Acta Polytechnica doi:10.14311/AP.2013.53.0462 Acta Polytechnica 53(5):462–469, 2013 © Czech Technical University in Prague, 2013 available online at http://ojs.cvut.cz/ojs/index.php/ap gln+1 ALGEBRA OF MATRIX DIFFERENTIAL OPERATORS AND MATRIX QUASI-EXACTLY-SOLVABLE PROBLEMS Yuri F. Smirnov (deceased), Alexander V. Turbiner∗ Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 México, D.F., Mexico ∗ corresponding author: turbiner@nucleares.unam.mx Abstract. The generators of the algebra gln+1 in the form of differential operators of the first order acting on Rn with matrix coefficients are explicitly written. The algebraic Hamiltonians for matrix generalization of 3−body Calogero and Sutherland models are presented. Keywords: algebra of differential operators, exactly-solvable problems. Submitted: 13 May 2013. Accepted: 6 June 2013. 1. Introduction This work has a certain history related to Miloslav Havlicek. On the important occasion of Miloslav’s 75th birthday, we think this story should be re- vealed. About 25 years ago, when quasi-exactly- solvable Schroedinger equations with the hidden alge- bra sl2 were discovered [1], one of the present authors (AVT) approached Israel M. Gelfand and asked about the existence of the algebra gln+1 of matrix differ- ential operators. Instead of giving an answer, Israel Moiseevich said that M. Havlicek knows the answer and that he must be asked. A set of Dubna preprints was given (see [2, 3] and reference therein). Then AVT studied them for many years, at first separately and then together with the first author (YuFS), who also happened to have the same set of preprints. The results of these studies are presented below. While carrying out these studies, we always kept in mind that a constructive answer exists and is known to Miloslav. Thus, we are certain that at least some of results presented here are known to Miloslav. Hav- ing difficulty to understand what is written in the texts we did not know what he really knew, and were therefore unable to indicate it in our text. Our main goal is to find a mixed representation of the algebra gln+1 which contains both matrices and differential operators in a non-trivial way. Then to generalize it to a polynomial algebra which we call g(m) (see below, Section 4). Another goal is to apply the obtained rep- resentations for a construction of the algebraic forms of (quasi)-exactly-solvable matrix Hamiltonians. 2. The algebra gln in mixed representation Let us take the algebra gln and consider the vector field representation Ẽij = xi∂j, i,j = 1, . . .n,x ∈ Rn. (1) It obeys the canonical commutation relations [Ẽij, Ẽkl] = δjkẼil − δilẼkj. (2) On the other hand, let us consider another representa- tion Mpm, p,m = 1, . . . ,n of the algebra gln in terms of some operators (matrix, finite-difference, etc) with the condition that all ‘cross-commutators’ between these two representations vanish [Ẽij,Mpm] = 0. (3) Let us choose Mpm to obey the canonical commutation relations [Mij,Mkl] = δjkMil − δilMkj, (4) (cf. (2)). It is evident that the sum of these two representations is also the representation, Eij ≡ Ẽij + Mij ∈ gln. (5) Now we consider an embedding of gln ⊂ gln+1 try- ing to complement the representation (1) of the alge- bra gln up to the representation of the algebra gln+1. In principle, this can be done due to the existence of the Weyl-Cartan decomposition, gln+1 = L⊕ (gln ⊕ I) ⊕U with the property gln+1 = L o (gln ⊕ I) n U, (6) where L(U) is the commutative algebra of the lowering (raising) generators with the property [L,U] = gln⊕I. Thus, it realizes a property of the Gauss decomposition of gln+1. It is worth emphasizing that dim(L) = dim(U) = n. Obviously, the lowering generators (of negative grad- ing) from L can be given by derivations T −i = ∂i, i = 1, . . . ,n, ∂i ≡ ∂ ∂xi , (7) (see e.g. [5]) when assuming that all commutators [T −i ,Mpm] = 0, (8) 462 http://dx.doi.org/10.14311/AP.2013.53.0462 http://ojs.cvut.cz/ojs/index.php/ap vol. 53 no. 5/2013 gln+1 Algebra of Matrix Differential Operators vanish. This probably implies that the only possible choice for Mpm exists when they are either given by matrices or act in a space which is a complement to x ∈ Rn. It is easy to check that [Eij,T −k ] = −δikT − j . Now we have to add the Euler-Cartan generator of the gln algebra, see (6) −E0 = n∑ j=1 xj∂j −k, (9) where k is arbitrary constant. Raising generators from U are chosen as −T +i = −xiE0 + n∑ j=1 xjMij = xi ( n∑ j=1 xj∂j −k ) + n∑ j=1 xjMij, i = 1, . . . ,n. (10) (cf. for instance [5]). Needless to say that one can check explicitly that T −i , Eij, E0, T + i span the algebra gln+1. In particular, [E,T +] = T +, and [T +i ,T − j ] = Eii −δijE0. If parameter k takes non-negative integer the alge- bra gln+1 spanned by the generators (5), (7), (9), (10) appears in a finite-dimensional representation. There exists a linear finite-dimensional space of polynomials of finite-order in the space of columns/spinors of finite length which is a common invariant subspace for all generators (5), (7), (9), (10). This finite-dimensional representation is irreducible. The non-negative integer parameter k has the mean- ing of the length of the first row of the Young tableau of gln+1, describing a totally symmetric representa- tion (see below). All other parameters are coded in Mij, which corresponds to an arbitrary Young tableau of gln. Thus, we have some peculiar splitting of the Young tableau. Each representation is characterized by the Gelfand- Tseitlin signature, [m1,n, . . .mnn], where min ≥ mi+1,n and their difference is positive integer. Each basic vector is characterized by the Gelfand-Tseitlin scheme. An explicit form of the representation is given by the Gelfand-Tseitlin formulas [4]. It can be demonstrated that all Casimir operators of gln+1 in this realization (5), (7), (9), (10) are expressed in Mij, and thus do not depend on x. They coincide with the Casimir operators of the gln-subalgebra real- ized by matrices Mij. 3. Example: the algebra gl3 in mixed representation In the case of the algebra gl3, the generators (5), (7), (9), (10) take the form E11 = x1∂1 + M11, E22 = x2∂2 + M22, E12 = x1∂2 + M12, E21 = x2∂1 + M21, E0 = k −x1∂1 −x2∂2, T −1 = ∂1, T − 2 = ∂2, T +1 = x1(k −x1∂1 −x2∂2) −x1M11 −x2M12, T +2 = x2(k −x1∂1 −x2∂2) −x1M21 −x2M22. (11) The Casimir operators of gl3 in this realization are given by C1 = E11 + E22 + E0 = k + M11 + M22 = k + C1(M), C2 = E12E21 + E21E12 + T +1 T − 1 + T − 1 T + 1 + T + 2 T − 2 + T −2 T + 2 + E 2 11 + E 2 22 + E 2 0 = k(k + 2) + M211 + M 2 22 + M12M21 + M21M12 −M11 −M22 = k(k + 2) + C2(M) −C1(M), and, finally, C3 = − 1 2 C31 + 3 2 C1C2 + 3C2 − 2C21 − 2C1. In this realization, the Casimir operator C3 is alge- braically dependent on C1 and C2. In fact, C1 and C2 are nothing but the Casimir operators of the gl2 sub-algebra. Therefore, the center of the gl3 universal enveloping algebra in realization (11) is generated by the Casimir operators of the gl2 sub-algebra realized by Mij. Thus, it seems natural that these reps are irreducible. Now we consider concrete matrix realizations of the gl2-subalgebra in our scheme. 3.1. Reps in 1 × 1 matrices This corresponds to the trivial representation of gl2, M11 = M12 = M21 = M22 = 0. This is [k, 0] or, in other words, a symmetric represen- tation (the Young tableau has two rows of length k and 0, correspondingly). We also can call it a scalar representation, since the generators E11 = x1∂1, E22 = x2∂2, E12 = x1∂2, E21 = x2∂1, E0 = k −x1∂1 −x2∂2, T −1 = ∂1, T − 2 = ∂2, T +1 = x1(k −x1∂1 −x2∂2), T +2 = x2(k −x1∂1 −x2∂2), (12) act on one-component spinors or, in other words, on scalar functions (see e.g. [5]). The Casimir operators are: C1 = k, C2 = k(k + 2). 463 Yu. F. Smirnov, A. V. Turbiner Acta Polytechnica If parameter k takes non-negative integer the algebra gl3 spanned by the generators (12) appears in finite- dimensional representation. Its finite-dimensional rep- resentation space is a space of polynomials Pk,0 = 〈 x1 p1x2 p2 ∣∣ 0 ≤ p1 +p2 ≤ k〉, k = 0, 1, 2, . . . . (13) Namely in this representation (12), the algebra gl3 appears as the hidden algebra of the 3-body Calogero and Sutherland models [5], BC2 rational and trigono- metric, and G2 rational models [6] and even of the BC2 elliptic model [7]. 3.2. Reps in 2 × 2 matrices Take gl2 in two-dimensional reps by 2 × 2 matrices, M11 = ( 1 0 0 0 ) , M22 = ( 0 0 0 1 ) , M12 = ( 0 1 0 0 ) , M21 = ( 0 0 1 0 ) , Then the generators (11) of gl3 are: T −1 = ( ∂1 0 0 ∂1 ) , T −2 = ( ∂2 0 0 ∂2 ) , E11 = ( x1∂1+1 0 0 x1∂1 ) , E12 = ( x1∂2 1 0 x1∂2 ) , E21 = ( x2∂1 0 1 x2∂1 ) , E22 = ( x2∂2 0 0 x2∂2+1 ) , E0 = ( A 0 0 A ) , T +1 = ( x1(A− 1) −x2 0 x1A ) , T +2 = ( x2A 0 −x1 x2(A− 1) ) , (14) where A = k−x1∂1−x2∂2. This is [k, 1]-representation (the Young tableau has two rows of length k and 1, correspondingly), and their Casimir operators are: C1 = k + 1, C2 = (k + 1)2. If parameter k takes non-negative integer the algebra gl3 spanned by the generators (14) appears in finite- dimensional representation. Let us consider several different values of k in detail. The case k = 1. Then three-dimensional represen- tation space V (2)1 appears to be spanned by: P− = [ 0 1 ] , P+ = [ 1 0 ] , Y1 = [ x2 −x1 ] . (15) This corresponds to antiquark multiplet in standard (fundamental) representation. The Newton polygon is a triangle with points P± as vortices at the base. Figure 1. Newton hexagon for the representation space V (2)4 of the [4, 1]-representation of dimension 24. The case k = 2. Then eight-dimensional represen- tation space V (2)2 appears to be spanned by: P− = [ 0 1 ] , P+ = [ 1 0 ] , P (1) − = [ 0 x2 ] , Y (1) 1 = [ 0 x1 ] , Y (2) 1 = [ x2 0 ] , P (1) + = [ x1 0 ] , Y2 = [ x22 −x1x2 ] , Y3 = [ x1x2 −x21 ] . (16) This corresponds to octet in standard (fundamental) representation. Space V (2)2 contains V (2) 1 as a sub- space, V (2)1 ⊂ V (2) 2 . It should be mentioned that Y1 = −Y (1) 1 + Y (2) 1 . Now the Newton polygon is a hexagon where the central point is doubled, being presented by Y (1,2)1 , and the lower (upper) base has length two being given by P± (Y2,3). The case k = 3. The representation space V (2)3 is 15-dimensional. In addition to P±,P (1) ± and Y (1,2) 1 (see (15) and (16)), it contains several vectors more, namely, P (2) − = [ 0 x22 ] , P (2) + = [ x21 0 ] , (17) which are situated on the ±-sides of the Newton hexagon, doubling the points corresponding to Y2,3 (see (16)) Y (1) 2 = [ 0 x1x2 ] , Y (2) 2 = [ x22 0 ] , Y (1) 3 = [ 0 x21 ] , Y (2) 3 = [ x1x2 0 ] , (18) plus three extra vectors on the boundary Y8 = [ x32 −x1x22 ] , Y9 = [ x1x 2 2 −x21x2 ] , Y10 = [ x21x2 −x31 ] . (19) It is clear that V (2)1 ⊂ V (2) 2 ⊂ V (2) 3 . All internal points of the Newton hexagon are double points, while the points on the boundary are single ones. 464 vol. 53 no. 5/2013 gln+1 Algebra of Matrix Differential Operators The general case. The finite-dimensional repre- sentation space V (2)k has dimension k(k + 2) and is presented by the Newton hexagon, which contains (k + 1) horizontal layers. The lower base has length two, while the upper base has length k (see Fig. 1 as an illustration for k = 4). All internal points of the Newton hexagon are double points, while the points on the boundary are single ones. Except for k vectors of the last (highest) layer of the Newton hexagon, the remaining k(k + 1) vectors span the space of all pos- sible two-component spinors with components given by the inhomogeneous polynomials in x1,x2 of degree not higher than (k − 1). We denote this space as Ṽ (2) k ⊂ V (2) k . The non-trivial task is to describe k vec- tors of the last (highest) layer of the hexagon. After some analysis one can find that they have the form Yk(k+1)+i = [ xk−i2 x i 1 −xk−i−12 x i+1 1 ] , i = 0, 1, 2, . . . , (k − 1), (20) hence they span a non-trivial k-dimensional subspace of spinors with components given by specific homoge- neous polynomials of degree k. 3.3. Reps in 3 × 3 matrices Take gl2 in three-dimensional reps by 3 × 3 matrices, M11 = 2 0 00 1 0 0 0 0 , M22 = 0 0 00 1 0 0 0 2 , M12 = 0 √ 2 0 0 0 √ 2 0 0 0 , M21 = 0 0 0√2 0 0 0 √ 2 0 . Then the generators (11) of gl3 are: T −1 = ∂1 0 00 ∂1 0 0 0 ∂1 , T −2 = ∂2 0 00 ∂2 0 0 0 ∂2 , E11 = x1∂1 + 2 0 00 x1∂1 + 1 0 0 0 x1∂1 , E12 = x1∂2 √ 2 0 0 x1∂2 √ 2 0 0 x1∂2 , E21 = x2∂1 0 0√2 x2∂1 0 0 √ 2 x2∂1 , E22 = x2∂2 0 00 x2∂2 + 1 0 0 0 x2∂2 + 2 , E0 = A 0 00 A 0 0 0 A , T +1 = x1(A− 2) − √ 2x2 0 0 x1(A− 1) − √ 2x2 0 0 x1A , T +2 = x2A 0 0−√2x1 x2(A− 1) 0 0 − √ 2x1 x2(A− 2) , (21) where A = k−x1∂1−x2∂2. This is [k, 2]-representation (the Young tableau has two rows of length k and 2, correspondingly) and their Casimir operators are: C1 = k + 2, C2 = (k + 1)2 + 3. As an illustration let us explicitly show finite- dimensional representation spaces for k = 2, 3. The case k = 2. Then the six-dimensional repre- sentation space V (3)2 appears to be spanned by: P− = 00 1 , P0 = 01 0 , P+ = 10 0 , Y1 = 0x2 − √ 2x1 , Y2 = − √ 2x2 x1 0 , Y3 = x22−√2x1x2 x21 . (22) This corresponds to ‘di-antiquark’ multiplet. The case k = 3. Then 15-dimensional representa- tion space V (3)3 appears to be spanned by: P− = 00 1 , P0 = 01 0 , P+ = 10 0 , Y (1) 1 = 0x2 0 , Y (2)1 = 00 x1 , Y (1)2 = x20 0 , Y (2) 2 = 0x1 0 , P (1)− = 00 x2 , P (1)+ = x10 0 , Y (1) 3 = − √ 2x22 x1x2 0 , Y (2)3 = 0x1x2 − √ 2x21 , Y4 = 0−√2x22 2x1x2 , Y5 = 2x1x2−√2x21 0 , Y6 = x32−√2x1x22 x21x2 , Y7 = x1x22−√2x21x2 x31 . (23) It is worth mentioning that as a consequence of a particular realization of the generators (11) of the gl3 algebra there exist a certain relations between generators other than those given by the Casimir operators. The first observation is that there are no linear relations between generators of such a type. Some time ago nine quadratic relations were found 465 Yu. F. Smirnov, A. V. Turbiner Acta Polytechnica m s0 1/2 1 3/2 2 5/2 3 l/2 Figure 2. Verma module with the lowest weight (057) for s = 5/2. between gl3 generators taken in scalar representation (12) other than Casimir operators [8]. Surprisingly, certain modifications of these relations also exist for [kn] mixed representations (11), −T +1 E22 + T + 2 E12 = x1 [ M22x1∂1 + M11x2∂2 + (M11 −k)M22 −M21M12 ] −x2(x1∂1 −k − 1)M12 −M21x21∂2 ≡−T̃ + 1 , (24) −T +2 E11 + T + 1 E21 = x2 [ M22x1∂1 + M11x2∂2 + (M22 −k)M11 −M12M21 ] −x1(x2∂2 −k − 1)M21 −M12x22∂1 ≡−T̃ + 2 , (25) −E12(E0 + 1) + T +1 T − 2 = M12(x1∂1 −k − 1) −M11x1∂2 ≡−Ẽ12, (26) −E21(E0 + 1) + T +2 T − 1 = M21(x2∂2 −k − 1) −M22x2∂1 ≡−Ẽ21, (27) T +1 T − 1 −E11(1 + E0) = M11x2∂2 −M12x2∂1 − (k + 1)M11 ≡−Ẽ11, (28) T +2 T − 2 −E22(1 + E0) = M22x1∂1 −M21x1∂2 − (k + 1)M22 ≡−Ẽ22, (29) E12E21 −E11E22 −E11 = M12x2∂1 + M21x1∂2 −M22x1∂1 −M11x2∂2 + M12M21 −M11M22 −M11 ≡−Ê11, (30) E22T − 1 −E21T − 2 = M22∂1 −M21∂2 ≡ T̃ − 1 , (31) E12T − 1 −E11T − 2 = M12∂1 −M11∂2 ≡−T̃ − 2 . (32) Not all these relations are independent. It can be shown that one relation is linearly dependent, since the sum of (28) + (29) + (30) gives the second Casimir operator C2. In scalar case, at least, we can assign a natural (vec- torial) grading to the generators. The above relations also reflect a certain decomposition of the gradings, (1, 0)(0, 0) = (0, 1)(1,−1) (0, 1)(0, 0) = (1, 0)(−1, 0) for the first two relations, (1,−1)(0, 0) = (1, 0)(0,−1) (−1, 1)(0, 0) = (0, 1)(−1, 0) for the second two, (1, 0)(−1, 0) = (0, 0)(0, 0) (0, 1)(0,−1) = (0, 0)(0, 0) (1,−1)(−1, 1) = (0, 0)(0, 0) for three before the last two, and (0, 0)(−1, 0) = (−1, 1)(0,−1) (0, 0)(0,−1) = (1,−1)(−1, 1) for the last two. 4. Algebra g(m) in mixed representation The basic property which was used to construct the mixed representation of the algebra gln+1 is the ex- istence of the Weyl-Cartan decomposition gln+1 = L⊕ (gln ⊕ I) ⊕U with property (6). One can pose a question about the existence of other algebras than gln+1 for which the Weyl-Cartan decomposition with property (6) holds. The answer is affirmative. Let us consider the important particular case of the Cartan 466 vol. 53 no. 5/2013 gln+1 Algebra of Matrix Differential Operators algebra gl2 ⊕ I, and construct a realization of a new algebra denoted g(m) with the property g(m) = Lm+1 o (gl2 ⊕ I) n Um+1, (33) where Lm(Um) is the commutative algebra of the lowering (raising) generators with the property [Lm,Um] = Pm−1(gl2 ⊕I) with Pm−1 as a polynomial of degree (m − 1) in generators of gl2 ⊕ I. Thus, it realizes a property of the generalized Gauss decompo- sition. The emerging algebra is a polynomial algebra. It is worth emphasizing that the realization we are go- ing to construct appears at dim(Lk) = dim(Um) = m. For m = 1 the algebra g(1) = gl3, see (6). Our final goal is to build the realization of (33) in terms of finite order differential operators acting on the plane R2. The simplest realization of the algebra gl2 by dif- ferential operators in two variables is the vector field representation, see (1) at n = 2. Exactly this rep- resentation was used to construct the representation of the gl3 algebra acting of R2, see (11), (12). In this case dim(Lm) = dim(Um) = 2. We are unable to find other algebras with dim(Lm) = dim(Um) > 2. However, there exists another representation of the algebra gl2 by the first order differential operators in two variables, J̃12 = ∂x, J̃ (k) 11 = −x∂x + k 3 , J̃ (k) 22 = −x∂x + sy∂y, J̃ (k) 21 = x 2∂x + sxy∂y −kx, (34) (see S. Lie, [9] at k = 0 and A. González-Lopéz et al, [10] at k 6= 0 (Case 24)), where s,k are arbitrary numbers. These generators obey the standard commu- tation relations (2) of the algebra gl2 in the vector field representation (1). It is evident that the sum of the two representations, J̃ij and the matrix representation Mij, is also a representation, Jij ≡ J̃ij + Mij ∈ gl2. (35) (cf. (5)). It is worth mentioning that the gl2 algebra commutation relations for Mpm are taken in a canon- ical form (4). The unity generator I in (33) is written in the form of a generalized Euler-Cartan operator J (k) 0 = x∂x + sy∂y −k. (36) Now let us assume that s is non-negative integer, s = m, m = 0, 1, 2, . . .. Evidently, the lowering gener- ators (of negative grading) from Lm+1 can be given by T −i = x i∂y, i = 0, 1, . . . ,m, (37) forming commutative algebra [T −i ,T − j ] = 0. (38) (cf. [9, 10]). Eventually, the generators of the algebra (gl2 ⊕ I) n Lm+1 take the form J12 = ∂x + M12, J (k) 11 = −x∂x + k 3 + M11, J (k) 22 = −x∂x + my∂y + M22, J (k) 21 = x 2∂x + mxy∂y −kx + M21, (39) with J(k)0 and T − i given by (36) and (37), respectively. Let us consider two particular cases of the general construction of the raising generators for the commu- tative algebra U. Case 1. For the first case we take the trivial matrix representation of the gl2, M11 = M12 = M21 = M22 = 0. One can check that one of the raising generators is given by U0 = y∂mx , (40) while all other raising generators are multiple commu- tators of J(k)21 with U0, Ui ≡ [J (k) 21 , [J (k) 21 , [· · ·J (k) 21 ,T0] · · · ] ]︸ ︷︷ ︸ i = y∂m−ix J (k) 0 (J (k) 0 + 1) . . . (J (k) 0 + i− 1) , (41) at i = 1, . . .m. All of them are differential operators of fixed degree m. The procedure for construction of the operators Ui has the property of nilpotency: Ui = 0, i > m. In particular, for m = 1, U0 = y∂x, U1 = yJ (k) 0 = y(x∂x + y∂y −k). Inspecting the generators T −0,1, Jij, J (n), U0,1 one can see that they span the algebra gl3, see (12). Hence, the algebra g(1) ≡ gl3. If parameter k takes non-negative integer the alge- bra g(m) spanned by the generators (39), (40), (41) appears in finite-dimensional representation. Its finite- dimensional representation space is a triangular space of polynomials Pk,0 = 〈 xp1yp2 ∣∣ 0 ≤ p1 + mp2 ≤ k〉, k = 0, 1, 2, . . . . (42) Namely in this representation, the algebra g(m) ap- pears as a hidden algebra of the 3-body G2 trigono- metric model [6] at m = 2 and of the so-called TTW model at integer m, in particular, of the dihedral I2(m) rational model [11]. 467 Yu. F. Smirnov, A. V. Turbiner Acta Polytechnica Case 2. The second case is a certain evident exten- sion when generators Mij are of an arbitrary matrix representation of the algebra gl2. Raising generators (40), (41) remain raising generators even if Cartan generators are given by (39) with arbitrary Mij ∈ gl2. However, the algebra is not closed: [T,U] 6= P (gl2⊕I). It can be fixed, at least, for the case m = 1. If Mij are generators of gl2 subalgebra of gl3. By adding to T,U generators (38), (40), (41) the appropriate matrix gen- erators from gl3, the algebra gets closed. We end up with the gl3 algebra of matrix differential operators other than (11). We are not aware of a solution to this problem for the case of m 6= 1 except for the case of trivial matrix representation, see Case 1. 5. Extension of the 3-body Calogero Model The first algebraic form for the 3-body Calogero Hamil- tonian [12] appears after gauge rotation with the ground state function, separation of the center-of- mass and changing the variables to elementary sym- metric polynomials of the translationally-symmetric coordinates [5], hCal = −2τ2∂2τ2τ2 − 6τ3∂ 2 τ2τ3 + 2 3 τ22 ∂ 2 τ3τ3 − [ 4ωτ2 + 2(1 + 3ν) ] ∂τ2 − 6ωτ3∂τ3. (43) These new coordinates are polynomial invariants of the A2 Weyl group. Its eigenvalues are − �p = 2ω(2p1 + 3p2), p1,2 = 0, 1, . . . . (44) As is shown in Ruhl and Turbiner [5], the operator (43) can be rewritten in a Lie-algebraic form in terms of gl(3)-algebra generators of the representation [k, 0]. The corresponding expression is hCal = −2E11T −1 − 6E22T − 1 + 2 3 E12E12 − 4ωE11 − 2(1 + 3ν)T −1 − 6ωE22 . (45) Now we can substitute the generators of the represen- tation [k,n] in the form (11) h̃Cal = −2τ2∂2τ2τ2 − 6τ3∂ 2 τ2τ3 + 2 3 τ22 ∂ 2 τ3τ3 − 2 [ 2ωτ2 + (1 + 3ν) + (n− 2M22) ] ∂τ2 − ( 6ωτ3 − 4 3 M12τ2 ) ∂τ3 + 2 3 M12M12 − 4ωn− 2ωM22. (46) This is an n×n matrix differential operator. It con- tains infinitely many finite-dimensional invariant sub- spaces which are nothing but finite-dimensional repre- sentation spaces of the algebra gl(3). This operator remains exactly-solvable with the same spectra as the scalar Calogero operator. This operator probably remains completely inte- grable. A higher-than-second-order integral is the differential operator of the sixth order (ω 6= 0) or of the third order (ω = 0), which takes an algebraic form after gauging away the ground state function in τ coordinates. It can be rewritten in terms of the gl(3)- algebra generators of the representation [k, 0], which then can be replaced by the generators of the repre- sentation [k,n]. Under such a replacement the spectra of the integral remain unchanged and algebraic. 6. Extension of the 3-body Sutherland Model The first algebraic form for the 3-body Sutherland Hamiltonian [13] appears after gauge rotation with the ground state function, separation of the center-of-mass and changing the variables to elementary symmetric polynomials of the exponentials of translationally- symmetric coordinates [5], hSuth = − ( 2η2 + α2 2 η22 − α4 24 η23 ) ∂2η2η2 − ( 6 + 4α2 3 η2 ) η3∂ 2 η2η3 + (2 3 η22 − α2 2 η23 ) ∂2η3η3 − [ 2(1+3ν)+2 ( ν+ 1 3 ) α2η2 ] ∂η2−2 ( ν+ 1 3 ) α2η3∂η3, (47) where α is the inverse radius of the circle on which the bodies are situated. These new coordinates are fundamental trigonometric invariants of the A2 Weyl group. As shown in [5], operator (47) can be rewritten in a Lie-algebraic form in terms of the gl(3)-algebra generators of the representation [k, 0], hSuth = −2E11T −1 − 6E22T − 1 + 2 3 E12E12 −2(1 + 3ν)T −1 + α4 24 E21E21− α2 6 [ 3E11E11 + 8E11E22 + 3E22E22 + (1 + 12ν)(E11 + E22) ] . (48) Now we can substitute the generators of the represen- tation [k,n] in the form (11) h̃Suth = − ( 2η2 + α2 2 η22 − α4 24 η23 ) ∂2η2η2 − ( 6 + 4α2 3 η2 ) η3∂ 2 η2η3 + (2 3 η22 − α2 2 η23 ) ∂2η3η3 − 2 [ (1 + 3ν) + ( ν + 1 3 ) α2η2 + (n− 2M22) ] ∂η2 + α4 24 M21η3∂η2 + [ 2 ( ν + 1 3 ) α2η3 − 4 3 M12η2 ] ∂η3 − α2 3 [ 3n(η2∂η2 + η3∂η3 ) + M11η3∂η3 + M22η2∂η2 ] + 2 3 M12M12 + α4 24 M21M21 − α2 6 [ 2M11M22 + (1 + 12ν + 3n)n ] . (49) 468 vol. 53 no. 5/2013 gln+1 Algebra of Matrix Differential Operators This is an n×n matrix differential operator. It con- tains infinitely-many finite-dimensional invariant sub- spaces which are nothing but finite-dimensional repre- sentation spaces of the algebra gl(3). This operator remains exactly-solvable with the same spectra as the scalar Sutherland operator. The operator (49) probably remains completely in- tegrable. A non-trivial integral is the differential op- erator of the third order, it takes the algebraic form after gauging away the ground state function in η coordinates. It can be rewritten in terms of the gl(3)- algebra generators of the representation [k, 0], which then can be replaced by the generators of the repre- sentation [k,n]. Under such a replacement the spectra of the integral remain unchanged and algebraic. 7. Conclusions The algebra gln of differential operators plays the role of a hidden algebra for all An,Bn,Cn,Dn,BCn Calogero-Moser Hamiltonians, both rational and trigonometric, with the Weyl symmetry of classical root spaces (see [14] and references therein). We have described a procedure which, in our opinion, should carry the name of the Havlicek procedure, to construct the algebra gln of the matrix differential operators. The procedure is based on a mixed, matrix-differential operator realization of the Gauss decomposition dia- gram. As for Hamiltonian reduction models with the ex- ceptional Weyl symmetry group G2,F4,E6,7,8, both rational and trigonometric, there exist hidden alge- bras of differential operators (see [14] and references therein). All these algebras are infinite-dimensional but finitely-generated. For generating elements of these algebras an analogue of the Weyl-Cartan de- composition exists but in the Gauss decomposition diagram, a commutator of the lowering and raising generators is a polynomial of the higher-than-one or- der in the Cartan generators. Matrix realizations of these algebras surely exist. Thus, the above mentioned procedure for building the mixed representations can be realized. It may lead to a new class of matrix exactly-solvable models with exceptional Weyl sym- metry. Acknowledgements It was planned long ago to dedicate this text to Miloslav Havlicek who has always been deeply respected by both authors as a scientist and also as a citizen. The text is based mainly on notes jointly prepared by two authors. It does not include results of the authors obtained separately (except for Section 4) and which the authors had no chance to discuss. Thus, the text will appear somehow incomplete. When the first author (YuFS) passed away, it took years for the second author (AVT) to return to the subject due to sad memories. Even now, almost a decade after the death of Yura Smirnov, the preparation of this text was quite difficult for AVT. AVT thanks CRM, Montreal for their kind hospitality extended to him. A part of this work was done there during his numerous visits. AVT is grateful to J C Lopez Vieyra for taking the interest in the work and for technical assistance. This work was supported in part by the Uni- versity Program FENOMEC, by PAPIIT grant IN109512, and by CONACyT grant 166189 (Mexico). References [1] A.V. Turbiner. Quasi-Exactly-Solvable Problems and the SL(2, R) Group, Comm.Math.Phys. 118: 467-474, 1988. [2] C. Burdik. Realisations of the real semisimple Lie algebras: a method of construction, J. Phys. A18: 3101-3111, 1985. [3] C. Burdik, M. Havlicek. Boson realization of the semi-simple Lie algebras, in Symmetry in physics: in memory of Robert T. Sharp, CRM Proceeding 2004, vol.34, p.87-98, edited by R.T. Sharp, P. Winternitz. [4] I.M. Gelfand and M.L. Tsetlin. Finite-dimensional representations of groups of orthogonal matrices, Dokl. Akad. Nauk SSSR 71: 1017–1020, 1950 (in Russian) English transl. in: I.M. Gelfand, Collected papers. Vol II, Berlin: Springer-Verlag 1988, pp. 657–661. [5] W. Rühl and A. V. Turbiner. Exact solvability of the Calogero and Sutherland models, Mod. Phys. Lett. A10: 2213–2222, 1995 arXiv:hep-th/9506105. 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From quantum AN (Calogero) to H4 (rational) model, SIGMA 7: 071, 2011; 20 pp From quantum AN (Sutherland) to E8 trigonometric model: space-of-orbits view, SIGMA 9: 003, 2013; 25 pp 469 http://arxiv.org/abs/hep-th/9506105 http://arxiv.org/abs/funct-an/9301001 http://arxiv.org/abs/0904.0738 Acta Polytechnica 53(5):462–469, 2013 1 Introduction 2 The algebra gl_n in mixed representation 3 Example: the algebra gl_3 in mixed representation 3.1 Reps in 1x1 matrices 3.2 Reps in 2x2 matrices 3.3 Reps in 3x3 matrices 4 Algebra g^(m) in mixed representation 5 Extension of the 3-body Calogero Model 6 Extension of the 3-body Sutherland Model 7 Conclusions Acknowledgements References