Segal–Sugawara vectors for the Lie algebra of type G 2 6 1 A. I. Molev, E. Ragoucy and N. Rozhkovskaya 0 2 n a J 8 2 ] T R Abstract . h Explicit formulas for Segal–Sugawara vectors associated with the simple Lie al- t a gebra g of type G are found by using computer-assisted calculations. This leads to m 2 a direct proof of the Feigin–Frenkel theorem describing the center of the correspond- [ ing affine vertex algebra at the critical level. As an application, we give an explicit 1 v solution of Vinberg’s quantization problem by providing formulas for generators of 8 maximal commutative subalgebras of U(g). We also calculate the eigenvalues of the 3 6 Hamiltonians on the Bethe vectors in the Gaudin model associated with g. 7 0 Preprint LAPTH-006/16 . 1 0 6 1 : v i X r a School of Mathematics and Statistics University of Sydney, NSW 2006, Australia [email protected] Laboratoire de Physique Th´eorique LAPTh, CNRS and Universit´e de Savoie BP 110, 74941 Annecy-le-Vieux Cedex, France [email protected] Department of Mathematics Kansas State University, USA [email protected] 1 1 Introduction For a simple Lie algebra g over C equipped with a standard symmetric invariant bilinear form, consider the corresponding (non-twisted) affine Kac–Moody algebra g g = g[t,t 1] CK. b (1.1) − ⊕ Theuniversal vacuummoduleV(g)bover gisthequotient ofU(g)bytheleftideal generated by g[t]. A Segal–Sugawara vector is any element S V(g) with the property b ∈ b g[t]S (K +h )V(g), ∨ ∈ where h is the dual Coxeter number for g. In particular, the canonical quadratic Segal– ∨ Sugawara vector is given by d S = X [ 1]2, (1.2) a − Xa=1 where X ,...,X is an orthonormal basis of g and we write X[r] = Xtr for X g. 1 d ∈ By an equivalent approach, Segal–Sugawara vectors are elements of the subspace z(g) of invariants of the vacuum module at the critical level b z(g) = v V(g) g[t]v = 0 , (1.3) cri { ∈ | } where V(g) isthequotient ofbV(g) by thesubmodule (K+h )V(g). The vacuummodule cri ∨ possesses a vertex algebra structure, and by the definition (1.3), z(g) coincides with the center of the vertex algebra V(g) . This induces a structure of commutative associative cri b algebra on the center which coincides with the one obtained via identification of z(g) with a subalgebra of U t 1g[t 1] . − − b The structure (cid:0)of z(g) wa(cid:1)s described by a theorem of Feigin and Frenkel [8], which states that z(g) is an algebra of polynomials in infinitely many variables; see [11] for a detailed b exposition of these results. The algebra z(g) is refereed to as the Feigin–Frenkel center. b Explicit formulas for generators of this algebra were found in [5] for type A and in [17] for b types B, C and D; see also [4] and [20] for simpler arguments in type A and extensions to Lie superalgebras. Our goalinthis paper isto give explicit formulas forgeneratorsof z(g) inthecase where g is the exceptional Lie algebra of type G . In particular, we obtain a direct proof of the 2 b Feigin–Frenkel theorem in this case. Furthermore, using the connections with the Gaudin model as discovered in [9], we get formulas for higher Gaudin Hamiltonians associated with g and calculate their eigenvalues on the Bethe vectors; see also [10]. In the classical types such formulas were given in a recent work [19]. As another application, following [10] and[25] wegive explicit formulasforalgebraically independent generators of maximal commutative subalgebras of U(g). These subalgebras 2 are parameterized by regular elements µ g , and their classical limits are Poisson µ ∗ µ A ∈ A commutative subalgebras of S(g) known as the Mishchenko–Fomenko or shift of argument subalgebras. The formulas for generators of thus provide an explicit solution of Vin- µ A berg’s quantization problem [27]. Our calculations of the explicit expressions for the Segal–Sugawara vectors and their Harish-Chandra images were computer-assisted. We gratefully acknowledge the use of the Symbolic Manipulation System FORM originally developed by Vermaseren [26]. This project was completed within the visitor program of the Center of Quantum Algebra of the South China University of Technology, Guangzhou, China. We would like to thank the Center and the School of Mathematical Sciences for the warm hospitality during our visits. 2 Lie algebra of type G and its matrix presentation 2 We start by recalling well-known matrix presentations of an arbitrary simple Lie algebra g; see, e.g., [6] and [16]. 2.1 Matrix presentations of simple Lie algebras Equip g with a symmetric invariant bilinear form , . Choose a basis X1,...,Xd of g h i and let X ,...,X be its dual with respect to the form. Let π be a faithful representation 1 d of g afforded by a finite-dimensional vector space V, π : g EndV. (2.1) → Introduce the elements d G = π(Xi) X EndV U(g) (2.2) i ⊗ ∈ ⊗ Xi=1 and d Ω = π(Xi) π(X ) EndV EndV. (2.3) i ⊗ ∈ ⊗ Xi=1 Note that G and Ω are independent of the choice of the basis Xi. In particular, d Ω = π(X ) π(Xi). (2.4) i ⊗ Xi=1 Consider the tensor product algebra EndV EndV U(g) andidentify Ω with the element ⊗ ⊗ Ω 1. Also, introduce its elements ⊗ d d G = π(Xi) 1 X and G = 1 π(Xi) X . (2.5) 1 i 2 i ⊗ ⊗ ⊗ ⊗ Xi=1 Xi=1 3 Write the commutation relations for g, d [X ,X ] = ck X (2.6) i j ij k Xk=1 with structure coefficients ck. We will regard the universal enveloping algebra U(g) as the ij associative algebra with generators X subject to the defining relations (2.6), where the i left hand side is understood as the commutator X X X X . i j j i − Proposition 2.1. The defining relations of U(g) are equivalent to the matrix relation G G G G = ΩG +G Ω. (2.7) 1 2 2 1 2 2 − − Proof. The left hand side of (2.7) reads d π(Xi) π(Xj) (X X X X ). i j j i ⊗ ⊗ − iX,j=1 For the right hand side we have d π(Xi) π [X ,Xk] X . (2.8) i k − ⊗ ⊗ iX,k=1 (cid:0) (cid:1) By the invariance of the form, we find [X ,Xk],X = Xk,[X ,X ] = ck. h i ji −h i j i − ij Hence (2.8) equals d ck π(Xi) π(Xj) X . ij ⊗ ⊗ k i,Xj,k=1 Since the representation π is faithful, we conclude that (2.7) is equivalent to the defining relations (2.6) of U(g). The defining relations (2.7) can be written in an equivalent form G G G G = ΩG G Ω, (2.9) 1 2 2 1 1 1 − − which is easily verified with the use of (2.4). The element G can be regarded as an n n × matrix (n = dimV) with entries in U(g). We point out another well-known property of this matrix which goes back to [14]. Corollary 2.2. All elements trGk with k > 1 belong to the center of U(g). 4 Proof. Relation (2.7) implies G Gk GkG = ΩGk +GkΩ. 1 2 − 2 1 − 2 2 By taking trace over the second copy of EndV and using its cyclic property, we get [G ,trGk] = 0 as required. 1 2 The Casimir elements trGk are widely used in representation theory, especially for the Lie algebras g of classical types. In those cases one usually takes V to be the first fundamental (or vector) representation. Note also that the relation (2.7) can be regarded as the ‘classical part’ of the RTT presentation of the Yangian Y(g) associated with g; see [6]. More precisely, Y(g) contains U(g) as a subalgebra, and (2.7) is recovered as a reduction of the RTT relation to the generators of this subalgebra. 2.2 Lie algebra of type G 2 The simple Lie algebra g of type G admits a few different presentations; see, e.g., [12], 2 [28]. It is well-known that it can be embedded into the orthogonal Lie algebras o and o ; 7 8 these embeddings were employed in [21] to construct the classical -algebra for g. We W will follow [12, Lect. 22] to realize g as the direct sum of vector spaces g = C3 sl (C3) . 3 ∗ ⊕ ⊕ The Lie bracket on g is determined by the conditions that sl is a subalgebra of g, the 3 vector spaces C3 and (C3) are, respectively, the vector representation of sl and its dual, ∗ 3 together with additional brackets C3 C3 (C3) , (C3) (C3) C3 and C3 (C3) sl . ∗ ∗ ∗ ∗ 3 × → × → × → To produce a matrix presentation of g as provided by Proposition 2.1, consider the 7- dimensional representation π : g EndV with V = C7 where the action is described ∼ → explicitly as follows. Write an arbitrary element of g as a triple (v,A,ϕ), where A sl is 3 ∈ a traceless 3 3 matrix, × v 1 v = v2 C3 and ϕ = ϕ1,ϕ2,ϕ3 (C3)∗. ∈ ∈ v (cid:2) (cid:3)  3 Then, under the representation π we have A v 1 B(ϕt) √2   π : (v,A,ϕ) ϕ 0 vt , 7→ −    1 B(v) ϕt At  √2 − −  5 where t denotes the antidiagonal matrix transposition, ϕ 3 vt = v ,v ,v , ϕt = ϕ , (At) = A 3 2 1 2 ij 4 j,4 i − − (cid:2) (cid:3) ϕ  1 and v v 0 2 1 − B(v) =  v 0 v . 3 1 − 0 v v  3 − 2 The representation π is faithful, so we may use it to identify the Lie algebra g with its image under π where elements of g can be regarded as 7 7 matrices. Then the bilinear × form on g defined by 1 X,Y = trXY (2.10) h i 6 is symmetric and invariant. Note that this form is proportional to the standard normalized Killing form 1 tr adXadY , 2h ∨ (cid:0) (cid:1) where h = 4 is the dual Coxeter number for g. We have ∨ 1 X,Y = tr adX adY . h i 24 (cid:0) (cid:1) The additional scalar factor 1/3 is meant to simplify our formulas for Segal–Sugawara vectors by avoiding fractions. Let e EndC7 denote the standard matrix units. For all 1 6 i,j 6 7 set f = ij ij ∈ eij ej′i′, where i′ = 8 i. The following elements form a basis of g: − − f f , f f , f with 1 6 i,j 6 3 and i = j, 11 22 22 33 ij − − 6 together with 1 1 1 f14 f3′2, f24 f1′3, f34 f2′1, − √2 − √2 − √2 and 1 1 1 f41 f23′, f42 f31′, f43 f12′. − √2 − √2 − √2 In the general setting of Sec. 2.1, these elements are understood as the basis X1,...,X14. The elements X ,...,X of the dual basis with respect to the form (2.10) are then given 1 14 by the following expressions, where we use the corresponding capital letters to think of the X as abstract generators of U(g) rather than matrices: i 2F F F , F +F 2F , 3F with 1 6 i,j 6 3 and i = j, 11 22 33 11 22 33 ji − − − 6 6 together with 2F41 √2F23′, 2F42 √2F31′, 2F43 √2F12′, − − − and 2F14 √2F3′2, 2F24 √2F1′3, 2F34 √2F2′1. − − − Using (2.2), we can now define the entries G of the matrix G from the expansion ij 7 G = e G EndC7 U(g). (2.11) ji ij ⊗ ∈ ⊗ iX,j=1 In particular, G = 2F F F , G = 2F F F , G = 2F F F , 11 11 22 33 22 22 11 33 33 33 11 22 − − − − − − so that G + G + G = 0. Also, for all 1 6 i,j 6 3 with i = j we have G = 3F . 11 22 33 ij ij 6 Furthermore, G14 = 2F14 √2F3′2, G24 = 2F24 √2F1′3, G34 = 2F34 √2F2′1, − − − and G41 = 2F41 √2F23′, G42 = 2F42 √2F31′, G43 = 2F43 √2F12′. − − − TheremainingentriesofthematrixGaredeterminedbythesymmetrypropertiesGt = G − which give Gij +Gj′i′ = 0 together with G14 = √2G3′2, G24 = √2G1′3, G34 = √2G2′1, − − − and G41 = √2G23′, G42 = √2G31′, G43 = √2G12′. − − − Note that the above formulas define an explicit embedding of g into the orthogonal Lie algebra o7 spanned by the elements Fij = Eij Ej′i′, where the Eij denote the standard − basis elements of gl . An expression for the element Ω defined in (2.3) can be given by 7 3 3 3 Ω = 3 f f f f +2 f f +f f ij ji ii jj 4i i4 i4 4i ⊗ − ⊗ ⊗ ⊗ iX,j=1 iX,j=1 Xi=1 (cid:0) (cid:1) + (cid:9)1,2,3 f12′ ⊗f2′1 +f2′1 ⊗f12′ (2.12) (cid:0) (cid:1) −√2 (cid:9)1,2,3 f14 ⊗f23′ +f23′ ⊗f14 +f41 ⊗f3′2 +f3′2 ⊗f41 , (cid:0) (cid:1) where the symbol (cid:9) indicates the summation over cyclic permutations of the indices 1,2,3 1,2,3 keeping all other symbols, including primes, at their positions; that is, (cid:9)1,2,3 X1,2′,3 = X1,2′,3 +X3,1′,2 +X2,3′,1. Proposition 2.1 provides a matrix form (2.7) of the defining relations of U(g) with the elements G and Ω defined in (2.11) and (2.12). 7 2.3 A formula for Ω as an element of the centralizer algebra By its definition (2.3), the element Ω can be viewed as an operator Ω : V V V V. ⊗ → ⊗ It is easily seen that this operator commutes with the action of the Lie algebra g on V V ⊗ given by X π(X) 1+1 π(X). 7→ ⊗ ⊗ This implies that Ω must be a linear combination of the projections of V V onto its ⊗ irreducible components. As a representation of the Lie algebra sl , the tensor product C7 C7 of two copies of 7 ⊗ V = C7 splits into the direct sum of two irreducible components C7 C7 = Λ2(C7) S2(C7), ⊗ ⊕ afforded by the exterior and symmetric square of V. The canonical projections onto the irreducible components are given by the respective operators (1 P)/2 and (1 + P)/2, − where P is the permutation operator 7 P = e e . ij ji ⊗ iX,j=1 The restriction of the representation Λ2(C7) to the subalgebra o remains irreducible, 7 whereas the restriction of S2(C7) splits into two irreducible components; each of them remains irreducible under the further restriction to the subalgebra g o of type G : 7 2 ⊂ S2(C7) = V V . ∼ 0 ⊕ 2ω1 The respective projections are given by the operators Q/7 and (1+P)/2 Q/7, where Q − is the operator 7 Q = eij ei′j′. ⊗ iX,j=1 It is obtained by applying the antidiagonal transposition t : EndC7 EndC7, (eij)t = ej′i′, → to the first or the second component of the permutation operator, Q = Pt1 = Pt2. Here V is the trivial one-dimensional representation of g, and V is the 27-dimensional repre- 0 2ω1 sentation corresponding to the double of the first fundamental weight ω . 1 8 When restricted to g, the exterior square Λ2(C7) splits into two irreducible components Λ2(C7) = V V ∼ ω1 ⊕ ω2 of the respective dimensions 7 and 14, associated with the fundamental weights. The respective projections are given by the operators T/6 and (1 P)/2 T/6, where the − − operator T is defined by means of a 3-form on C7 as follows. The 3-form β is defined in the canonical basis e ,...,e of C7 by 1 7 3 β = ei e4 ei′ +√2e1 e2 e3 +√2e3′ e2′ e1′. ∧ ∧ ∧ ∧ ∧ ∧ Xi=1 One easily verifies that β is invariant under the action of g, which provides a g-module embegging C7 ֒ Λ2(C7) defined as contraction with β. Explicitly, the operator T can be → written as 7 7 T = β β e e , ika jla ij kl ⊗ i,jX,k,l=1Xa=1 where the coefficients of the form β are defined by 7 β = β e e e ijk i j k ⊗ ⊗ i,Xj,k=1 via the standard embedding Λ3(C7) ֒ (C7) 3 such that ⊗ → e e e = sgnσ e e e . i1 ∧ i2 ∧ i3 · iσ(1) ⊗ iσ(2) ⊗ iσ(3) σXS3 ∈ Thus, we have an expansion Ω = aΠ +bΠ +cΠ +dΠ 0 ω1 ω2 2ω1 into a linear combination of the projections to the respective irreducible components of the decomposition C7 C7 = V V V V . ⊗ ∼ 0 ⊕ ω1 ⊕ ω2 ⊕ 2ω1 The coefficients in the expansion are found by applying Ω to particular vectors to give the formula Ω = 1+P 2Q T. (2.13) − − Since Π , Π , Π andΠ arepairwise orthogonalprojections, wederive thefollowing 0 ω1 ω2 2ω1 relations for the operators P, Q and T. They commute pairwise and P2 = 1, Q2 = 7Q, T2 = 6T, PQ = Q, PT = T, QT = 0. (2.14) − 9 By (2.13) the matrix form (2.7) of the defining relations implies a uniform expression for the commutators of the generators of g, [Gij,Gkl] = δkjGil δilGkj 2δki′Gj′l +2δj′lGki′ − − 7 + β β G β β G . (2.15) iab jlb ka ikb jab al − aX,b=1(cid:16) (cid:17) Remark 2.3. It is known by Ogievetsky [22] that a rational R-matrix associated with g can be given by the formula P 2Q T R(u) = 1 + + . − u u 6 u 4 − − Its expansion into a power series in u 1 takes the form − R(u) = 1 (Ω 1)u 1+... − − − so that the classical limit of the RTT relations defining the Yangian Y(g) reproduces the defining relations (2.7) for U(g); see [6]. 2.4 Isomorphism with the Chevalley presentation The simple Lie algebra g of type G is associated with the Cartan matrix A = [a ], 2 ij 2 1 A = − . (cid:20) 3 2(cid:21) − Its Chevalley presentation is defined by generators e ,h ,f with i = 1,2, subject to the i i i defining relations [e ,f ] = δ h , [h ,h ] = 0, i j ij i i j [h ,e ] = a e , [h ,f ] = a f , i j ij j i j ij j − together with (the Serre relations) (ade )2e = 0, (ade )4e = 0, (adf )2f = 0, (adf )4f = 0. 1 2 2 1 1 2 2 1 We let α and β denote the simple roots. The set of positive roots is α, β, α+β, α+2β, α+3β, 2α+3β. For each positive root γ we let e and f denote the root vectors associated with γ and γ, γ γ − respectively. In particular, e = e and e = e . We have the triangular decomposition 1 α 2 β g = n h n , (2.16) + − ⊕ ⊕ 10