⊗ V N , i.e. V ad =¯ I · V ⊗ N . As a result, we obtain g ij,kℓ = 12 N (cid:18) δ jk δ iℓ − N δ ij δ kℓ (cid:19) = g kℓ,ij . (3.1.7) Remark.
Strictly speaking, V ad is the subspace of second rank traceless tensors in V N ⊗ ¯ V N , where ¯ V N is the space of the contragradient representation of sℓ ( N ). In other words, V ad is the space of tensors with the components ψ ik that satisfy the traceless property ψ ii = 0. Further, for technical reasons, we treat ¯ V N as V N and consider V ad as the spaceof traceless tensors in V N ⊗ V N with the components ψ ik such that ψ ii = 0. The caseswhere the difference between V N and ¯ V N is important are specially negotiated.The matrix T ⊗ ( b C ) for the split Casimir operator (2.1.3) of the algebra sℓ ( N ) withbasis (3.1.1) in the defining representation T is written as T ⊗ ( b C ) i i j j = g ij,kℓ ( T ij ⊗ T kℓ ) i i j j = g ij,kℓ ( T ij ) i j ( T kℓ ) i j = 12 N (cid:16) δ i j δ i j − N δ i j δ i j (cid:17) , (3.1.8)and in the index-free notation we have T ⊗ ( b C ) = 12 N ( P − N I ) ≡ b C T , (3.1.9)where I = I ⊗ N is the unit operator in V ⊗ N and P is the permutation operator acting inthe space V ⊗ N . Let e i ( i = 1 , ..., N ) be the basis vector in V N , then the operator P isdefined as follows P ( v ⊗ u ) = u ⊗ v ( ∀ v, u ∈ V N ) ⇒ P ( e k ⊗ e m ) = e m ⊗ e k = ( e i ⊗ e j ) P ijkm , (3.1.10)8.e. the operator P has the components P ijkm = δ im δ jk in the basis ( e m ⊗ e k ) ∈ V N ⊗ V N .We note that in view of (3.1.6) the equality holdsdim C (cid:0) sℓ ( N ) (cid:1) ≡ g ij,kℓ g ij,kℓ = ( N − . (3.1.11)In addition, setting j = i in (3.1.8) and summing over i , we obtain the value of thequadratic Casimir operator in the defining representation. Besides, setting j = i in(3.1.8) and summing over i , we obtain the value of the quadratic Casimir operator in thedefining representation T ( C (2) ) = g ij,kℓ ( T ij T kℓ ) i j = N − N δ i j ⇒ c ( T )2 = N − N . (3.1.12)This corresponds to (2.3.3) if we fix metric (2.1.11) in the root space of the algebra sℓ ( N )so that the square of the lengths of all roots of sℓ ( N ) is equal to 1 /N ; in particular forthe highest root θ := λ ad we also have (cf. (2.3.8))( θ, θ ) = 1 /N . (3.1.13)Finally, for the split Casimir operator (3.1.9) of the algebra sℓ ( N ) in the defining repre-sentation we obtain the characteristic identity b C T + 1 N b C T + 1 − N N = 0 ⇔ (cid:16) b C T + 1 + N N (cid:17)(cid:16) b C T + 1 − N N (cid:17) = 0 , (3.1.14)here and below the identity operator I in V N ⊗ V N is replaced with 1 for simplicity. Identity(3.1.14) is consistent with formula (2.3.4) when the root space metric is normalized inaccordance with (3.1.13). In view of (3.1.14) the projectors onto eigen-spaces of theoperator b C T in V N ⊗ V N have the form P ± = ± (cid:16) N b C T + 1 ± N N (cid:17) = 12 (1 ± P ) , (3.1.15)where P + and P − denote a symmetrizer and an antisymmetrizer, respectively. Finally, the sℓ ( N )-symmetric solution of the Yang-Baxter equation (1.0.2) in the defining representa-tion (which is called the Yang solution) is written in several equivalent ways (includingthe form of R ( u ) written in terms of the operator b C T ): R ( u ) = u + P − u = ( u + 1)(1 − u ) P + − P − ⇔ R ( u ) = P + + uP + − u = N b C T + N N + uN b C T + N N − u , (3.1.16)where u is the spectral parameter. Solution (3.1.16) is unitary P R ( u ) P R ( − u ) = R ( u ) R ( − u ) = 1 and is defined up to multiplication by an arbitrary function f ( u ) thatsatisfies f ( u ) f ( − u ) = 1. b C for sℓ ( N ) in the adjoint representation. In this paper we use the standard index-free matrix notation. Namely, let A be an operatorin V N ⊗ V N , where V N = C N is the space of the defining representation of the algebra sℓ ( N ). The operator A is defined by the relations A ( e k ⊗ e l ) = e i ⊗ e j A ij kl , A ij kl are the components of A in the basis { e i ⊗ e j } Ni,j =1 ∈ V ⊗ N . Then, A ab denotesthe action of the operator A in the space V ⊗ N so that it is nontrivial only in the a -th and b -th factors in the product V N ⊗ V N ⊗ V N ⊗ V N (cf. (2.1.9)). For example, the operator A has the components ( A ) i i i i j j j j = A i i j j δ i j δ i j in the basis e i ⊗ e i ⊗ e i ⊗ e i ∈ V ⊗ N and so on. In addition to the permutation operator P , which is defined in (3.1.10), we need one more operator K acting in V ⊗ N : K · ( e m ⊗ e n ) = ( e i ⊗ e j ) δ ij δ mn ⇒ K ijmn = δ ij δ mn . (3.1.17)The operators P ab and K ab acting in the space V ⊗ N satisfy the following useful relations: K ab = K ba , K ab = N K ab , P ab K ab = K ab , K ab K bc = K ab P ac = P ac K bc ,K ab K bc K ab = K ab = K ab P bc K ab , P ab K ad K bc = P cd K ad K bc , (3.1.18)which are specific to matrix representations of the Brauer algebra generators (see e.g. [8],[18]).The split Casimir operator b C ad for the algebra sℓ ( N ) in the adjoint representation isgiven by formula (2.2.1) and acts in V ad ⊗ V ad ⊂ V ⊗ N . Taking into account the definitions(3.1.3) and (3.1.7), we have( b C ad ) i i i i j j j j = g k k ,k k ad( T k k ) i i j j ad( T k k ) i i j j == g k k ,k k C i i k k ,j j C i i k k ,j j = N ( P + P − K − K ) i i i i j j j j . (3.1.19)Note that here the operators ad( T ik ) act in the adjoint representation space V ad , which weidentify with the space of the second rank traceless tensors V ad ≡ ¯ I · V ⊗ N = ( I − N K ) · V ⊗ N ,where the projector ¯ I was introduced in (3.1.6). It means that the indices of the adjointrepresentation are associated in formula (3.1.19) with the pairs of indices ( i i ), ( j j )etc., possessing the traceless property, i.e. the contraction of the indices in each such pairgives zero. Setting in (3.1.19) j = i and j = i and summing over i and i , we obtainthe value of the quadratic Casimir operator (2.1.4) of the algebra sℓ ( N ) in the adjointrepresentationad( C (2) ) = Tr ( b C ad ) = 12 N Tr (cid:0) ( P + P − K − K ) P P (cid:1) = ¯ I , i.e. c ad(2) = 1, which is consistent with the general formula (2.2.9).Next, we need three more operators K , P ( ad ) and P , where the first two act in V ⊗ ,and the last one acts in V ⊗ N . The operator K is defined as follows (cf. (3.1.17)): K i i i i j j j j = g i i i i g j j j j = (cid:0) δ i i δ i i − N δ i i δ i i (cid:1) ( δ j j δ j j − N δ j j δ j j ) == (cid:0) K K − N P K K − N P K K + N K K (cid:1) i i i i j j j j . (3.1.20)The operator P permutes in the tensor product V ⊗ N the first factor with the third oneand the second factor with the fourth one and has an explicit form P i i i i j j j j = δ i j δ i j δ i j δ i j ⇒ P = P P ⇒ P = I , (3.1.21)10here I is the unit operator in V ⊗ N . Finally, the operator P ( ad ) ≡ ¯ I ¯ I P , (3.1.22)plays the role of the permutation operator in the space V ad ⊗ V ad ⊂ V ⊗ N . We stress that P commutes with both b C ad and KP b C ad = b C ad P , P K = K = K P , (3.1.23)and therefore P can be diagonalized simultaneously with b C ad and K .We also note that one cannot choose in the definition of the permutation (3.1.22)in V ⊗ instead of P = P P another operator P ′ = P P . This is because actually V ad = ¯ I · ( V N ⊗ ¯ V N ), where ¯ V N is the space of the contragradient representation of sℓ ( N )(see Remark after (3.1.7)), and the element A ∈ SL ( N ) acts in the space V ad ⊗ V ad asfollows: V ad ⊗ V ad → ( A ⊗ A − T ⊗ A ⊗ A − T ) V ad ⊗ V ad , (3.1.24)and this action commutes with P but does not commute with P ′ .Using the permutations P and P ( ad ) , we define the symmetrizer P ( ad )+ and the anti-symmetrizer P ( ad ) − in the space ( V ad ) ⊗ : P ( ad ) ± ≡ ( I ± P ( ad ) ) = ( I ± P ) ¯ I ¯ I = ¯ I ¯ I ( I ± P ) , P ( ad ) − = (1 − P P )(1 − N ( K + K )) , P ( ad )+ = (1 + P P )(1 − N ( K + K ) + N K K ) , (3.1.25)where I = ¯ I ¯ I is the unit operator in ( V ad ) ⊗ (we often write unit 1 instead of the unitoperator I in V ⊗ N ). We define, respectively, the symmetrized and antisymmetrized partsof the Casimir operator (3.1.19) b C + = P ( ad )+ b C ad = (1 + P ) b C ad = N (1 + P P )(2 P − K − K ) == N (cid:16) P + 2 P − (1 + P P ) K − (1 + P P ) K (cid:17) , (3.1.26) b C − = P ( ad ) − b C ad = 12 ( I − P ) b C ad = 14 N ( P P − K + K ) , (3.1.27)where we used the equations ¯ I b C ad = b C ad = ¯ I b C ad , which are easily checked with thehelp of explicit formula (3.1.19). For b C + and b C − in view of (3.1.23) we have the followingrelations: b C + + b C − = b C ad , P b C ± = b C ± P , b C + b C − = 0 = b C − b C + . (3.1.28)In addition, we have K b C − = 0 = b C − K , K b C + = − K = b C + K , K b C ad = − K = b C ad K , (3.1.29)which are nothing but formulas (2.2.6) valid for all simple Lie algebras. The second chainof equalities in (3.1.29) is derived by means of relations (3.1.18).11 roposition 3.1.1 The antisymmetrized b C − and symmetrized b C + parts of the splitCasimir operator of the Lie algebra sℓ ( N ) (defined in (3.1.27) and (3.1.26)) satisfy theidentities b C − + 12 b C − = 0 ⇒ b C − ( b C − + 12 ) = 0 . (3.1.30) b C + 12 b C − N b C + − N ( I ( ad ) + P ( ad ) − K ) = 0 , (3.1.31) b C + ( b C + + 12 )( b C + − N )( b C + + 1 N ) = 12 N K , (3.1.32)( b C + + 1)( b C + + 12 )( b C + − N )( b C + + 1 N ) P ( ad )+ = 0 , (3.1.33) b C + ( b C + + 1)( b C + + 12 )( b C + − N )( b C + + 1 N ) = 0 , (3.1.34) The split Casimir operator b C ad = b C + + b C − satisfies the characteristic identity (cf. (2.3.5)) b C ad ( b C ad + 12 )( b C ad + 1)( b C ad − N )( b C ad + 1 N ) = 0 . (3.1.35) Proof.
Identity (3.1.30) follows from the general statement (2.2.5), which is valid for allsimple Lie algebras. Further, the identities (3.1.31) – (3.1.34) can be proved by using thediagram technique developed in [4]. Nevertheless, we give here a direct algebraic proofof these identities that uses relations (3.1.18) arising in the matrix representations of theBrauer algebra (see e.g. [8], [18]). First, we calculate b C = ( I + P ) b C ad ( I + P ) b C ad = ( I + P ) b C == N (1 + P P ) (cid:16) − P ( K , + K , ) + N K , + 2 K K (cid:17) , (3.1.36)where we introduce the notation K ij,kℓ ≡ K ij + K kℓ . Multiply the left and right sides of(3.1.36) by b C + . As a result, we get b C = N (1 + P ) (cid:16) P − K , + K , )++2 N P ( K , + K , ) + 4 P ( K K + K K ) − N K , − N K K (cid:17) == − b C + N b C + + N (1 + P ) (cid:16) N − K , + P ( K K + K K ) − N K K (cid:17) == − b C + N b C + + N ( I ( ad ) + P ( ad ) ) − N K , where in the last equality we used formulas (3.1.20), (3.1.25) and( N − K , ) = ( N ¯ I ¯ I − N K K ) , ¯ I ≡ I − N K , P ≡ P P . Thus, identity (3.1.31) is proved. Again, we multiply both sides of equality (3.1.31) by b C + and obtain b C + 12 b C − N b C − N b C + = 12 N K (3.1.37)that is equivalent to (3.1.32). Substitution of the expression (3.1.37) of the operator K into relation (3.1.31) gives (3.1.33). Identity (3.1.34) is obtained either by multiplying123.1.33) by b C + , or by multiplying both sides of equality (3.1.32) by ( b C + + 1) and takinginto account relation K ( b C + + 1) = 0, which follows from (3.1.29).Note that the antisymmetric part b C − of the Casimir operator, in view of the relation(3.1.30), satisfies the same relation (3.1.34) as the symmetric part of b C + . Hence, takinginto account the last relation in (3.1.28), it follows that the complete Casimir operator b C ad = b C + + b C − for the algebra sℓ ( N ) will obey a characteristic identity (3.1.35) similarto (3.1.34).Now we show that in the sℓ ( N ) case, in addition to the invariant operators in V ⊗ represented as polynomials in b C ad , b C ± , there is one more invariant operator Q − in V ⊗ ,which commutes with b C ad , b C ± but is not expressed as a polynomial in b C ad , b C ± . To con-struct such an operator, note that the generators T a = T ( X a ) of sℓ ( N ) in the definingrepresentation T together with the unit matrix I N form a basis in the space of all N × N matrices; therefore, along with the defining relation (2.1.1), there is one more relation[ T a , T b ] + = D cab T c + α I N g ab , α ≡ c ( T ) N . (3.1.38)Here the parameter α is fixed by the condition (2.3.1), and D cab are new structure constantsof the algebra sℓ ( N ), symmetric with respect to permutation of subscript indices a and b . Now we define the operators Q and Q − in the space V ⊗ (cf. (2.2.1) and (2.2.4)) Q a a b b ≡ g df C a db D a fb , Q − ≡ N I − P ) Q ( I − P ) , (3.1.39)where I and P are the unit matrix and the permutation matrix in V ⊗ , respectively. Notethat operator Q − acts nontrivially in the antisymmetric part P ( ad ) − V ⊗ of the space V ⊗ . Proposition 3.1.2
The operator Q − given in (3.1.39) is written as an operator in thespace V ⊗ = ¯ I ¯ I ( V ⊗ N ) as follows: ( Q − ) = 12 ( P − P ) (cid:16) − N ( K + K + K + K ) (cid:17) . (3.1.40) The operator Q − satisfies the relations P ( ad )+ Q − = 0 = Q − P ( ad )+ , Q − b C − = b C − Q − = 0 , Q − = 2 b C − + P ( ad ) − , (3.1.41) Q − ( Q − + 1)( Q − −
1) = 0 , (3.1.42) where b C − and P ( ad ) ± are given in (3.1.27) and (3.1.25). Proof.
Relations (3.1.38) in the basis (3.1.1) are represented as[ T ij , T km ] + = D rsij,km T rs + 1 N g ij,km I N , (3.1.43)where for the structure constants we have the explicit expressions D rsij,km ≡ ( δ rℓ δ sn − N K rsℓn ) ¯ D ℓnij,km , ¯ D rsij,km ≡ δ jk δ ri δ sm + δ im δ rk δ sj − N (cid:0) δ ij δ rk δ sm + δ km δ ri δ sj (cid:1) . Q and Q − , given in (3.1.39), are equal to Q i i i i j j j j = g k k ,k k C i i k k ,j j D i i k k ,j j == N (cid:0) ¯ I (cid:0) P − P + K − K + N ( P − P ) K (cid:1)(cid:1) i i i i j j j j , Q − = N (1 − P ) ¯ I ¯ I Q ¯ I ¯ I (1 − P ) == (1 − P ) (cid:0) P − P − N K ( P − P ) − N ( P − P ) K (cid:1) (1 − P ) , (3.1.44)and the right-hand side of (3.1.44) after the substitution P = P P matches the right-hand side of (3.1.40). Relations (3.1.41) and (3.1.42) are verified by direct calculations. Remark.
Characteristic identity (3.1.42) for the operator Q − allows us to build threemutually orthogonal projectors: e P ( − )0 = − ( Q − + 1)( Q − − P ( ad ) − = − b C − , e P ( − ) ± = Q − ( Q − ±
1) = b C − + P ( ad ) − ± Q − . (3.1.45)Due to the relation P ( ad ) − = e P ( − )+1 + e P ( − ) − + e P ( − )0 , the projectors e P ( − )0 , e P ( − )+1 , e P ( − ) − decomposethe space P ( ad ) − ( V ad ⊗ V ad ) of the antisymmetric part A (ad ⊗ ad) of the representation(ad) ⊗ into eigenspaces of the operator Q − with eigenvalues 0, +1, − b C ad in ( V ad ⊗ V ad )by means of the standard methods (see Section 3.5 in [4] and Section 4.6.4 in [8]):P ( a j ) = Y i =1 i = j b C ad − a i I a j − a i , (3.1.46)where a i are the roots of the characteristic equation (3.1.35) a = 0 , a = − / , a = − , a = 1 /N , a = − /N . Note that the case N = 2 is special since in this case we have ( a − a ) = 0, and projectorsP ( a ) = P ( − / and P ( a ) = P ( − /N ) are not defined (see below (3.1.47)). In addition,we note that in general the projectors (3.1.46) are not primitive and extract invariantsubspaces in ( V ad ) ⊗ ⊂ V ⊗ N , which are not the spaces of the irreducible representationsof sℓ ( N ). First of all, this is due to the presence of the invariant permutation operator P that commutes with b C ad (see (3.1.23)) and allows us to split the projectors into two partsP ( ± )( a j ) = P ( ad ) ± · P ( a j ) , where P ( ad ) ± ≡ ( I ± P ) ¯ I ¯ I . From the condition (3.1.30), which canbe written as P ( ad ) − b C ad ( b C ad + ) = 0, it immediately follows thatP ( − )( − = P ( ad ) − P ( − = 0 , P ( − )( ± /N ) = P ( ad ) − P ( ± /N ) = 0 , Moreover, due to relation (3.1.33) for the symmetrized part of P (0) we obtainP (+)(0) = ( b C ad + 1)( b C ad + )( b C ad + N )( b C ad − N )(+1)(+ )( N )( − N ) P ( ad )+ = 0 , ( − )(0) = ( b C ad + 1)( b C ad + )( b C ad + N )( b C ad − N )(+1)(+ )( N )( − N ) P ( ad ) − = 2 b C − + P ( ad ) − ≡ e P ( − )(+1) + e P ( − )( − , is not primitive since it is equal to the sum of the projectors e P ( − )( ± from (3.1.45).As a result, for N > e P ( − )(+1) , e P ( − )( − , P ( − )( − ) , P (+)( − ) , P (+)( − = P ( − , P ( − N ) = P (+)( − N ) , P (+ N ) = P (+)(+ N ) , which extract invariant subspaces in ( V ad ) ⊗ and by construction form a complete andmutually orthogonal system. Due to (3.1.45) and (3.1.46), these projectors have the form(cf. projectors in [4], Section 9.12) e P ( − )(+1) = b C − + P ( ad ) − + Q − , dim = ( N − N − , e P ( − )( − = b C − + P ( ad ) − − Q − , dim = ( N − N − , P ( − )( − ) = b C ad ( b C ad +1)( b C ad + N )( b C ad − N )( − )( )( − + N )( − − N ) P − = − b C − ≡ e P ( − )(0) , dim = N − , P (+)( − ) = b C ad ( b C ad +1)( b C ad + N )( b C ad − N )( − )( )( − + N )( − − N ) P + = N − ( N b C − P ( ad )+ − K ) , dim = N − , P (+)( − = b C ad ( b C ad + )( b C ad + N )( b C ad − N )( − − )( − N )( − − N ) P + = N − K , dim = 1 , P (+)( N ) = b C ad ( b C ad +1)( b C ad + )( b C ad + N )( N )( N +1)( N + )( N ) P + == − N N +1)( N +2) K + N ( N +2) b C + N b C + + N N +2) P ( ad )+ , dim = N ( N − N +3)4 , P (+)( − N ) = b C ad ( b C ad +1)( b C ad + )( b C ad − N )( − N )( − N +1)( − N + )( − N ) P + == N N − N − K − N ( N − b C − N b C + + N N − P ( ad )+ , dim = N ( N +1)( N − , (3.1.47)where we used the property b C ad P ( ad ) ± = b C ad P ± = b C ± and identities (3.1.30), (3.1.31),(3.1.32), (3.1.37). Note that the projector P ( − )( − ) is the same as the projector e P ( − )(0) givenin (3.1.45). The right column in formula (3.1.47) shows the dimensions of the invariantsubspaces in V ⊗ , which are extracted by the corresponding projectors. The way tocalculate these dimensions is shown below.It is well known that the tensor product of two adjoint representations of the algebra sℓ ( N ) for N > , N − ] ⊗ [2 , N − ] = [ ∅ ] + [2 , N − ] + [3 , N − ] + [3 , N − ] + [4 , N − ] + 2 · [2 , N − ] , where the diagram [2 , N − ] corresponds to the adjoint representation anddim [2 , N − ] = N − , dim [ ∅ ] = 1 , dim [2 , N − ] = N ( N +1)( N − , dim [3 , N − ] = dim [3 , N − ] = ( N − N − , dim [4 , N − ] = N ( N − N +3)4 . (3.1.48)15omparing the dimensions in (3.1.47) and (3.1.48), we conclude that seven mutuallyorthogonal and nontrivial projectors (3.1.47), which form a complete system in the space( V ad ) ⊗ , select in ( V ad ) ⊗ the subspaces of all irreducible representations of sℓ ( N ). Toverify this fact, we need to calculate the dimensions of the invariant subspaces e V ( − )( b i ) = e P ( − )( b i ) ( V ad ⊗ V ad ) and V (+)( a i ) = P (+)( a i ) ( V ad ⊗ V ad ), which are given in (3.1.47), and comparethese dimensions with (3.1.48). A way to calculate these dimensions is to find traces ofthe projector (3.1.47):dim( e V ( − )( b i ) ) = Tr ( e P ( − )( b i ) ) , dim( V (+)( a i ) ) = Tr (P (+)( a i ) ) , (3.1.49)where we introduce the notation Tr ≡ Tr Tr Tr Tr for the trace in ( V ad ) ⊗ ⊂ ( V N ) ⊗ .For this calculation, we use the traces (2.2.10) of the basic operators that make up theprojectors (3.1.47): Tr ( K ) = N − , Tr ( P ( ad ) − ) = 12 ( N − N − , Tr ( P ( ad )+ ) = 12 N ( N − , Tr ( b C ± ) = ± ( N − , Tr ( b C ) = ( N − , Tr ( b C ) = 34 ( N − , Tr ( e C − ) = 0 . Substituting these traces into expressions (3.1.49), where the projectors e P ( − )( b i ) , P (+)( a i ) aredefined in (3.1.47), we obtain the dimensions indicated in (3.1.47) that coincide with thedimensions (3.1.48).So using the projectors P ( ad )+ and P ( ad ) − , the representation (ad) ⊗ of the algebra sℓ ( N )is decomposed into the symmetric S (ad ⊗ ) and antisymmetric A (ad ⊗ ) parts. In turn,for all simple Lie algebras (see Section ) the antisymmetric part A (ad ⊗ ) splits intothe sum of two subrepresentations X and X , which in the case of the algebra sℓ ( N )correspond to the projectors P ( − )( − ) and P ( − )(0) and have dimensions (2.2.12):dim( X ) = N − sℓ ( N )) , dim( X ) = 12 ( N − N − . Moreover, in the case of the Lie algebra sℓ ( N ), the representation X associated with theprojector P ( − )(0) turns out to be reducible and expands into the sum of two inequivalentirreducible representations associated with the projectors e P ( − )(+1) , e P ( − )( − and having the samedimensions: dim e V ( − )( ± = ( N − N − sℓ ( N ), when N > N = 2 , S (ad ⊗ ) decomposes into the sum of four irreducible representations, one of which X associated with the projector P (+)( − is trivial and has dimension 1, and three otherrepresentations Y , Y ′ and Y ′′ associated with the projectors P (+)( − / , P (+)(1 /N ) and P (+)( − /N ) have the corresponding dimensionsdim Y = N − , dim Y ′ = 14 N ( N − N + 3) , dim Y ′′ = 14 N ( N + 1)( N − . Note that for N = 2 the projectors P (+)( − / and P (+)( − /N ) in (3.1.47) are not defined, thedimension of the representation Y ′′ becomes negative and the above expansion does notwork. For N = 3 we have dim Y ′′ = 0 and, therefore, in the case of the algebra sℓ (3), the16epresentation Y ′′ dose not appear in S (ad ⊗ ), and the corresponding projector P (+) − N in(3.1.47) for N = 3 must vanish P (+) − N (cid:12)(cid:12)(cid:12) N =3 = 0, which gives b C = − b C + + 112 ( I ( ad ) + P ( ad ) + K ) , (3.1.50)i.e. in this case, the symmetric part b C + of the split Casimir operator satisfies the secondorder identity (3.1.50), and the third order identity (3.1.31) for N = 3 is a consequenceof (3.1.50). To derive characteristic identities in the case N = 3, we multiply both partsof (3.1.50) by b C + and then we multiply the resulting relation by ( b C + + 1) and use theequality ( b C + + 1) K = 0. As a result, we obtain b C + 16 b C − b C + = − K ⇒ b C + ( b C + + 1)( b C + + 12 )( b C + −
13 ) = 0 ⇒ (3.1.51) b C ad ( b C ad + 12 )( b C ad + 1)( b C ad −
13 ) = 0 , (3.1.52)which for the case of sℓ (3) replace (3.1.32), (3.1.34) and (3.1.35).Finally, the decomposition of the product of two adjoint representations of the algebra sℓ ( N ) for N > N − ⊗ [ N −
1] = A ([ N − ⊗ [ N − S ([ N − ⊗ [ N − , A ([ N − ⊗ [ N − N − ⊕ [ ( N − N − ] ⊕ [ ( N − N − ] , S ([ N − ⊗ [ N − ⊕ [ N − ⊕ [ N ( N − N +3)4 ] ⊕ [ N ( N +1)( N − ] . This decomposition is well known and is in accordance with the general theory of theuniversal description of all simple Lie algebras using the Vogel parameters [11] (see also[12], [13]). We will discuss this universal description below in Section . b C for Lie algebras so ( N ) and sp (2 n ) b C for so ( N ) and sp (2 n ) in the defining representation In this Subsection, to fix the notation, we give the well-known definition of the Lie algebras so ( N ) and sp (2 n ) which one can find in many monographs and textbooks (see e.g. [4],[5] and [7]). We prefer to give here a natural unified definition [7, 21] of these algebrassince it will be useful for us below.We introduce the metric || c ij || i,j =1 ,...,N which is equal to the unit matrix || δ ij || in thecase of the so ( N ) algebras and equal to the matrix || c ij || = (cid:18) I n − I n (cid:19) (3.2.1)in the case of the algebras sp (2 n ). Thus, we have c ij = ǫc ji with ǫ = ± so ( n ) /sp (2 n ) cases, respectively. The inverse metric ¯ c ij is defined in a standard wayas ¯ c ik c kj = δ ij . We denote the space C N of the defining representation of so ( N ) and sp ( N )as V N . 17sing the matrix units ( e st ) ik = δ tk δ is , or ( e st ) ik = c tk δ is with lowered indices, one maydefine the generators of so ( N ) and sp ( N ) as M ij = e ij − ǫe ji , ( M ij ) kl = c jl δ ki − ǫc il δ kj = 2 δ k [ i c j ) l , (3.2.2)where the notation [ ij ) means (anti-)symmetrization for the ( so ( N )-) sp ( N ) algebras. Thecommutation relations for both algebras acquire the generic form[ M ij , M kl ] = c jk M il − ǫc ik M jl − ǫc jl M ik + c il M jk = X ij,klmn M mn , (3.2.3)with the structure constants given by X mnij,kl = c jk δ [ mi δ n ) l − ǫc ik δ [ mj δ n ) l − ǫc jl δ [ mi δ n ) k + c il δ [ mj δ n ) k = 4 δ [ m [ i c j )[ k δ n ) l ) . (3.2.4)In this basis the Cartan-Killing metric reads g i i ,j j = 2( N − ǫ )( c i j c j i − ǫc i j c j i ) ≡ ( N − ǫ ) Tr( M i i M j j ) , (3.2.5)while the inverse metric has the form g i i ,j j = 18( N − ǫ ) ( ǫ ¯ c i j ¯ c i j − ¯ c i j ¯ c i j ) . (3.2.6)This inverse metric is defined by the equation g ij,kℓ g kℓ,mn = ( P ( ǫ ) ) mnij , where ( P ( ǫ ) ) mnij ≡ ( δ mi δ nj − ǫδ ni δ mj ) is the projector on the (anti)symmetric part of V ⊗ N .Now it is easy to calculate the split Casimir operator b C for the algebras so ( N ) and sp ( N ) in the defining representation [7]( b C T ) k k ℓ ℓ ≡ T ⊗ ( b C ) k k ℓ ℓ = g ij,nm ( M ij ) k ℓ ( M nm ) k ℓ == N − ǫ ) (cid:0) δ k ℓ δ k ℓ − ǫ ¯ c k k c ℓ ℓ (cid:1) , (3.2.7)or in the index-free matrix notation we have b C T = 12( N − ǫ ) ( P − ǫK ) . (3.2.8)Here, P is the permutation operator acting in the space V ⊗ N (see (3.1.10)), while theoperator K acting in the same space V ⊗ N has the following components: K i i j j = ¯ c i i c j j . Proposition 3.2.1
The characteristic identity for the split Casimir operator (3.2.8) forthe algebras so ( N ) and sp ( N ) in the defining representation reads (cid:16) b C T + d (cid:17)(cid:16) b C T − d (cid:17)(cid:16) b C T + d N − ǫ ) (cid:17) = 0 , (3.2.9) where d = 1 / ( N − ǫ ) . b C T in V ⊗ N have the form [4], [8] P a = (cid:0) b C T − a (cid:1)(cid:0) b C T − a (cid:1) ( a − a )( a − a ) = ( I + P ) − (1+ ǫ )2(1+ N − ǫ ) K ≡ P ( ǫ )+ ,P a = (cid:0) b C T − a (cid:1)(cid:0) b C T − a (cid:1) ( a − a )( a − a ) = ( I − P ) − (1 − ǫ )2(1 − N + ǫ ) K ≡ P ( ǫ ) − ,P a = (cid:0) b C T − a (cid:1)(cid:0) b C T − a (cid:1) ( a − a )( a − a ) = ǫN K ≡ P ( ǫ )0 . (3.2.10)Here, a = d , a = − d , a = − d ( N − ǫ ) are the roots of the characteristic equation(3.2.9), while P a and P a are, respectively, the symmetrization and antisymmetrizationoperator for the algebras so ( N ) ( ǫ = +1) and sp ( N ) ( ǫ = − , N = 2 r ).Finally note that the so ( N ) (or sp ( N ))-symmetric solution of the Yang-Baxter equa-tion in the defining representation (the so-called Zamolodchikov solution) can be writtenas follows (see e.g. [7]): R ( u ) = 1 ǫ − u (cid:16) u + P − ǫu ( u + N/ − ǫ ) K (cid:17) = (3.2.11)= ( u + 1)( ǫ − u ) P ( ǫ )+ + ( u − ǫ − u ) P ( ǫ ) − + ( N/ − ǫ − u )( N/ − ǫ + u ) P ( ǫ )0 , P R ( u ) P R ( − u ) = 1 . It is quite intriguing that the solution (3.2.11) can be elegantly represented as a rationalfunction of the split Casimir operator R ( u ) = b C T + d ( ǫ/ u ) b C T + d ( ǫ/ − u ) , (3.2.12)where the constant d was introduced in (3.2.9) (see also (2.3.1) and (3.2.5)). b C for so ( N ) and sp (2 n ) in the adjoint representation The split Casimir operator b C ad for the algebras so ( N ) and sp (2 n ) in the adjoint represen-tation together with the construction of the projectors on the irreducible representationsin ad ⊗ ad were considered in detail in [21]. Thus, for completeness, we will present hereonly a short review of the results discussed in [21].The split Casimir operator b C ad for the algebras g = so ( N ) , sp ( N ) in the adjointrepresentation can be expressed through the split Casimir operator in the defining repre-sentation b C T (3.2.7) as [21]:( b C ad ) k k k k j j j j = 4 δ [ k [ j ( b C T ) k )[ k j )[ j δ k ) j ) = 4 (cid:0) P ( ǫ )12 , ( b C T ) P ( ǫ )12 , (cid:1) k k k k j j j j , (3.2.13)where we introduce the projectors P ( ǫ )12 , = P ( ǫ )12 P ( ǫ )34 , P ( ǫ ) ab ≡
12 ( I − ǫP ab ) , and use the index-free matrix notation explained at the beginning of Section . Usingthe known expression (3.2.8) for b C T , we obtain b C ad = 2( N − ǫ ) P ( ǫ )12 , ( P − ǫK ) P ( ǫ )12 , . (3.2.14)19ue to the existence of accidental isomorphisms so (3) ≃ sℓ (2) ≃ sp (2), so (4) ≃ sℓ (2) + sℓ (2), so (5) ≃ sp (4) and so (6) ≃ sℓ (4), in what follows we limit ourselves toconsidering the algebras so ( N ) with N ≥ sp ( N ) with N = 2 n ≥
4, only.Let us define the following operators acting in the space V ǫ ad ⊗ V ǫ ad ⊂ V ⊗ N : I ≡ P ( ǫ )12 , = P ( ǫ )12 P ( ǫ )34 , P ≡ P ( ǫ )12 , P P P ( ǫ )12 , , K ≡ P ( ǫ )12 , K K P ( ǫ )12 , . (3.2.15)These operators obey useful relations: I = I P P = P P I , P = P P I = I P P , P = I , K P = P K = K , K = M ( M − K , M ≡ ǫN , (3.2.16) b C ad P = P b C ad , b C ad K = K b C ad = − K , (3.2.17)It proved useful to define the symmetric b C + and antisymmetric b C − parts of the splitCasimir operator b C ad as b C + = 12 ( I + P ) b C ad , b C − = 12 ( I − P ) b C ad . (3.2.18)In the paper [21], the following proposition was proven. Proposition 3.2.2
The characteristic identities for the operators b C − , b C + and b C ad for thealgebras so ( N ) and sp ( N ) read b C − + 12 b C − = 0 ⇔ b C − ( b C − + 12 ) = 0 , (3.2.19) b C = − b C − M − M − b C + + M − M − ( I + P − K ) , (3.2.20) b C + ( b C + + 1) (cid:16) b C + − M − (cid:17)(cid:16) b C + + 2( M − (cid:17)(cid:16) b C + + ( M − M − (cid:17) = 0 , (3.2.21) b C ad ( b C ad + 12 )( b C ad + 1)( b C ad − M − b C ad + 2 M − b C ad + M − M −
2) ) = 0 , (3.2.22) where the parameter M = ǫN is supposed to obey M ≥ , M = 8 for the algebras so ( N ) ,and M ≤ − for the algebras sp ( N ) . Remark 1.
Identity (3.2.20) is derived from the intermediate formula [21]: b C = 1( M − ( I + P + K ) − M − b C + + ( M − M − P ( ǫ )12 , K (1 + ǫ P ) P ( ǫ )12 , , which is simplified for M = 8 and instead of (3.2.20) we obtain the identity on b C + of thesecond order b C = − b C + + 136 ( I + P + K ) . (3.2.23)That is why the case M = 8 was excluded from Proposition .20he characteristic identity (3.2.22) can be used to construct a complete system oforthogonal projectors on the invariant subspaces in V ad ⊗ V ad , which are simultaneouslyare eigenspaces of the operator b C ad . They are defined in a standard way (see e.g. [4, 8]):P j := P a j = Y i =1 i = j b C ad − a i I a j − a i , (3.2.24)where a i are the roots of the characteristic equation (3.2.22): a = 0 , a = − , a = − , a = 1 M − , a = − M − , a = − M − M − , (3.2.25)and the characteristic identity (3.2.22) is written in the form (cid:16) b C ad − a (cid:17) (cid:16) b C ad − a (cid:17) (cid:16) b C ad − a (cid:17) (cid:16) b C ad − a (cid:17) (cid:16) b C ad − a (cid:17) (cid:16) b C ad − a (cid:17) = 0 . (3.2.26)There are some special cases in which some of the roots (3.2.25) coincide: • M = 4 - algebra so (4). In this case, a = a = − a = a = 0. The correctcharacteristic identity reads b C ad (cid:18) b C ad + 12 (cid:19) (cid:16) b C ad + 1 (cid:17) (cid:18) b C ad − (cid:19) = 0 . In other words, the characteristic identity contains each factor b C ad and (cid:16) b C ad + 1 (cid:17) only once. • M = 6 - algebra so (6). Now, a = a = − and, therefore, the characteristicidentity is of the fifth order: b C ad (cid:18) b C ad + 12 (cid:19) (cid:16) b C ad + 1 (cid:17) (cid:18) b C ad − (cid:19) (cid:18) b C ad + 14 (cid:19) = 0 . • M = 8 - algebra so (8). Now a = a = − and again the characteristic identity isof the fifth order (this case will be considered in detail below): b C ad (cid:18) b C ad + 12 (cid:19) (cid:16) b C ad + 1 (cid:17) (cid:18) b C ad − (cid:19) (cid:18) b C ad + 13 (cid:19) = 0 . (3.2.27) • M = 5. In virtue of the accidental automorphism so (5) = sp (4), this case is identicalto the case with M = − , ..., P in (3.2.24) were calculated in [21, 4]P ≡ P − = 12 ( I − P ) + 2 b C − , P ≡ P − = − b C − , P ≡ P +3 = 2 K ( M − M ≡ K dim g , P ≡ P +4 = 23 ( M − b C + M b C + + ( M − I + P )3( M − − M − K M − M − , (3.2.28)P ≡ P +5 = − M − M − b C − ( M − M − M − b C + + ( M − I + P )6( M −
8) + 2 K M − , P ≡ P +6 = 4( M − M − b C + 4 M − b C + − I + P )( M − M − − M − K M ( M − M − . V a ) = Tr P = 18 M ( M − M + 2)( M − , dim( V a ) = Tr P = 12 M ( M − , dim( V a ) = Tr P = 1 , (3.2.29)dim( V a ) = Tr P = 112 M ( M + 1)( M + 2)( M − , dim( V a ) = Tr P = 124 M ( M − M − M − , dim( V a ) = Tr P = 12 ( M − M + 2) . Remark 2.
The characteristic identities (3.2.22) and dimensions (3.2.29) for so ( N ) and sp ( N ) are related by replacement N → − N . This fact manifests the duality betweencertain formulas in the representation theories of the algebras sp ( N ) and so ( N ) (see [16],[4], [17] and references therein). so (8)For the algebra so (8) we have (3.2.19), (3.2.23): b C − = − b C − , b C = − b C + + 136 ( I + P + K ) (3.2.30)and, therefore, the characteristic identity has the fourth order (cf. (3.2.21)) b C + ( b C + + 1)( b C + −
16 )( b C + + 13 ) = 0 . (3.2.31)All these imply the existence of the characteristic identity for the full split Casimir oper-ator b C ad of the fifth order (3.2.27). Thus, the operator b C ad has the following eigenvalues: a = 0 , a = − / , a = − , a = 1 / , a = − / . All projectors P ′ k ≡ P ′ a k on the eigenspaces of the operator b C ad corresponding to the eigen-values a k have been constructed in [21]. However, not all P ′ k are projectors onto irreduciblerepresentations, because of the different expansion of the tensor product ad ⊗ ( so (8)) intothe irreducible representations [4, 22, 24]:ad ⊗ ( so (8)) = [28] = [1] + [28] + [35] + [35 ′ ] + [35 ′′ ] + [300] + [350] ⇒ (3.2.32) V ⊗ = V + V + V + V ′ + V ′′ + V + V . (3.2.33)The projector P ′ = (P + P ) | M =8 with the eigenvalue ( − /
3) corresponds to the spacewith dimension 105 (see (3.2.29)). Therefore, P ′ is not primitive, and it can be furthersplit into three projectors P ′ = P + P + + P − , so (8), the complete system of primitive projectors reads [21]:P ′ = P | M =8 = ( I − P ) + 2 b C − , dim = 350 , P ′ = P | M =8 = − b C − , dim = 28 , P ′ = P | M =8 = K , dim = 1 , P ′ = P | M =8 = ( I + P ) + 2 b C + + K , dim = 300 , P = P ′ − A = ( I + P ) − b C + − K − A , dim = 35 , P + = ( A + E ) , dim = 35 , P − = ( A − E ) , dim = 35 . (3.2.34)Here, A is the antisymmetrizer in V ⊗ and E is the invariant operator in V ⊗ withthe components ( E ) i ...i j ...j = (4!) − ε i ...i j ...j , where ε i ...i j ...j = ε i ...i j ...j is a fullyantisymmetric rank-eight tensor ε ... = 1. The operators A and E obey the conditions A = A , A E = E A = E , E = A , and P + and P − are the projectors onto the self-dual and anti-self-dual parts of V ∧ ≡ A ( V ⊗ ). We note that the operator E is independent of the operators b C ± , i.e., itcannot be expressed as a polynomial function of b C ± . b C + in thecase of Lie algebras of classical series For the algebras of the classical series A n , B n , C n , D n the characteristic identities (3.1.31)and (3.2.20) for the operator b C + in the adjoint representation can be written in a genericform b C + 12 b C = µ b C + + µ ( I ( ad ) + P ( ad ) − K ) , (3.3.1)where µ and µ are the parameters we define at the moment. Multiplying both sides ofequation (3.3.1) by K and using the relations K ( I ( ad ) + P ( ad ) ) = 2 K , K b C + = − K , K · K = dim g · K , one may express the dimension of the Lie algebra g through the parameters µ and µ dim g = 2 µ − µ + 1 / µ . (3.3.2)Then, we multiply both sides of (3.3.1) by b C + ( b C + + 1) and deduce the characteristicidentity for b C + : b C + ( b C + + 1)( b C + 12 b C − µ b C + − µ ) = 0 . (3.3.3)which can be written in a factorized form b C + ( b C + + 1)( b C + + α t )( b C + + β t )( b C + + γ t ) = 0 ⇔ Y i =1 ( b C + − a i ) = 0 . (3.3.4)23ere we introduce the notation for the roots of the identity (3.3.3) a = 0 , a = − , a = − α t , a = − β t , a = − γ t , t = α + β + γ , (3.3.5)and the last equation follows from the condition ( a + a + a ) = − /
2. The parameter t normalizes the eigenvalues of the operator b C + . For each simple Lie algebra g we choose t − such that ( θ, θ ) = 1 t , (3.3.6)where θ is the highest root of g . Thus, t coincides with the dual Coxeter number h ∨ ofthe algebra g . The parameters α, β, γ were introduced by Vogel [11]. The values of theseparameters for the algebras A n , B n , C n , D n are extracted from identities (3.1.34), (3.2.21),and we summarize them in Table 3. Table 3. sℓ ( n + 1) so (2 n + 1) sp (2 n ) so (2 n ) t n + 1 2 n − n + 1 2 n − α t − / ( n + 1) − / (2 n − − / ( n + 1) − / (2 n − β t / ( n + 1) 2 / (2 n −
1) 1 / (2 n + 2) 1 / ( n − γ t / n − / (4 n −
2) ( n + 2) / (2 n + 2) ( n − / (2 n − Comparison of equations (3.3.3) and (3.3.4) implies that the parameters µ and µ areexpressed via the Vogel parameters as µ = − αβ + αγ + βγ t , µ = − αβγ t , (3.3.7)and the dimensions (3.3.2) of the simple Lie algebras acquire a remarkable universal formobtained by Deligne and Vogel [10],[11]:dim g = ( α − t )( β − t )( γ − t ) αβγ . (3.3.8)Now by using the characteristic identity (3.3.4), one can obtain the universal form ofthe projectors P (+)( a i ) on the invariant subspaces V ( a i ) in the symmetrized space ( I (ad) + P (ad) ) ( V ⊗ ):P (+)( − α t ) = t ( β − α )( γ − α ) (cid:16) b C + (cid:0) − α t (cid:1) b C + + βγ t (cid:0) I (ad) + P (ad) − α ( α − t ) K (cid:1) (cid:17) ≡ P (+) ( α | β, γ ) , P (+)( − β t ) = P (+) ( β | α, γ ) , P (+)( − γ t ) = P (+) ( γ | α, β ) , P (+)( − = 1dim g K . The irreducible representations that act in the subspaces V ( − , V ( − α t ) , V ( − β t ) , V ( − γ t ) wererespectively denoted in [11] as X , Y ( α ), Y ( β ), Y ( γ ); see Section below. Finally,we calculate (by means of trace formulas (2.2.10)) the universal expressions [11] for thedimensions of the invariant eigenspaces V ( a i ) :dim V ( − = Tr P (+)( − = 1 , dim V ( − α t ) = Tr P (+)( − α t ) = − (3 α − t )( β − t )( γ − t ) t ( β + t )( γ + t ) α ( α − β ) β ( α − γ ) γ , (3.3.9)dim V ( − β t ) = Tr P (+)( − β t ) = − (3 β − t )( α − t )( γ − t ) t ( α + t )( γ + t ) β ( β − α ) α ( β − γ ) γ , (3.3.10)dim V ( − γ t ) = Tr P (+)( − γ t ) = − (3 γ − t )( β − t )( α − t ) t ( β + t )( α + t ) γ ( γ − β ) β ( γ − α ) α . (3.3.11)24ere we encounter an interesting nonlinear Diophantine problem of finding all integerdim g in (3.3.8) for which the parameters α, β , γ and dim V ( a i ) are integers. The partialsolutions of this problem are given in Table 3. The analogous Diophantine problems wereconsidered in [19], [20]. b C for exceptional Lie alge-bras R -matrices in thefundamental representation Let T be the minimal fundamental representation of the exceptional Lie algebras g = g , f , e , e , e , acting in the space V . Let us choose the basis elements X a of the algebras g sothat the Cartan-Killing metric g ab is proportional to δ ab , i.e.Tr ( T a T b ) = d g ab = − δ ab , (4.1.1)where T a ≡ T ( X a ). For this normalization the split Casimir operator in the representation T reads ˆ C = g ab T a ⊗ T b = − d T a ⊗ T a . Below we omit the constant parameter d and usethe following definition for the split Casimir operatorˆ C i i j j = g ab d ( T a ) i j ( T b ) i j = − ( T a ) i j ( T a ) i j . (4.1.2)In what follows we also need the identity I and permutation P operators acting in( V ⊗ V ) and defined as: I i i j j = δ i j δ i j , P i i j j = δ i j δ i j . (4.1.3)Using these operators one may define the symmetric SC and anti-symmetric AC parts ofthe split Casimir operator ˆ C in the fundamental representation SC i i j j = (cid:0) ( I + P ) ˆ C (cid:1) i i j j = (cid:16) ˆ C i i j j + ˆ C i i j j (cid:17) , AC i i j j = (cid:0) ( I − P ) ˆ C (cid:1) i i j j = (cid:16) ˆ C i i j j − ˆ C i i j j (cid:17) , ˆ C = SC + AC . (4.1.4)Note that by definition we have P · ˆ C = ˆ C · P , ( I ± P ) · ˆ C · ( I ∓ P ) = 0 , SC · AC = 0 = AC · SC , (4.1.5)and, in accordance with (4.1.1) and (4.1.2), one obtainsTr ( SC ) = − Tr ( AC ) = 12 dim g . (4.1.6)Besides the standard Yang-Baxter equation for R matrix (1.0.2), we also need theYang-Baxter equation for the twisted R -matrix: ˇ R ( u ) ≡ P R ( u ). This equation followsfrom (1.0.2) and is written in the form of the braid group relationsˇ R ( u ) ˇ R ( u + v ) ˇ R ( v ) = ˇ R ( v ) ˇ R ( u + v ) ˇ R ( u ) , (4.1.7)25r in the componentsˇ R ( u ) i i k k ˇ R ( u + v ) k i j l ˇ R ( v ) k j l l = ˇ R ( v ) i i k k ˇ R ( u + v ) i k l j ˇ R ( u ) j k l l . (4.1.8)Below we always require the unitarity conditionˇ R ( u ) i i k k ˇ R ( − u ) k k j j = δ i j δ i j ⇒ ˇ R ( u ) ˇ R ( − u ) = P R ( u ) P R ( − u ) = 1 (4.1.9)for the solutions of the Yang-Baxter equations (4.1.8). R -matrix for the algebra g The dimension of the minimal fundamental (defining) representation of the Lie algebra g is equal to 7. It is known that the algebra g is embedded into so (7) (see e.g. [6], [7]).This embedding can be constructed by using the definition of the algebra g as the algebraof differentiations of the octonions. The procedure looks as follows [26]. We consider thealgebra O of octonions with the generators e = 1, e i ( i = 1 , . . . ,
7) obeying the followingmultiplication rules: e i · e j = − δ ij + f ijk e k , i, j, k = 1 , , . . . , , (4.1.10)where the structure constants f ijk are the components of the completely anti-symmetric3-rd rank tensor. The non-zero components (including index permutations) of this tensorare f = f = f = f = f = f = f = 1 . (4.1.11)Let D be the differentiation of the algebra O : D ( a · b ) = D ( a ) · b + a · D ( b ) , ∀ a, b ∈ O , D (1) = 0 , D ( e i ) = e k D ki , (4.1.12)where D ik is the matrix of the differentiation operator. If we differentiate the relation(4.1.10), then using (4.1.12) we obtain D ij = − D ji , D im f mjk + D jm f imk + D km f ijm = 0 . (4.1.13)In other words, the matrix D ij is antisymmetric and, therefore, it belongs to the algebra so (7), while the second condition in (4.1.13) shows that the tensor f ijk is invariant underthe action of the elements D ∈ so (7). This second condition produces 7 additionalrelations on the matrix D ij D + D = D , D + D = D , D + D = D , D + D = D ,D + D = D , D + D = D , D + D = D , which reduce the number of its independent components to 14. For example, one canexpress the components D i and D through the other ones and expand the antisymmetricmatrix D over the remaining 14 free parameters to obtain the basis in the algebra g .The useful identities for f ijk follow from the definition (4.1.11): f ijk f jkℓ = 6 δ iℓ , (4.1.14) f ijk f kℓm f mri = +3 f jℓr . (4.1.15)26he split Casimir operator (4.1.2) of the algebra g in the minimal fundamental rep-resentation [ ] acts in the reducible 49-dimensional space [7] × [7] which can be expandedin the irreducible representations as follows:[7] × [7] = S ([7] × [7]) + A ([7] × [7]) = ([1] + [27]) + ([7] + [14]) . (4.1.16)Here the fundamental [7] and adjoint [14] representations embed into the antisymmetricpart of [7] × [7], while the representations [1] and [27] compose the symmetric part of[7] × [7].In the space of the representation [7] × [7] we define four operators( I ) i i j j = δ i j δ i j , ( P ) i i j j = δ i j δ i j , ( K ) i i j j = δ i i δ j j , ( F ) i i j j = f i i m f mj j , (4.1.17)invariant with respect to the action of the algebra g , i.e., for any linear combination X of the operators (4.1.17) we have( a ⊗ I + I ⊗ a ) · X = X · ( a ⊗ I + I ⊗ a ) , ∀ a ∈ g . (4.1.18)Using operators (4.1.17) one may construct four mutually orthogonal projectors [25]: P [1] = K , P [7] = F , P [27] = ( I + P ) − K ,P [14] = ( I − P ) − F , (4.1.19)which form the complete system P [1] + P [7] + P [27] + P [14] = I and extract irreduciblerepresentations in the tensor product of two defining representations [7] × [7].Note that the explicit form of the projector P [7] leads to the interpretation of thestructure constants f i i m as Clebsch-Gordan coefficients describing the fusion of twofundamental representations [7] × [7] into one such representation [7].The symmetric part SC of the split Casimir operator is expressed through the invariantstructures (4.1.17) as SC = 16 ( I + P ) − K. (4.1.20)while the projectors P (7) and P (14) are related to the antisymmetric part AC of the splitCasimir operator P (7) = − AC , P (14) = 12 ( I − P ) + AC . (4.1.21)Therefore, the split Casimir operator (4.1.2) for the algebra g readsˆ C = 16 ( I + P − K − F ) . (4.1.22)In addition, we have the following useful relationˆ C i j i j = − ( T a ) i i ⊗ ( T a ) j j = − ( P [14] ) i i j j . (4.1.23) Proposition 4.1.1
The characteristic identity for the operator ˆ C reads ˆ C ( ˆ C − / C + 1)( ˆ C + 2) = 0 . (4.1.24)27 roof. The spectral decomposition for the split Casimir operator (4.1.22) follows fromthe definitions of the projectors in (4.1.19)ˆ C = 13 P [27] − P [7] − P (1) ⇒ ˆ C P [14] = 0 . (4.1.25)Thus, the operator ˆ C has four eigenvalues a = 0, a = 1 / a = − a = − P [14] , P [27] , P [7] , P [1] , respectively), which immediately leadsto the identity (4.1.24).For completeness, we give explicit formulas for the projectors P a i in terms of the operatorˆ C : P = − ( ˆ C − / C + 1)( ˆ C + 2) ≡ P [14] , P / = ˆ C ( ˆ C + 1)( ˆ C + 2) ≡ P [27] ,P − = ˆ C ( ˆ C − / C + 2) ≡ P [7] , P − = − ˆ C ( ˆ C − / C + 1) ≡ P [1] . These formulas are obtained by means of identity (4.1.24) via the standard procedure.The g -invariant solution R ( u ) of the Yang-Baxter equation (1.0.2) in the definingrepresentation was found in [25],[26]. The braid form of this solution isˇ R ( u ) = 11 − u (cid:18) I − u P − u ( u − K + u ( u − F (cid:19) . (4.1.26)Using formulas (4.1.19) we obtain the spectral decomposition of this solutionˇ R ( u ) = ( u + 1)( u + 6)( u − u − P [1] − ( u + 4)( u − P [7] − ( u + 1)( u − P [14] + P [27] , (4.1.27)after which the fulfillment of the unitarity condition (4.1.9) for this R -matrix becomesevident.Finally, the standard g -invariant R -matrix acquires the form [26], [28] R ( u ) = P ˇ R ( u ) = ( u + 6)( u + 1)( u − u − P [1] + u + 4 u − P [7] + u + 1 u − P [14] + P [27] == 1 u − (cid:18) u − P + 2 u ( u − K + u ( u − F (cid:19) . (4.1.28)Remarkably, this g -invariant solution of the Yang-Baxter equation can be rewritten as therational function of the symmetric and antisymmetric parts of the split Casimir operator R ( u ) = (3 SC − u )(3 SC + u ) · (3 AC − − u )(3 AC − u ) . (4.1.29) R -matrix for the algebra f The minimal fundamental (defining) representation T of the Lie algebra f has dimension26. In this representation the algebra f can be embedded into the algebra so (26).We follow the approaches of [31] and [32] to define the basis of the algebra f inthe representation T . For this, we consider the Jordan algebra J that consists of thehermitian 3 × o αβ = ¯ o βα ∈ O ( α, β = 1 , , A = x o o ¯ o x o ¯ o ¯ o x , x α ≡ o αα = ¯ o αα ∈ R . (4.1.30)28hus, the matrices A are defined by three octonions ¯ o , ¯ o , ¯ o and three real numbers x α , which form the (3 · J is defined as A ◦ B ≡ [ A, B ] + ( ∀ A, B ∈ J ). It is easy to check that if A and B belongto J , then [ A, B ] + also belongs to J .Let us choose in J the basis e = I , e i ( i = 1 , . . . , I – is the 3 × e i are the basis elements in the space of traceless matrices in J e = − , e = 1 √ − , e a = e a e a , (4.1.31) e a = e a e a , e a = e a e a , ( a = 1 , , . . . . Here e a ∈ O are the basis octonions (see (4.1.10)). The nomalization of elements (4.1.31)is chosen so that Tr( e i ◦ e j ) = 2 δ ij and the structure relations read e i ◦ e j ≡
12 [ e i , e j ] = 23 δ ij I − d i,j,k e k , ( i, j, k = 1 , . . . , , (4.1.32)Tr( e i ◦ e j ) = 2 δ ij . Here, the structure constants d i,j,k form the completely symmetric 3-rd rank tensor andone can extract the explicit values of these constants from equation (4.1.32).Let D be the differentiation of the algebra J and D ( e i ) = e k D ki . Acting by D on thebasic relation (4.1.32) we obtain D ij = − D ji , D im d mjk + D jm d imk + D km d ijm = 0 . (4.1.33)Thus, the matrices D ij of the differentiations of J belong to the algebra so (26) andtheir action on the tensor d ijm preserves it. The second condition in (4.1.33) reduces thenumber of independent matrices D ij to 52, which is the dimension of the algebra f .The split Casimir operator of the algebra f in the representation T acts in the reducible676-dimensional representation [ ] × [ ] which can be expanded over the irreduciblerepresentations as follows:[26] × [26] = S ([26] × [26]) + A ([26] × [26]) = ([1] + [26] + [324]) + ([52] + [273]) . (4.1.34)Here the antisymmetric part of [ ] × [ ] is decomposed into the representation [ ] andthe adjoint representation [ ], while the symmetric part of [ ] × [ ] is decomposed intorepresentations [ ] , [ ] and [ ].In the space of the [ ] × [ ] representation one may defined five operators invariantrespect to the algebra f ( I ) i i j j = δ i j δ i j , ( P ) i i j j = δ i j δ i j , ( K ) i i j j = δ i i δ j j , ( D ) i i j j = d i i m d j j m , ( F ) i i j j = T i i a T aj j , (4.1.35)where a = 1 , , . . . ,
52 and T aij ≡ δ ik ( T a ) kj – are the generators of the algebra f inthe fundamental representation. With our definitions, the structure constant d ijk andgenerators T aij obey the conditions: d ijk d kℓm d mri = − d jℓr , d i i ,m d i i ,ℓ = 563 δ mℓ , Tr( T a T b ) = − δ ab . (4.1.36)29sing the operators (4.1.35), one may construct five mutually orthogonal projectors: P [1] = K , P (26) = D = ( I + P ) − SC − K ,P [324] = (cid:16) ( I + P ) + 2 SC + K (cid:17) ,P [273] = ( I − P ) + 2 AC , P [52] = F = − AC , (4.1.37)which form the complete system P [1] + P [26] + P [52] + P [273] + P [324] = I and which single outirreducible representations in the tensor product of two defining representations [26] × [26].Note that symmetric and anti-symmetric parts of the split Casimir operator can berepresented as SC = 112 ( I + P − K − D ) , AC = − F (4.1.38)while the full split Casimir operator reads b C = SC + AC = 112 ( I + P − K ) − D − F . (4.1.39)
Proposition 4.1.2
The split Casimir operator ˆ C for the algebra f in the defining rep-resentation satisfies the characteristic identity ˆ C ( ˆ C + 1)( ˆ C + 2)( ˆ C + 1 / C − /
6) = 0 . (4.1.40) Proof.
The spectral decomposition of the split Casimir operator (4.1.39) follows fromthe definitions of the projectors in (4.1.37)ˆ C = 16 P [324] − P [26] − P [1] − P [52] ⇒ ˆ C P [273] = 0 . (4.1.41)Thus, the operator ˆ C has five eigenvalues a = 0, a = − a = − a = − / a = 1 / f -invariant solution ˇ R ( u ) of the Yang-Baxter equa-tion (4.1.8) in the defining representation was obtained in [26] and has the braid formˇ R ( u ) = ( u + 9)( u + 4)( u − u − P [1] + ( u + 6)( u + 1)( u − u − P [26] − u + 4 u − P [52] − u + 1 u − P [273] + P [324] . (4.1.42)It is clear that the solution ˇ R ( u ) (4.1.42) automatically obeys the unitarity condition.Finally, the standard f -invariant R -matrix can be written in terms of the invariants(4.1.35) and acquires the form R ( u ) = P ˇ R ( u ) == ( u + 9)( u + 4)( u − u − P [1] + ( u + 6)( u + 1)( u − u − P [26] + u + 4 u − P [52] + u + 1 u − P [273] + P [324] == 1 u − (cid:18) u − P + u ( u − u − u − K + 6 u ( u − F + 3 u u − D (cid:19) . (4.1.43) There is a missprint in the form of the f -invariant solution presented in [26]. R ( u ) = (6 SC ′ − u )(6 SC ′ + u ) · (6 AC ′ − − u )(6 AC ′ − u ) . (4.1.44)Here we introduced the notation SC ′ ≡ SC + βP [1] , AC ′ ≡ AC − βP [1] and β = 1 /
2, or β = 4 / β the operators (4.1.44) coincide in view of thecharacteristic identities for AC and SC ). R -matrix for the algebra e The algebra e , with dimension equal to 78, has two inequivalent minimal fundamentalrepresentations [ ] and [ ]. We only consider the split Casimir operator of the algebra e in the minimal fundamental representation which acts in the reducible [ ] × [ ]-dimensional space. This space can be expanded in the following irreducible representations[ ] × [ ] = S ([ ] × [ ]) + A ([ ] × [ ]) = ([ ] + [ ] ) + ([ ] ) . (4.1.45)The mutually orthogonal projectors on these irreducible representations look as follow: P [27] = 115 ( I + P ) − SC , P [351](1) = − AC = 12 ( I − P ) , P [351](2) = 1330 ( I + P ) + 35 SC , (4.1.46)where SC and AC are respectively the symmetric and antisymmetric parts of the splitCasimir operator ˆ C for the e algebra in the representation [ ]. Proposition 4.1.3
The operator ˆ C for e algebra in the representation [ ] satisfies thecharacteristic identity ( ˆ C + 139 )( ˆ C + 19 )( ˆ C −
29 ) = 0 . (4.1.47) Proof.
The spectral decomposition for the operator b C follows from the definitions of theprojectors in (4.1.46): b C = 19 (cid:0) P [351](2) − P [351](1) − P [27] (cid:1) . (4.1.48)Thus, the operator ˆ C has three eigenvalues a = − , a = − , a = , which immediatelyleads to identity (4.1.47).The e -invariant solution ˇ R ( u ) of the Yang-Baxter equation (4.1.8) in the definingrepresentation [ ] has the formˇ R ( u ) = u − u + 4 P [27] + u + 1 u − P [351](2) − P [351](1) . (4.1.49)It is clear that the solution ˇ R ( u ) (4.1.49) automatically obeys the unitarity condition.Finally, the standard e -invariant R -matrix acquires the form [26] R ( u ) = P ˇ R ( u ) = u − u + 4 P [27] + u + 1 u − P [351](2) + P [351](1) . (4.1.50)The e -invariant solution R ( u ) in (4.1.50) can be elegantly written as a rational func-tion of b C (compare with (3.1.16), (3.2.12)) R ( u ) = − (3 b C + 1 / u )(3 b C + 1 / − u ) . (4.1.51)31 .1.5 Split Casimir operator and R -matrix for the algebra e The dimension of the exceptional algebra e is 133. The tensor product of its minimal56-dimensional fundamental representations has the following expansion into irreducibleones[ ] × [ ] = S ([ ] × [ ]) + A ([ ] × [ ]) = ([ ] + [ ]) + ([ ] + [ ]) . (4.1.52)The mutually orthogonal projectors on these irreducible representations read: P [1] = − ( I − P ) − AC , P [133] = ( I + P ) − SC ,P [1463] = ( I + P ) + SC , P [1539] = ( I − P ) + AC , (4.1.53)where SC and AC are respectively the symmetric and antisymmetric parts of the splitCasimir operator ˆ C for the e algebra in the representation [ ]. Proposition 4.1.4
The operator ˆ C for the e algebra in the representation [ ] satisfiesthe characteristic identity (cid:18) ˆ C − (cid:19) (cid:18) ˆ C + 78 (cid:19) (cid:18) ˆ C + 198 (cid:19) (cid:18) ˆ C + 124 (cid:19) = 0 . (4.1.54) Proof.
The following spectral decomposition for the split Casimir operator follows fromthe definitions of the projectors in (4.1.53) b C = SC + AC = 18 (cid:0) P [1463] − P [133] (cid:1) + 124 (cid:0) − P [1] − P [1539] (cid:1) , (4.1.55)Thus, the operator ˆ C has four eigenvalues a = , a = − , a = − , a = − , whichimmediately leads to identity (4.1.54).The decomposition (4.1.52) of the antisymmetric part A ([ ] × [ ]) of the tensor prod-uct [ ] × [ ] contains the singlet representation [ ] . This means that the correspondingprojector can be rewritten in the form( P [1] ) i i j j = − J i i J j j , J ik = − J ki , J ik J kj = δ ij , where J ik and J ik are the invariant antisymmetric metrics. With the help of these metricsone may raise and lower indices of tensors. The existence of these metrics indicates thatthe e algebra in the representation [ ] is embedded as a subalgebra in the symplecticalgebra sp (56).The e -invariant solution ˇ R ( u ) of the Yang-Baxter equation (4.1.7) in the definingrepresentation has the formˇ R ( u ) = − ( u − u − u + 9)( u + 5) P [1] + u − u + 5 P [133] + u + 1 u − P [1463] − P [1539] , (4.1.56)The solution ˇ R ( u ) (4.1.56) obviously obeys the unitarity condition.Finally, the standard e -invariant R -matrix acquires the form [26] R ( u ) = P ˇ R ( u ) = ( u − u − u + 9)( u + 5) P [1] + u − u + 5 P [133] + u + 1 u − P [1463] + P [1539] . (4.1.57)32ote that this solution of the Yang-Baxter equation can also be written as a rationalfunction of the ”shifted” symmetric and antisymmetric parts of the split Casimir operator(cf. (4.1.44)) R ( u ) = ( u + 6 SC ′ + 1 / u − SC ′ − / · ( u + 6 AC ′ )( u − AC ′ ) . (4.1.58)Here, we introduced the ”shifted” symmetric and antisymmetric parts SC ′ ≡ SC − βP [1] , AC ′ ≡ AC + βP [1] , where β = 37 /
4, or β = 21 /
4. For both values of the parameter β theoperators (4.1.57) coincide in view of the characteristic identities for AC and SC . e The exceptional Lie algebra e has dimension 248. Its minimal fundamental representationhas also dimension 248 and it appears to be an adjoint representation of the algebra e . The split Casimir operators b C and their characteristic identities for all exceptionalLie algebras g in the adjoint representation are discussed in the next section . Sowe postpone the consideration of the operator b C for the Lie algebra e in the minimalfundamental (adjoint) representation to subsection .It is known that there are no solutions of the Yang-Baxter equation ( R - matrices) forthe simple Lie algebras in the adjoint representation besides the sℓ -series of the Lie alge-bras. This is the consequence of the fact that the adjoint representation of the simple Liealgebras g (except sℓ algebras) can not be extended to the representation of the Yangian Y ( g ) (see [33], [34]). However, the reducible representation g ⊕ C , which is a direct sumof the adjoint and trivial representations, can be extended to the representation of theYangian Y ( g ). The g -invariant solution of the Yang-Baxter equation can be constructedjust within such extended adjoint representation. For the extended adjoint (minimal fun-damental) representation [ ] of the algebra e , this solution has been constructed in [34]and [35]. The explicit form of this solution, written in the form of spectral decompositionover projectors, turns out to be rather cumbersome and we will not present it here. Weassume that writing this solution in terms of the symmetrized SC and antisymmetrized AC parts of the operator b C will result in a more visual and compact formula. b C and invariant pro-jectors for exceptional Lie algebras in the adjoint represen-tations. In this section, we will find characteristic identities for the split Casimir operator b C in theadjoint representations for the exceptional algebras g = g , f , e , e and e . As we notedat the end of theprevious section, the solutions of the Yang-Baxter equation that areinvariant with respect to actions of exceptional Lie algebras in the adjoint representationdo not exist, so this topic is not covered here. Let us define the normalization of the generators X a of the exceptional Lie algebra g sothat the Cartan-Killing metric (2.1.2) looks likeTr (cid:0) ad( X a ) ad( X d ) (cid:1) = dim g X c,b =1 ( C a ) cb ( C d ) bc = − δ ad . (4.2.1)33here ( C d ) bc ≡ C bdc are the structure constants of the Lie algebra g . The split Casimiroperator in the adjoint representation reads( b C ad ) a a b b = − X d ( C d ) a b ( C d ) a b . (4.2.2)We will also need the identity I and permutation P operators defined as: I a a b b = δ a b δ a b , P a a b b = δ a b δ a b , (4.2.3)together with the operator (2.2.2), which in the normalization (4.2.1)reads K a a b b = δ b b δ a a . (4.2.4)In what follows, similarly to the previous consideration, it proved useful to define thesymmetric b C + and antisymmetric b C − parts of the split Casimir operators in the adjointrepresentation b C + = P + b C ad ⇒ ( b C + ) a a b b = (cid:16) ( b C ad ) a a b b + ( b C ad ) a a b b (cid:17) , b C − = P − b C ad ⇒ ( b C − ) a a b b = (cid:16) ( b C ad ) a a b b − ( b C ad ) a a b b (cid:17) , (4.2.5)where P ± = ( I ± P ). g The tensor product of two adjoint 14-dimensional representations of the algebra g hasthe following decomposition into irreducible representations [4], [24]:[ ] × [ ] = S ([ ] × [ ]) + A ([ ] × [ ]) = ([1] + [27] + [77]) + ([14] + [77 ⋆ ]) . (4.2.6)The dimensions of two representations appearing in the decomposition of A ([ ] × [ ])are given by (2.2.12). The antisymmetric b C − and symmetric b C + parts of the split Casimiroperator b C ad in the adjoint representation obey the following identities: b C − (cid:16) b C − + 12 (cid:17) = 0 , b C = − b C + + 596 ( I + P + K ) . (4.2.7)Here the first identity is fulfilled for all simple Lie algebras, while the second one has beenobtained by direct explicit calculations with the help of the M athematica
T M package (fordetails, see [36]).Multiplying both parts of equation (4.2.7) by b C + and using the relations (2.2.4),(2.2.7),one can obtain b C + ( b C + −
14 )( b C + + 512 ) = − K ⇒ b C + ( b C + + 1)( b C + −
14 )( b C + + 512 ) = 0 , (4.2.8)where the second identity follows from the first ones after multiplying it by ( b C + + 1) andtaking into account (2.2.7).The characteristic identity for the complete Casimir operator b C ad = b C + + b C − can beobtained from identities (4.2.7) and (4.2.8): b C ad (cid:16) b C ad + 1 (cid:17) (cid:18) b C ad + 12 (cid:19) (cid:18) b C ad − (cid:19) (cid:18) b C ad + 512 (cid:19) = 0 . (4.2.9)34sing this identity together with the relation (4.2.7), one can find the projectors P dim( V i ) ≡ P ( a i ) on the eigenspaces V i of the operator b C ad with the eigenvalues a i = ( − , − , − , , P = P ( − ) = − b C − , P ⋆ = P (0) = 12 ( I − P ) + 2 b C − , P = P ( − = 114 K , P = P ( − ) = 316 ( I + P ) − b C + − K , (4.2.10) P = P ( ) = 516 ( I + P ) + 32 b C + + 116 K , where the first two and the last three projectors act on the antisymmetrized P − (14 ⊗ ), andthe symmetrized P + (14 ⊗ ) parts of the representation 14 ⊗ , respectively. The dimensionsdim( V i ) of the representations corresponding to the projectors (4.2.10) are calculated usingformulas (2.2.10). f The exceptional Lie algebra f has dimension 52. The tensor product of its two adjoint52-dimensional representations has the following decomposition into irreducible represen-tations [4, 24][ ] × [ ] = S ([ ] × [ ]) + A ([ ] × [ ]) = ([1] + [324] + [1053]) + ([52] + [1274]) . (4.2.11)The dimensions of two representations in the decomposition of A ([ ] × [ ]) are calculatedby means of formulas (2.2.12). The antisymmetric b C − and symmetric b C + parts of thesplit Casimir operator b C for the algebra f in the adjoint representation obey the followingidentities : b C − (cid:16) b C − + 12 (cid:17) = 0 , b C = − b C + + 5324 ( I + P + K ) . (4.2.12)Multiplying both parts of equation (4.2.12) by b C + and using the relations (2.2.4),(2.2.7),one obtains b C + ( b C + −
19 )( b C + + 518 ) = − K ⇒ b C + ( b C + + 1)( b C + −
19 )( b C + + 518 ) = 0 , (4.2.13)Thus, the characteristic identity for the full split Casimir operator b C ad = ( b C + + b C − ) reads b C ad (cid:16) b C ad + 1 (cid:17) (cid:18) b C ad + 12 (cid:19) (cid:18) b C ad − (cid:19) (cid:18) b C ad + 518 (cid:19) = 0 . (4.2.14)Finally, the projectors P dim( V i ) ≡ P ( a i ) onto the representations counted in the decompo-sition (4.2.11), or in other words, onto the representations acting in the eigenspaces V i of b C ad with the eigenvalues a i = ( − , − , − , ,
0) can be found to be P = P ( − ) = − b C − , P = P (0) = 12 ( I − P ) + 2 b C − , P = P ( − = 152 K , P = P ( − ) = 17 ( I + P ) − b C + − K , (4.2.15) P = P ( ) = 514 ( I + P ) + 187 b C + + 128 K . The second identity for the symmetric part of the Casimir operator here and for all other exceptionalalgebras below were obtained by direct explicit calculations with the help of the
M athematica
T M package(for details, see [36]). V i ) of the representations corresponding to the projectors (4.2.15)are calculated using formulas (2.2.10). e The exceptional Lie algebra e has dimension 78. The tensor product of its two adjoint78-dimensional representations has the following decomposition into irreducible represen-tations [4, 24]:[ ] × [ ] = S ([ ] × [ ]) + A ([ ] × [ ]) = ([1] + [650] + [2430]) + ([78] + [2925]) . (4.2.16)The dimensions of two representations in the decomposition of A ([ ] × [ ]) are calculatedby means of (2.2.12). The antisymmetric b C − and symmetric b C + parts of the split Casimiroperator in the adjoint representation obey the following identities: b C − (cid:16) b C − + 12 (cid:17) = 0 , b C = − b C + + 196 ( I + P + K ) . (4.2.17)From these identities, similarly to the previously considered cases of the g and f algebras,one obtains b C + ( b C + + 14 )( b C + −
112 ) = − K ⇒ b C + ( b C + + 1)( b C + + 14 )( b C + −
112 ) = 0 , (4.2.18)and, therefore, b C ad (cid:16) b C ad + 1 (cid:17) (cid:18) b C ad + 12 (cid:19) (cid:18) b C ad + 14 (cid:19) (cid:18) b C ad − (cid:19) = 0 . (4.2.19)Thus, the projectors P dim( V i ) ≡ P ( a i ) on the representations listed in (4.2.16) and corre-sponding to the eigenvalues a i = ( − , − , − , − ,
0) read (cf. projectors in [4], Table18.5) P = P ( − ) = − b C − , P = P (0) = 12 ( I − P ) + 2 b C − , P = P ( − = 178 K , P = P ( − ) = 18 ( I + P ) − b C + − K , (4.2.20) P = P ( ) = 38 ( I + P ) + 3 b C + + 3104 K . These projectors are built using the standard method with the help of identity (4.2.19) andrelations (4.2.17), (4.2.18). The dimensions dim V i of the representations correspondingto the projectors (4.2.20) are calculated using formulas (2.2.10). e The exceptional Lie algebra e has dimension 133. The tensor product of its two adjoint133-dimensional representations has the following decomposition into irreducible repre-sentations [4, 24]:[ ] × [ ] = S ([ ] × [ ])+ A ([ ] × [ ]) = ([1] + [1539] + [7371])+([133] + [8645]) . (4.2.21)36he dimensions of two representations in the decomposition of A ([ ] × [ ]) are givenby formula (2.2.12). The antisymmetric b C − and symmetric b C + parts of the split Casimiroperator in the adjoint representation obey the following identities: b C − (cid:16) b C − + 12 (cid:17) = 0 , b C = − b C + + 1162 ( I + P + K ) . (4.2.22)From these identities we obtain b C + ( b C + + 29 )( b C + −
118 ) = − K ⇒ b C + ( b C + + 1)( b C + + 29 )( b C + −
118 ) = 0 , (4.2.23)and, therefore, b C ad (cid:16) b C ad + 1 (cid:17) (cid:18) b C ad + 12 (cid:19) (cid:18) b C ad + 29 (cid:19) (cid:18) b C ad − (cid:19) = 0 . (4.2.24)Finally, the projectors P dim( V i ) ≡ P ( a i ) on the representations appearing in the decompo-sition (4.2.21) and corresponding to the eigenvalues a i = ( − , − , − , ,
0) read P = P ( − ) = − b C − , P = P (0) = 12 ( I − P ) + 2 b C − P = P ( − = 1133 K , P = P ( − ) = 110 ( I + P ) − b C + − K , (4.2.25) P = P ( ) = 25 ( I + P ) + 185 b C + + 295 K . The dimensions dim V i of the representations related to the projectors (4.2.25) are calcu-lated by formulas (2.2.10). e The exceptional Lie algebra e has dimension 248. The tensor product of its two adjoint248-dimensional representations has the following decomposition into irreducible repre-sentations [4, 24]: [ ] × [ ] = S ([ ] × [ ]) + A ([ ] × [ ]) == ([ ] + [ ] + [ ]) + ([ ] + [ ]) . (4.2.26)The dimensions of two representations in the decomposition of A ([ ] × [ ]) are cal-culated by formula (2.2.12). The antisymmetric b C − and symmetric b C + parts of the splitCasimir operator b C ad in the adjoint representation obey the following identities: b C − (cid:16) b C − + 12 (cid:17) = 0 , b C = − b C + + 1300 ( I + P + K ) . (4.2.27)From these identities one can obtain b C + ( b C + + 15 )( b C + −
130 ) = − K ⇒ b C + ( b C + + 1)( b C + + 15 )( b C + −
130 ) = 0 . (4.2.28)The characteristic identity for the full split Casimir operator b C ad = ( b C + + b C − ) reads b C ad (cid:16) b C ad + 1 (cid:17) (cid:18) b C ad + 12 (cid:19) (cid:18) b C ad + 15 (cid:19) (cid:18) b C ad − (cid:19) = 0 . (4.2.29)37he projectors P dim( V i ) ≡ P ( a i ) on the representations in the decomposition (4.2.26), whichcorrespond to the eigenvalues a i = ( − , − , − , , P = P ( − ) = − b C − , P = P (0) = ( I − P ) + 2 b C − , P = P ( − = K , P = P ( − ) = ( I + P ) − b C + − K , P = P ( ) = ( I + P ) + b C + + K . (4.2.30)The dimensions dim V i of the representations related to the projectors (4.2.30) are calcu-lated by formulas (2.2.10). In the adjoint representations the antisymmetric parts of the split Casimir operators hC − for all simple Lie algebras obey the same identity b C − (cid:18) b C − + 12 (cid:19) = 0 . (4.3.1)The symmetric parts of the split Casimir operators b C + in the adjoint representation for theexceptional Lie algebras obey identities (4.2.7), (4.2.12), (4.2.17), (4.2.22) and (4.2.27),which have a similar structure b C = − b C + + µ ( I + P + K ) , (4.3.2)where the universal parameter µ is fixed as follows: µ = 56(2 + dim( g )) . (4.3.3)Note that identities (3.1.50) and (3.2.23) for the algebras sℓ (3) and so (8) have the samestructure.From (4.3.2) one can obtain the universal characteristic identity on the symmetricpart of the split Casimir operator b C + b C + ( b C + + 1)( b C + 16 b C + − µ ) ≡ b C + ( b C + + 1)( b C + + α t )( b C + + β t ) = 0 , (4.3.4)where we introduced the notation for two eigenvalues of the b C + : α t = 1 − µ ′ , β t = 1 + µ ′ , µ ′ := p µ = s dim g + 242dim g + 2 . (4.3.5)These parameters are related as 3( α + β ) = t. The universal formulae (4.3.2) was obtained in [4], eq. (17.10), under the assumption that b C isexpressed as a linear combination of g -invariant operators ( I + P ), K and b C + . We explicitly checked thisassumption for all exeptional Lie algebras. α , this relation defines the line of the exceptionalLie algebras on the β, t plane (see eq.(5.1.15) below). Following [4], note that µ ′ is arational number only for a certain sequence of dimensions dim g . It turns out that thissequence is finite :dim g = 3 , , , , , , , , , , , , , , , , , , , , (4.3.6)which includes the dimensions 14 , , , ,
248 of the exceptional Lie algebras g , f , e , e , e , and the dimensions 8 and 28 of the algebras sℓ (3) and so (8), which aresometimes also referred to as exceptional. Thus, for these algebras, using (4.3.5), wecalculate the values of the parameters α t , β t given in Table 4.Table 4. sℓ (3) so (8) g f e e e α t − / − / − / − / − / − / − / β t / / /
12 5 /
18 1 / / / These values are in agreement with the formulas (3.1.51), (3.2.31), (4.2.8), (4.2.13),(4.2.18), (4.2.23), (4.2.28). Taking into account that b C − satisfies (2.2.5) and b C + sat-isfies (4.3.4), we obtain the following identities for the total split Casimir operator b C ad = ( b C + + b C − ) in the case of the exceptional Lie algebras: b C ad (cid:18) b C ad + 12 (cid:19) (cid:16) b C ad + 1 (cid:17) (cid:18) b C + 16 b C ad − µ (cid:19) = 0 ⇒ (4.3.7) b C ad (cid:18) b C ad + 12 (cid:19) (cid:16) b C ad + 1 (cid:17) (cid:16) b C ad + α t (cid:17) (cid:18) b C ad + β t (cid:19) = 0 . (4.3.8)Here µ is defined in (4.3.3) and α t , β t are given in Table 4. Remark.
The sequence (4.3.6) contains dimensions dim g ∗ = (10 m −
122 + 360 /m ),( m ∈ N ) referring to the adjoint representations of the so-called E family of algebras g ∗ ;see [4], eq. (21.1). For these dimensions we have the relation µ ′ = | ( m + 6) / ( m − | .Two numbers 47 and 119 from the sequence (4.3.6) do not belong to the sequence dim g ∗ .Thus, the interpretation of these two numbers as dimensions of some algebras is missing.Moreover, for values dim g given in (4.3.6), using (4.3.5), one can calculate dimensions(3.3.9) of the corresponding representations Y ( α ):dim V ( − α t ) = (cid:8) , , , , , , , , , , , , , , , , , , , (cid:9) Since dim V ( − α t ) should be integer, we conclude that no Lie algebras exist with dimen-sions 47 , , , , , , , , We thank D.O.Orlov who proved the finiteness of this sequence. Universal characteristic identities for operator b C for simple Lie algebras in the adjoint representa-tion and Vogel parameters In the previous sections, the projectors were constructed onto the spaces of irreduciblesubrepresentations in the representation ad ⊗ for all simple complex Lie algebras (Lie al-gebras of classical series A n , B n , C n , D n and the exceptional Lie algebras). In all cases theconstruction was carried out by finding the characteristic identities for the split Casimiroperators. In this regard, it should be noted that certain results of this work, namely theconstruction of projectors in terms of the split Casimir operator and finding their dimen-sions can be obtained by using the Vogel parameters α, β and γ , which were introducedin [11] (see also [12, 14]). The specific values of these parameters correspond to eachsimple complex Lie algebra. These values and the value of t = ( α + β + γ ) are given inTable 5 (see below). Since all universal formulas for the simple Lie algebras are written ashomogeneous functions of the parameters α, β and γ , and these formulas are independentof all permutations of α, β, γ one can consider simple Lie algebras as points in the space RP / S . It is convenient to choose normalization in which one of the parameters is fixed,for example α = −
2, which is already done in Table 5. Note that the data in the first sixlines of Table 5 coincide with the data given in Table 3 of Section . We indicate theVogel parameters for the algebras sℓ (3) and so (8) in the separate lines of Table 5, sincethe characteristic identities (3.1.50), (3.1.51) and (3.2.30), (3.2.27) for the symmetric part b C + of the split Casimir operator in the adjoint representations have the same order andthe same structure as for the exceptional Lie algebras (cf. (4.3.2), (4.3.4)).Table 5.Type Lie algebra α β γ t − α t = t − β t − γ t A n sℓ ( n + 1) − n + 1 n + 1 n +1 − n +1 − / B n so (2 n + 1) − n − n − n − − n − − n − n − C n sp (2 n ) − n + 2 n + 1 n +1 − n +1) − n +22( n +1) D n so (2 n ) − n − n − n − − n − − n − n − A sℓ (3) − / − / − / D so (8) − / − / − / G g − / / / − / − / F f − / − / − / E e − / − / − / E e − / − / − / E e − / − / − / It proved useful to split the tensor product of two adjoint representations into thesymmetric and antisymmetric partsad ⊗ ad = S (ad ⊗ ad) + A (ad ⊗ ad) . (5.1.9)In the general case of the Lie algebras of the classical series , the symmetric part S (ad ⊗ )decomposes into 4 irreducible representations: a singlet, denoted as X , with zero eigen-value of the quadratic Casimir operator C (2) (which corresponds to the eigenvalue ( − The algebras sℓ (3) and so (8) are exeptional cases. b C ), and 3 representations which we denote as Y ( α ) , Y ( β ) , Y ( γ ).Their dimensions, as well as the corresponding values c ( λ )(2) and ˆ c ( λ )(2) (here λ = µ, µ ′ , µ ′′ arethe highest weights of the representations Y ( α ) , Y ( β ) , Y ( γ )) of the quadratic Casimiroperator C (2) (defined in (2.1.4)) and split Casimir operator b C are equal to:dim Y ( α ) = dim V ( − α t ) , c ( µ )(2) = 2 − αt , ˆ c ( µ )(2) = − α t , (5.1.10)dim Y ( β ) = dim V ( − β t ) , c ( µ ′ )(2) = 2 − βt , ˆ c ( µ ′ )(2) = − β t , (5.1.11)dim Y ( γ ) = dim V ( − γ t ) , c ( µ ′′ )(2) = 2 − γt , ˆ c ( µ ′′ )(2) = − γ t . (5.1.12)where the explicit expressions for dim V ( − α t ) , dim V ( − β t ) , dim V ( − γ t ) are given in (3.3.9)–(3.3.11) and the eigenvalues c ( λ )(2) and ˆ c ( λ )(2) of the operators C (2) and b C are related by thecondition (2.3.6): ˆ c ( λ )(2) = 12 ( c ( λ )(2) − c ad(2) ) = 12 c ( λ )(2) − . (5.1.13)The eigenvalues ˆ c ( λ )(2) of the operator b C on the representations Y ( α ) , Y ( β ) , Y ( γ ) in SC (ad × ad) are presented in three last columns of Table 5. Therefore, taking into accountthat b C + has four eigenvalues ( − , − α t , − β t , − γ t ) and b C − has two eigenvalues (0 , − ), thegeneric characteristic identity for the split Casimir operator reads b C ad ( b C ad + 12 )( b C ad + 1)( b C ad + α t )( b C ad + β t )( b C ad + γ t ) = 0 . (5.1.14)In the case of the sℓ ( N ) algebras, the eigenvalue ( − /
2) of the operator b C ad is doublydegenerated, since γ t = 1 /
2; therefore, in identity (5.1.14) one should keep only one factor( b C ad + ) of two (compare the identities (3.1.35) and (5.1.14)).We now turn to the discussion of the case of the exceptional Lie algebras. Note thatall exceptional Lie algebras are distinguished in Table 5 by the value of the parameter − γ/ (2 t ) equals to − / ). Thus, all exceptionalLie algebras in the three-dimensional space of the Vogel parameters ( α, β, γ ) lie in theplane α = − γ = 2 t ⇒ γ = 2 β − . (5.1.15)We chose the coordinates ( β, γ ) on this plane. When the condition (5.1.15) is fulfilled,the dimension (3.3.11),(5.1.12) of the space of the representation Y ( γ ) is zero in view ofthe factor (3 γ − t ) in the numerator of (3.3.11). So the corresponding projector P ( − γ t ) on this space is also equal to zero and the parameter − γ/ (2 t ) cannot be an eigenvalueof b C ad (in the case of the non-exceptional Lie algebras, this parameter is the eigenvalueof the operator b C ad on the representation Y ( γ ); see Subsection ). In this case, in thegeneral characteristic identity (5.1.14) for the operator b C ad = (ad ⊗ ad)( b C ), the last factor( b C ad + γ t ) will be absent and the universal characteristic identity coincides with (4.3.8): b C ad ( b C ad + 12 )( b C ad + 1)( b C ad + α t )( b C ad + β t ) = 0 . (5.1.16)As we showed in Subsection , identity (5.1.16) for the values of the parameters α, β given in Table 4 and Table 5 exactly reproduces the characteristic identities (4.2.9),414.2.14), (4.2.19), (4.2.24) and (4.2.29) for the split Casimir operator b C ad in the case ofthe exceptional Lie algebras. Note that both algebras so (8) and sℓ (3) (for the latterone has to replace the parameters β ↔ γ ) lie on the line (5.1.15) and the characteristicidentities (3.2.27) and (3.1.52) are also given by the generic formula (5.1.16). Indeed, forthe algebra sℓ (3) we have γ t = ; therefore, the eigenvalue ( − /
2) of the operator b C ad is doubly degenerated and one of the factors ( b C ad + 1 /
2) in (5.1.16) must be omitted.Wherein, for the algebra so (8) both parameters β t and γ t are equal to the critical value , which gives zero in denominators of the expressions (3.3.10), (5.1.11) and (3.3.11),(5.1.12) for the dimensions dim V ( − β t ) and dim V ( − γ t ) of the representations Y ( β ) and Y ( γ ). However, these zeros are canceled with zeros coming from the terms (3 β − t )and (3 γ − t ) in the numerators of the expressions for dim V ( − β t ) , dim V ( − γ t ) and thesedimensions turn out to be 35, which is consistent with (3.2.32). Since the eigenvalue − β t = − γ t = − of the operator b C ad is doubly degenerated, we must omit one of thefactors ( b C ad + 1 /
3) in (5.1.14) and this identity is transformed into identity (5.1.16).The antisymmetric part A (ad ⊗ ad) decomposes for all simple Lie algebras into a directsum of two terms X and X (see Section ), one of which X is the adjoint representationad with the value of the quadratic Casimir c (ad)(2) = 1, and the other representation X hasthe value of the quadratic Casimir c ( X )(2) = 2. The representation X is reducible for thecase of algebras sℓ ( N ) (see Subsection ) and irreducible for all other simple Liealgebras. The dimension of the representations X , X and the corresponding eigenvaluesˆ c (ad)(2) and ˆ c ( X )(2) are equal to (cf. (2.2.12))dim X = dim g , ˆ c (ad)(2) = − / , dim X = dim g (dim g − , ˆ c ( X )(2) = 0 . The values ˆ c (ad)(2) and ˆ c ( X )(2) agree with the characteristic identity (2.2.5) for the antisym-metrized part of b C − , which is valid for all simple Lie algebras. Acknowledgements
The authors are thankful to O.V. Ogievetsky who draw our attention to the relation ofthe Vogel parametrization and characteristic identities of the split Casimir operator inthe adjoint representation and to P. Cvitanovi´c, R.L.Mkrtchyan, M.A.Vasiliev for usefulcomments. We are thankful to D. Lezin for the help with calculations of the projectors(3.1.47) at the initial stage. The authors are also grateful to D.O. Orlov and N.A. Tyurinfor the explanation of the methods for solving the nonlinear Diophantine equations.
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