Consider a field \(K\) of characteristic zero, an element \(q \in K\) such that \(q^8 \neq 1\) and a positive integer \(N\), the Drinfeld-Jimbo quantum group of type \(A_N\) is denoted by \(U_q(A_N)\). Denote by \(U^+_q(A_N)\) the \((+)\)-part and by \(U^-_q(A_N)\) the \((-)\)-part of the group.
Both \(U^+_q(A_N)\) and \(U^-_q(A_N)\) have a standard PBW \(K\)-basis. In this paper it is proved that for a suitable monomial ordering on their basis, \(U^+_q(A_N)\) and \(U^-_q(A_N)\) are solvable polynomial algebras.
This allows to deduce in a computational way, further properties of such algebras. Both algebras

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are Noetherian domains;

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have Gelfand-Kirillov dimension equal to \(\frac{N(N+1)}{2}\);

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have the global homological dimension smaller than or equal to the Gelfand-Kirillov dimension;

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are non-homogeneous 2-Koszul algebras;

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are Auslander regular algebras satisfying the Cohen-Macaulay property;

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their \(K_0\)-group is isomorphic to \((\mathbb{Z},+)\);

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have the elimination property for one-sided ideals, as defined in [H. Li, Commun. Algebra 46, No. 8, 3520–3532 (2018; Zbl 1391.13054)].

Moreover there are properties related to modules:

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given a nonzero left ideal of one of these algebras and the quotient \(Q\) of the same algebra modulo the ideal, the Gelfand-Kirillov dimension of \(Q\) is smaller than \(\frac{N(N+1)}{2}\);

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for a finitely generated module \(M\) over the algebra, one can find with an algorithm both the projective dimension and a finite free resolution.