$gl_{n+1}$ algebra of Matrix Differential Operators and Matrix Quasi-exactly-solvable Problems

The generators of the algebra $gl_{n+1}$ in a form of differential operators of the first order acting on ${\bf R}^n$ with matrix coefficients are explicitly written. The algebraic Hamiltonians for a matrix generalization of $3-$body Calogero and Sutherland models are presented.


Dedicated to Miloslav Havlicek on the occasion of his 75th birthday
Introduction This work has a certain history related to Miloslav Havlicek. Due to an important occasion of Miloslav's 75th birthday we think this story has to be revealed. About 25 years ago when quasi-exactly-solvable Schroedinger equations with the hidden algebra sl 2 were discovered [1] one of the present authors (AVT) approached to Israel M Gelfand and asked about the existence of the algebra gl n+1 of the matrix differential operators. Instead of giving an answer Israel Moiseevich said that M Havlicek knows the answer and he must be asked. A set of the Dubna preprints was given (see [2]- [3] and reference therein). Then AVT studied them for many years firstly separately and then together with YuFS who also happened to have the same set of preprints. The results of those studies are presented below. Doing these studies we always kept in mind that the constructive answer exists and is known to Miloslav. Thus, we are certain, at least, some of presented results are known to Miloslav. We were simply unable to understand and then indicate where they can be found. Our main goal is to find a mixed representation of the algebra gl n+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). While the another goal is to apply obtained representations for a construction of the algebraic forms of (quasi)-exactly-solvable matrix Hamiltonians.

I. THE ALGEBRA gl n+1 IN MIXED REPRESENTATION
Let us take the algebra gl n and consider the vector field representatioñ It obeys the canonical commutation relations On the other hand, let us consider another representation M pm , p, m = 1, . . . n of the algebra gl n in terms of some operators (matrix, finite-difference etc) with a condition that all 'cross-commutators' between these two representations vanish Let us choose M pm to obey canonical commutation relations (cf. (2)). It is evident that the sum of these two representations is also representation, Now we consider an embedding of gl n ⊂ gl n+1 trying to complement the representation (1) of the algebra gl n up to the representation of the algebra gl n+1 . In principle, it can be done due to the existence of the Weyl-Cartan decomposition, where L(U) is the commutative algebra of the lowering (raising) generators with a property [L, U] = gl n ⊕ I. Thus, it realizes a property of the Gauss decomposition of gl n+1 . It is worth emphasizing that the dim(L) = dim(U) = n.
Obviously, the lowering generators (of negative grading) from L can be given by derivations (see e.g. [5]) when assuming that all commutators vanish. Likely, it implies that the only possible choice for M pm exists when they are given either by matrices or act in a space which is a complement to the x ∈ R n . It is easy to check that Now we have to add the Euler-Cartan generator of the gl n algebra, see (6) where k is arbitrary constant. Raising generators from U are chosen as (cf. e.g. [5]). Needless to say that one can check explicitly that T − i , E ij , E 0 , T + i span the algebra gl n+1 . In particular, If the parameter k takes non-negative integer the algebra gl n+1 spanned by the generators (5), (7), (9) -(10) appears in finite-dimensional representation. There exists a linear finitedimensional space of polynomials of a finite-order in the space of columns/spinors of a finite length which is common invariant subspace for all generators (5), (7), (9) -(10). This finite-dimensional representation is irreducible.
The non-negative integer parameter k has a meaning of the length of the first row of the Young tableau of gl n+1 , describing the totally symmetric representation (see below). All other parameters are coded in M ij , which corresponds to arbitrary Young tableau of gl n .
Thus, we have some peculiar splitting of the Young tableau. In the case of the algebra gl 3 the generators (5), (7), (9) -(10) take the form The Casimir operators of gl 3 in this realization are given by and, finally, In this realization the Casimir operator C 3 is algebraically dependent on C 1 and C 2 . In fact, C 1 and C 2 are nothing but the Casimir operators of the gl 2 sub-algebra. Therefore, the center of the gl 3 universal enveloping algebra in the realization (11) is generated by the Casimir operators of the gl 2 sub-algebra realized by M ij . Thus, it seems natural that these reps are irreducible.
Now we consider concrete matrix realizations of the gl 2 -subalgebra in our scheme.
(0). Reps in 1 × 1 matrices. It corresponds to the trivial representation of gl 2 , This is [k, 0] or, in other words, the symmetric representation (the Young tableau has two rows of the length k and 0, correspondingly). We also can call it a scalar representation, since the generators act on one-component spinors or, in other words, on scalar functions (see e.g. [5]). The Casimir operators are: If the parameter k takes non-negative integer the algebra gl 3 spanned by the generators (12) appears in finite-dimensional representation. Its finite-dimensional representation space is a space of polynomials Namely in this representation (12) the algebra gl 3 appears as the hidden algebra of the 3body Calogero and Sutherland models [5], BC 2 rational and trigonometric, and G 2 rational models [6] and even of the BC 2 elliptic model [7].
Take gl 2 in two-dimensional reps by 2 × 2 matrices, Then the generators (11) of gl 3 are: This is [k, 1]−representation (the Young tableau has two rows of the length k and 1, correspondingly) and their Casimir operators are: If the parameter k takes non-negative integer the algebra gl 3 spanned by the generators (14) appears in finite-dimensional representation.
Let us consider several different values of k in detail.
(i) k = 1. Then three-dimensional representation space V (2) 1 appears to be spanned by: It corresponds to antiquark multiplet in standard (fundamental) representation. The Newton polygon is triangle with points P ± as vortices at base.
appears to be spanned by: It corresponds to octet in standard (fundamental) representation. The space V as a subspace, V 2 . It should be mentioned that 1 . Now the Newton polygon is the hexagon where the central point is doubled being presented by Y (1,2) 1 and lower (upper) base has a length two being given by P ± (Y 2,3 ).
is 15-dimensional. In addition to P ± , P ± and Y (1,2) 1 (see (15) and (16)) it contains several vectors more, namely, which are situated on the ±-sides of the Newton hexagon, doubling of the points correspond- and plus extra three vectors on the boundary 3 . All internal points of the Newton hexagon are double points, while the points on the boundary are single ones.
(iv) ∀k. The finite-dimensional representation space V (2) k has dimension k(k + 2) and is presented by the Newton hexagon which contains (k + 1) horizontal layers, lower base has length two, while the upper one 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 possible two-component spinors with components given by inhomogeneous polynomials in x 1 , x 2 of degree not higher than (k − 1).
We denote this space asṼ k . The non-trivial task is to describe k vectors of the last (highest) layer of the hexagon. After some analysis one can find out that they have a form hence span a non-trivial k-dimensional subspace of spinors with components given by specific homogeneous polynomials of degree k.
(II) Reps in 3 × 3 matrices. Take gl 2 in three-dimensional reps by 3 × 3 matrices, Then the generators (11) of gl 3 are: This is [k, 2]−representation (Young tableau has two rows of the length k and 2, correspondingly) and their Casimir operators are: As an illustration let us show explicitly finite-dimensional representation spaces for k = 2, 3.
appears to be spanned by: It is worth mentioning that as a consequence of a particular realization of the generators (11) of the gl 3 algebra there exist a certain relations between generators other than ones given by the Casimir operators. The first observation is that there are no linear relations between generators of such a type. Some time ago there were found nine quadratic relations between gl 3 generators taken in scalar representation (12) other than Casimir operators [8]. Surprisingly, a certain modifications of these relations also exist for [kn] mixed representations (11), Not all these relations are independent. It can be shown that one relation is linearly dependent since the sum (Art.5)+(Art.6)+(Art.7) gives the second Casimir operator C 2 .
At least, in scalar case we can assign a natural (vectorial) grading to generators. Above relations reflect a certain decomposition of gradings as well, where L m (U m ) is the commutative algebra of the lowering (raising) generators with a property [L m , U m ] = P m−1 (gl 2 ⊕ I) with P m−1 as a polynomial of degree (m − 1) in generators of gl 2 ⊕ I. Thus, it realizes a property of the generalized Gauss decomposition. The emerging algebra is a polynomial algebra. It is worth emphasizing that the realization we are going to construct appears at dim(L k ) = dim(U m ) = m. For m = 1 the algebra g (1) = gl 3 , see (6).
Our final goal to build the realization of (22) in terms of finite order differential operators acting on the plane R 2 .
The simplest realization of the algebra gl 2 by differential operators in two variables is the vector field representation, see (1) at n = 2. It was exactly this representation which was used to construct the representation of gl 3 acting of R 2 , see (11), (12). In this case dim(L m ) = dim(U m ) = 2. We are unable to find other algebras with dim(L m ) = dim(U m ) > 2. However, there exists another representation of the algebra gl 2 by the first order differential operators in two variables,J (see S. Lie, [9] at k = 0, W. Miller [10] and A. González-Lopéz et al, [11] at k = 0 (Case 24)), where s, k are arbitrary numbers. These generators obey the standard commutation relations (2) the algebra gl 2 in the vector field representation (1). It is evident that the sum of the two representations,J ij and the matrix one M ij is also representation, (cf. (5)). It is worth mentioning that the gl 2 algebra commutation relations for M pm are taken in a canonical form (4). The unity generator I in (22) is written in a form of generalized Euler-Cartan operator Now let us assume that s is non-negative integer, s = m, m = 0, 1, 2, . . .. Evidently, the lowering generators (of negative grading) from L m+1 can be given by forming commutative algebra (cf. [9], [10], [11]). Eventually, the generators of the algebra (gl 2 ⊕ I) ⋉ L m+1 take the form with J (k) 0 , T − i given by (25), (26), respectively. Let us consider two particular cases of the general construction of the raising generators for the commutative algebra U.
(i) For the first case we take the trivial matrix representation of the gl 2 , One can check that one of the raising generators is given by while all other raising generators are multiple commutators of J at i = 1, . . . m. All of them are differential operators of the fixed degree m. The procedure of construction of the operators U i has a property of nilpotency: In particular, for m = 1, Inspecting the generators T − 0,1 , J ij , J (n) , U 0,1 one can see that they span the algebra gl 3 , see (12). Hence, the algebra g (1) ≡ gl 3 .
If the parameter k takes non-negative integer the algebra g (m) spanned by the generators (28), (29), (30) appears in finite-dimensional representation. Its finite-dimensional representation space is a triangular space of polynomials Namely in this representation the algebra g (m) appears as the hidden algebra of the 3-body G 2 trigonometric model [6] at m = 2 and of the so-called TTW model at integer m, in particular, of the dihedral I 2 (m) rational model [12].

IV. EXTENSION OF THE 3-BODY CALOGERO MODEL
The first algebraic form for the 3-body Calogero Hamiltonian [13] appears after gauge rotation with the ground state function, separation of the center-of-mass and change the variables to elementary symmetric polynomials of the translationally-symmetric coordinates [5], These new coordinates are polynomial invariants of the A 2 Weyl group. Its eigenvalues are − ǫ p = 2ω(2p 1 + 3p 2 ) , p 1,2 = 0, 1, . . .
As is shown in Ruhl and Turbiner [5], the operator (32) can be rewritten in a Lie-algebraic form in terms of the gl(3)-algebra generators of the representation [k, 0]. The corresponding expression is Now we can substitute the generators of the representation [k, n] in the form (11) This is n × n matrix differential operator. It contains infinitely-many finite-dimensional invariant subspaces which are nothing but finite-dimensional representation spaces of the algebra gl (3). This operator remains exactly-solvable with the same spectra as the scalar Calogero operator.
Probably, this operator remains completely integrable. A non-trivial integral is the differen-

V. EXTENSION OF THE 3-BODY SUTHERLAND MODEL
The first algebraic form for the 3-body Sutherland Hamiltonian [14] appears after gauge rotation with the ground state function, separation of the center-of-mass and change the variables to elementary symmetric polynomials of the exponentials of translationally-symmetric coordinates [5], where α is the inverse radius of the circle on where the bodies are situated. These new coordinates are fundamental trigonometric invariants of the A 2 Weyl group.
As shown in [5], the operator (36) can be rewritten in a Lie-algebraic form in terms of the Now we can substitute the generators of the representation [k, n] in the form (11) This is n × n matrix differential operator. It contains infinitely-many finite-dimensional invariant subspaces which are nothing but finite-dimensional representation spaces of the algebra gl (3). This operator remains exactly-solvable with the same spectra as the scalar Sutherland operator.
Probably, the operator (38) remains completely integrable. A non-trivial integral is the differential operator of the third order takes the algebraic form after the gauging away

Conclusions
The algebra gl n of differential operators plays a role of the hidden algebra for all A n , B n , C n , D n , BC n Calogero-Moser Hamiltonians, both rational and trigonometric, with the Weyl symmetry of classical root spaces (see [15] and references therein). We described a procedure, which to our opinion should carry the name of the Havlicek procedure, to construct the algebra gl n of the matrix differential operators. The procedure is based on a mixed, matrix-differential operators realization of the Gauss decomposition diagram.
As for the Hamiltonian reduction models with the exceptional Weyl symmetry group G 2 , F 4 , E 6,7,8 , both rational and trigonometric, there exist hidden algebras of differential operators (see [15] and references therein). All these algebras are infinite-dimensional but finitely-generated. For generating elements of those algebras an analogue of the Weyl-Cartan decomposition exists but the Gauss decomposition diagram: a commutator of the lowering and raising generators is a polynomial of the higher-than-one order in the Cartan generators. Surely, matrix realizations of these algebras exist. The first attempt to construct such a realization was made in Section 4. Thus, the above mentioned procedure of building the mixed representations can be realized. It may lead to a new class of matrix exactly-solvable models with exceptional Weyl symmetry.

Acknowledgements (AVT)
The present text was planned long ago to be dedicated to Miloslav Havlicek who always caused for both authors a deep respect as the scientist and the citizen.
The text is based mainly on the notes jointly prepared by the first and the second author.
It does not include the results of the authors obtained separately (except for Section 4) and which the authors had no chance to discuss. Thus, it looks incomplete. When the first author (YuFS) passed away it took years for the second author (AVT) to return to the subject due to a sad memory. 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 where a part of this work was done during his numerous visits. AVT is grateful to J C Lopez Vieyra for the interest to the work and technical assistance. This work was supported in part by the