Noncommutative Polynomial Algebras of Solvable Type and Their Modules

Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

“…It is demonstrated that these representations share some similarities with those of su(3) and that states of given energy can be decomposed into sum of irreducible representations which take the form of su(3)-like triangular multiplets. In recent years, it was discovered how quadratic, cubic and more generally polynomial algebras appear in different contexts of mathematical physics [67,68,69,70,71,72,73,74]. The class of polynomial algebras and related representations we have obtained in this paper are of interest for further applications from both mathematical and physical perspectives.…”

The Standard Quantized Matrix Algebra $M_q(n)$ is A Solvable Polynomial Algebra

“…At this point, we specify several applications of Theorem 2.3 in this section, so as to show how Theorem 2.3 may bring some perspective of establishing and realizing further structural properties of M q (n) and their modules in a constructive-computational way. For more details on the basic constructive-computational theory and methods for solvable polynomial algebras and their modules, one is referred to [Li6]. All notations used in Section 2 are maintained.…”

“…In [Tu], it was shown that every left ideal L of M q (n) with Gelfand-Kirillov dimension GK.dimM q (n)/L < GK.dimM q (n) has the elimination property in the sense of [Li5]. Now that we know that M q (n) is a solvable polynomial algebra, [Li6,Section 1.6] tells us that this elimination property can be strengthened, and may be realized in a computational way. To see this clearly, let us first recall the Elimination Lemma given in [8].…”

On The Algebras $U_q^{\pm}(A_N)$: From A Constructive-Computational Viewpoint

“…U − q (A N ) and their modules in a constructive-computational way. For more details on the basic constructive-computational theory and methods for solvable polynomial algebras and their modules, one is referred to [Li6]. All notions and notations used in previous sections are retained.…”

“…It is also well known that nowadays the noncommutative Buchberger Algorithm for solvable polynomial algebras and their modules has been successfully implemented in the computer algebra system Plural [LS]. Concerning basic constructive-computational theory and methods for solvable polynomial algebras and their modules, one is referred to [Li6] for more details. Now, we aim to prove the following result.…”