Acta Polytechnica https://doi.org/10.14311/AP.2025.65.0562 Acta Polytechnica 65(5):562–565, 2025 © 2025 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague gl3 ALGEBRA IN MIXED MATRIX REPRESENTATIONS Alexander V. Turbiner Universidad Nacional Autónoma de México, Instituto de Ciencias Nucleares, Apartado Postal 70-543, 04510 México, Mexico correspondence: turbiner@nucleares.unam.mx Abstract. It is demonstrated that the so-called mixed realization of the gl3 algebra generators in terms of matrix differential operators in two variables, as presented by Smirnov-Turbiner (2013), can be “lifted” into the action in the Fock space associated with the five-dimensional Heisenberg algebra h5. A realization of the gl3 generators in terms of matrix finite-difference (translation-invariant) operators, matrix discrete (dilatation-invariant) operators, matrix complex operators in (z, z̄), and their mixtures is presented. Keywords: Mixed representations in Fock space, matrix differential operators, matrix finite-difference (discrete) operators, matrix complex operators in (z, z̄). To the memory of Miloslav Havlíček 1. Introduction As a result of numerous discussions with Miloslav Havlíček (Havlicek)1, which ran over many years about mixing the representations of Lie algebras [1], as dis- cussed between Yu. F. Smirnov (1935–2008) and the present author, in the article [2] there were constructed the so-called mixed representations of the gln+1 alge- bra realized by differential operators of the first order in n variables with matrix coefficients2. In the partic- ular case of the gl3 algebra, these generators 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, (1) see [2], Section 3, where M11, M12, M21, M22 are the generators of the gl2 algebra realized by n×n matrices, here, the notations ∂1 ≡ ∂ ∂x1 , ∂2 ≡ ∂ ∂x2 are used. For a non-negative integer k, this representation is finite- dimensional with marks/spins [k, n], it is characterized by the Young tableau with two rows of length k and n, correspondingly. One can check explicitly that T − i , Eij , E0, T + i span the algebra gl3. In particular: [E, T +] = T +, 1In many instances in the scientific literature the family name of Miloslav is written as Havlicek, to avoid confusions since now on we will use this name. 2For convenience, we will always assume the canonical com- mutation relations: [Ẽij , Ẽkl] = δjkẼil − δilẼkj , for the gln+1 generators if they are not specified otherwise. symbolically, while: [T + i , T − j ] = Eii − δijE0. The generator E0 is called the Euler-Cartan generator or number operator. It plays the role of a constant having the grading zero in whatever sense. The rep- resentation (1) acts in the space of n-tuples, with columns of a size n. The Casimir operators of gl3 algebra in this realiza- tion 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 + E2 11 + E2 22 + E2 0 = k(k + 2) + M2 11 + M2 22 + M12M21 + M21M12 − M11 − M22 ≡ k(k + 2) + C2(M) − C1(M), and, finally: C3 = −1 2C3 1 + 3 2C1C2 + 3C2 − 2C2 1 − 2C1, where C1(M), C2(M) are the Casimir operators of the gl2 algebra. In this realization (1), the Casimir operator C3 is algebraically dependent on C1 and C2. In fact, C1 and C2 are nothing but the Casimir opera- tors C1(M), C2(M) of the gl2 sub-algebra. Therefore, the center of the gl3 universal enveloping algebra in realization (1) is generated by the Casimir operators of the gl2 sub-algebra realized by Mij . Thus, it seems natural that these reps should be irreducible. In this short note we will show that the mixed rep- resentation (1) can be converted into a representation acting on a Fock space associated with the Heisenberg algebra h5. In turn, the Fock space can be realized by finite-difference (on the uniform lattice), discrete (on 562 https://doi.org/10.14311/AP.2025.65.0562 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 65 no. 5/2025 gl3 algebra in mixed matrix representations the exponential lattice) operators, and by complex op- erators in (z, z̄). This leads to matrix finite-difference (translation-invariant) operators, to matrix discrete (dilatation-invariant) operators, or to matrix complex operators in (z, z̄) and their mixtures as the generators of the gl3 algebra. 2. gl3 mixed representation in a Fock space Let us take the 5-dimensional Heisenberg algebra h5 spanned by the generators p1, p2, q1, q2, I, which obey the commutation relations: [p1, q1] = 1, [p2, q2] = 1, [p1, q2 ] = 0, [p2, q1] = 0, [p1, p2] = 0, [q1, q2] = 0, [p1,2, I] = 0, [q1,2, I] = 0. (2) The universal enveloping algebra of the algebra h5: Uh5 , is spanned by all ordered monomials in p1, p2, q1, q2. Introducing the vacuum |0⟩ as an object anni- hilated by p-operators: p1 |0⟩ = 0, p2 |0⟩ = 0, in addition to the universal enveloping algebra Uh5 , leads to the definition of a Fock space. It can be easily shown by direct calculation that: E11 = q1p1 + M11, E22 = q2p2 + M22, E12 = q1p2 + M12, E21 = q2p1 + M21, E0 = k − q1p1 − q2p2, T − 1 = p1, T − 2 = p2, T + 1 = q1(k − q1p1 − q2p2) − q1M11 − q2M12, T + 2 = q2(k − q1p1 − q2p2) − q1M21 − q2M22, (3) span the gl3 algebra for any k ∈ R, here M11, M12, M21, M22 are again generators of the gl2 algebra obey- ing canonical commutation relations, see Footnote 1. The representation (3) is the main result of the present paper. By taking the Heisenberg algebra (2) realized in the coordinate-momentum representation: q1 = x1, p1 = ∂1, q2 = x2, p2 = ∂2, with M11, M12, M21, M22 given by n × n matrices which spans gl2-algebra the representation (3) is re- duced to (1). It is also worth noting that in the one-dimensional representation of gl2, when: M11 = M12 = M21 = M22 = 0, the representation (3) realizes the hidden algebra of the A2/3-body Calogero rational/Tremblay-Turbiner- Winternitz (at index 3, degenerate, see [3]) model in the Fock space [4]. 3. Three canonical pairs Two operators a, b form a canonical pair if their com- mutator: [a, b] = 1, where [a, b] = ab − ba. The simplest example of a canonical pair is given by the coordinate-momentum representation: [∂x, x] = 1. 3.1. Translation-invariant canonical pair Let us take the shift operator: Tδf(x) = f(x + δ), Tδ = eδ∂x , where δ ∈ C is a parameter, which is called the spacing, and construct the pair of shift operators (see e.g. [5]): Dδ = Tδ − 1 δ , Xδ = xT−δ = x(1 − δD−δ), (4) where the operator Dδ is defined as: Dδf(x) = f(x + δ) − f(x) δ , sometimes, it is called the Norlund derivative. The operators Dδ, Xδ are translation-invariant. The vac- uum is chosen to be one, |0⟩ = 1. This canonical pair acts naturally on the space of polynomials (in x). In the limit δ → 0, this degenerates into the coordinate- momentum representation, Dδ → ∂x and Xδ → x. It is easy to check that the commutator [Dδ, Xδ] = 1, hence, Dδ, Xδ form a canonical pair. 3.2. Dilatation-invariant canonical pair Let us introduce the dilatation operator: Tq f(x) = f(qx), Tq = qA, A ≡ x ∂x, where q ∈ C, and construct a canonical pair of dilatation-invariant operators: Dq = x−1 Tq − 1 q − 1 , Xq = A(q − 1) Tq − 1 x, (5) see [6]. It can be easily checked that [Dq, Xq] = 1 for any q, thus, Dq, Xq form a canonical pair. Usu- ally, the operator Dq is called the Jackson symbol (or the Jackson derivative). The vacuum is equal to one, |0⟩ = 1. This canonical pair acts naturally on the space of polynomials (in x). Both operators Xq, Dq are pseudodifferential operators which action on monomials as follows: Dqxn = {n}q xn−1, Xqxn = n + 1 {n + 1}q xn+1, where {n}q = 1−qn 1−q is the so called q-number n. 563 Alexander V. Turbiner Acta Polytechnica 3.3. Complex (z, z̄) canonical pair Take the space L2(C, dµ) of square-integrable func- tions on C with the Gaussian measure: dµ(z) = π−1 e−z·z̄dv(z), where dv(z) = dxdy is the Euclidean volume element on C = R2. Let us consider the following lowering and raising operators [7], for discussion see [8]: a = ∂ ∂z̄ , a† = − ∂ ∂z + z̄. (6) They are unitary-conjugated (adjoint). The vacuum vector |0⟩, defined by: a |0⟩ = 0, is any analytic function. It is easy to check that: [a, a†] = I, where I is the unit operator, thus, a canonical pair is formed. 4. gl3 mixed representations in matrix operators It is evident that the representation (3) acting in the Fock space of columns allows us to construct the gl3 mixed representations in matrix operators. 4.1. Matrix representation of the gl3 algebra of finite-difference operators Let us take the translation-invariant canonical pair (4) and construct a realization of the Heisenberg genera- tors of h5: p1, p2, q1, q2, I in the following way: p1 = Dδ1(x), p2 = Dδ2(y), q1 = Xδ1(x), q2 = Xδ2(y). Substituting these into (3), we arrive at a repre- sentation of the gl3 algebra in the form of matrix finite-difference operators acting in the (x, y) space of columns/n-tuples on the rectangular lattice with spacings δ1, δ2. 4.2. Matrix representation of the gl3 algebra of discrete operators Let us take the dilatation-invariant canonical pair (5) and construct a realization of the Heisenberg genera- tors of h5 : p1, p2, q1, q2, I in the following way: p1 = Dq1(x), p2 = Dq2(y), q1 = Xq1(x), q2 = Xq2(y). Substituting these into (3), we arrive at a representa- tion of the gl3 algebra in the form of discrete operators with matrix coefficients acting in the (x, y) space of columns/n-tuples on rectangular lattice with dilations q1, q2 in the x, y directions, respectively. 4.3. Matrix representation of the gl3 algebra of mixed finite-difference/discrete operators Let us take the translation-invariant canonical pair (4) acting in the x-direction and the dilatation-invariant canonical pair (5) acting in the y-direction and con- struct a realization of the Heisenberg generators of h5 : p1, p2, q1, q2, I in the following way: p1 = Dδ(x), p2 = Dq(y), q1 = Xδ(x), q2 = Xq(y). Substituting this realization into (3) we arrive at a representation of the gl3 algebra in the form of matrix finite-difference/discrete operators acting in the (x, y) space of columns/n-tuples on a rectangular uniform/exponential lattice with spacings δ in the x direction and dilation q in the y direction, respec- tively. 4.4. Matrix representation of complex (z, z̄) generators By taking two (z1,2, z̄1,2) representations (6) acting on C2(z1, z2) complex space in the form: p1 = a1(z1, z̄1), q1 = a† 1(z1, z̄1), p2 = a2(z2, z̄2), q2 = a† 2(z2, z̄2), (7) and the unit generator I we construct a realiza- tion of the five-dimensional Heisenberg algebra h5 in a 2D complex space. By taking (7) and substitut- ing it into (3), we arrive at the matrix representation of the gl3 algebra in the C2(z1, z2) complex space of columns/n-tuples. 5. Conclusion In this paper, we were able to construct the Fock space representation [n, k] of the gl3 algebra (3) acting in the Fock space of columns/n-tuples. This representation becomes finite-dimensional when k is a non-negative integer and integer n is related to the n-dimensional representation of the gl2 algebra. This representation can be converted to the representations in terms of the first-order differential, finite-difference, discrete, and complex (z, z̄) operators with matrix coefficients. All these representations are alternative to the standard gl3 algebra representations acting in three-dimensional space (on flag manifold). Acknowledgements The author (AVT) partially supported by DGAPA grant IN104125 (Mexico). This work is dedicated to the memory of Miloslav Havlíček – an exemplary scientist and citizen. The present author thinks that the representation (1) should be called the Havlíček representation. 564 vol. 65 no. 5/2025 gl3 algebra in mixed matrix representations References [1] M. Havlíček. Personal communications, 1988–2010. [2] Y. F. Smirnov, A. 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Complex Analysis and Operator Theory 15(7):110, 2021. https://doi.org/10.1007/s11785-021-01154-y 565 https://doi.org/10.14311/AP.2013.53.0462 https://doi.org/10.1088/1751-8113/42/24/242001 https://doi.org/10.3842/SIGMA.2024.012 https://doi.org/10.1142/S0217732395001927 https://doi.org/10.1088/0305-4470/34/48/312 https://doi.org/10.1007/978-3-0348-8403-7_28 https://doi.org/10.1007/s11785-021-01154-y Acta Polytechnica 65(5):562–565, 2025 1 Introduction 2 gl3 mixed representation in a Fock space 3 Three canonical pairs 3.1 Translation-invariant canonical pair 3.2 Dilatation-invariant canonical pair 3.3 Complex (z,) canonical pair 4 gl3 mixed representations in matrix operators 4.1 Matrix representation of the gl3 algebra of finite-difference operators 4.2 Matrix representation of the gl3 algebra of discrete operators 4.3 Matrix representation of the gl3 algebra of mixed finite-difference/discrete operators 4.4 Matrix representation of complex (z,) generators 5 Conclusion Acknowledgements References