The q-Riccati Algebra

for all A, B  g. Note that the expression AB only makes sense as a matrix product in a representation. For example, if A and B are antisymmetric matrices, then AB-BA is skew-symmetric, but AB may not be antisymmetric. The possible irreducible representations of complex Lie algebras are determined by the classification of the semi simple Lie algebras. Any irreducible representation V of a complex Lie algebra g is the tensor product V = V0L, where V0 is an irreducible representation of the quotient gss|Rad(g) of the algebra g and its Lie algebra radical and L is a one-dimensional representation. In the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with the Lie algebra plays an important role. The Riccati algebra is a finite-dimensional linear space that is closed under commutator, that is R is a Lie algebra. In recent years the q-deformation of the Heisemburg commutation relation has drawn attention. Leeuwen and Maassen (1995) and many of other researcher like (Altoum, 2018a; 2018b; Rguigui, 2015a; 2015b; 2016a; 2016b; 2018a; 2018b; Altoum et al., 2017), the purpose is to study the probability distribution of a noncommutative random variable a + a, where a is a bounded operator on some Hilbert space satisfying:


Introduction
In the mathematical field of representation theory, the representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. More precisely, a representation of a Lie algebra g is a linear transformation: where, M(V) is the set of all linear transformations of a vector space V. In particular, if V = n , then M(V) is the set of n  n square matrices. The map  is required to be a map of Lie algebras so that: for all A, B  g. Note that the expression AB only makes sense as a matrix product in a representation. For example, if A and B are antisymmetric matrices, then AB-BA is skew-symmetric, but AB may not be antisymmetric. The possible irreducible representations of complex Lie algebras are determined by the classification of the semi simple Lie algebras. Any irreducible representation V of a complex Lie algebra g is the tensor product V = V0L, where V0 is an irreducible representation of the quotient gss|Rad(g) of the algebra g and its Lie algebra radical and L is a one-dimensional representation. In the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with the Lie algebra plays an important role. The Riccati algebra is a finite-dimensional linear space that is closed under commutator, that is R is a Lie algebra.
In recent years the q-deformation of the Heisemburg commutation relation has drawn attention. Leeuwen and Maassen (1995) and many of other researcher like (Altoum, 2018a;Rguigui, 2015a;2015b;2016a;2016b;Altoum et al., 2017), the purpose is to study the probability distribution of a noncommutative random variable a + a * , where a is a bounded operator on some Hilbert space satisfying: for some q  [-1, 1). The calculation is inspired by the case, q = 0, where a and a * turn out to be the left and right shift on l 2 (N): In this case a and a * can be quite nicely represented as operators on the Hardy class  2 of all analytic functions on the unit disk with L 2 limits toward the boundary. Subsequently, they find a measure q, q[0, 1), on the complex plane that replaces the Lebesgue measure on the unit circle in the above: q is concentrated on a family of concentric circle, the largest of which has the radius 1 1 q  . Their representation space (Leeuwen and Maassen, 1995) will be H 2 (Dq, q), the completion of the analytic functions on with respect to the inner product defined by q. In this space annihilation operator a is represented by a q difference operator Dq. As q tends to 1, q will tend to the Gauss measure on and Dq becomes differentiation. We recall some basic notations of the language of q-calculus (Abdi, 1962;Adams, 1929;Gasper and Rahman, 1990;Jackson, 1910;Leeuwen and Maassen, 1995). For q(0, 1) and analytic f:  define operators Z and Dq as follows (Gasper and Rahman, 1990;Jackson, 1910;Leeuwen and Maassen, 1995): In this paper, we introduce the q-Riccati Algebra. This paper is organized as follows: In Section 1, we present preliminaries include q-calculus. In Section 2, we introduce the q-Riccati algebra. In section 3, we give a representation of this algebra.

Representation of the q-Riccati Algebra
Let q(0, 1). Then, we define the q-Riccati Lie algebra as follows: , , ,

Representation of the q-Riccati Algebra
Let M0,q, M1,q and M2,q given by: where, Dq and X are defined as follows: Then, we obtain: Similarly, we get: Then, we get:

Proof
The proof follows from Proposition 3.1 and Proposition 3.2.