arXiv subjects
Publications and source records attributed to Samuel A. Lopes.
We investigate the combinatorics of the general formulas for the powers of the operator $h \partial^k$, where $h$ is a central element of a ring and $\partial$ is a differential operator. This generalizes previous work on the powers of operators $h \partial$. New formulas for the generalized Stirling numbers are obtained.
The Green ring of the half quantum group $H=U_n(q)$ is computed in [Chen, Van Oystaeyen, Zhang]. The tensor product formulas between indecomposables may be used for a generalized subgroup depth computation in the setting of quantum groups -- to compute depth of the Hopf subalgebra $H$ in its Drinfeld double $D(H)$. In this paper the Hopf subalgebra quotient module $Q$ (a generalization of the permutation module of cosets for a group extension) is computed and, as $H$-modules, $Q$ and its second tensor power are decomposed into a direct sum of indecomposables. We note that the least power $n$, referred to as depth, for which $Q^{\otimes (n)}$ has the same indecomposable constituents as $Q^{\otimes (n+1)}$ is $n = 2$, since $ Q^{\otimes (2)}$ contains all $H$-module indecomposables, which determines the minimum even depth $d_{ev}(H,D(H)) = 6$.
In this note we classify the non-Noetherian generalized Heisenberg algebras H(f) introduced by Rencai Lü and Kaiming Zhao [Linear Algebra Appl., 2015]. In case the polynomial f has degree greater than 1, we determine all locally finite and also all locally nilpotent derivations of H(f) and describe the automorphism group of these algebras.
We construct a family of irreducible representations of the quantum plane and of the quantum Weyl algebra over an arbitrary field, assuming the deformation parameter is not a root of unity. We determine when two representations in this family are isomorphic, and when they are weight representations, in the sense of Bavula.
An Ore extension over a polynomial algebra F[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra A_h generated by elements x,y, which satisfy yx-xy = h, where h is in F[x]. When h is nonzero, these algebras are subalgebras of the Weyl algebra A_1 and can be viewed as differential operators with polynomial coefficients. In previous work, we investigated the structure of A_h, determined its automorphisms and their invariants, and studied the irreducible A_h-modules. Here we determine the derivations of A_h over an arbitrary field.
An Ore extension over a polynomial algebra F[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra A_h generated by elements x,y, which satisfy yx-xy = h, where h is in F[x]. When h is nonzero, these algebras are subalgebras of the Weyl algebra A_1 and can be viewed as differential operators with polynomial coefficients. In previous work, we studied the structure of A_h and determined its automorphism group and the subalgebra of invariants under the automorphism group. Here we determine the irreducible A_h-modules. In a sequel to this paper, we completely describe the derivations of A_h over any field.
An Ore extension over a polynomial algebra $\mathbb{F}[x]$ is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra $\mathsf{A}_h$ generated by elements $x,y$, which satisfy $yx-xy = h$, where $h\in \mathbb{F}[x]$. We investigate the family of algebras $\mathsf{A}_h$ as $h$ ranges over all the polynomials in $\mathbb{F}[x]$. When $h \neq 0$, these algebras are subalgebras of the Weyl algebra $\mathsf{A}_1$ and can be viewed as differential operators with polynomial coefficients. We give an exact description of the automorphisms of $\mathsf{A}_h$ over arbitrary fields $\mathbb{F}$ and describe the invariants in $\mathsf{A}_h$ under the automorphisms. We determine the center, normal elements, and height one prime ideals of $\mathsf{A}_h$, localizations and Ore sets for $\mathsf{A}_h$, and the Lie ideal $[\mathsf{A}_h,\mathsf{A}_h]$. We also show that $\mathsf{A}_h$ cannot be realized as a generalized Weyl algebra over $\mathbb{F}[x]$, except when $h \in \mathbb{F}$. In two sequels to this work, we completely describe the derivations and irreducible modules of $\mathsf{A}_h$ over any field.
We determine when a generalized down-up algebra is a Noetherian unique factorisation domain or a Noetherian unique factorisation ring.
Motivated by the construction of new examples of Artin-Schelter regular algebras of global dimension four, J.J. Zhang and J. Zhang (2008) introduced an algebra extension $A_P[y_1,y_2;σ,δ,τ]$ of $A$, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over $A$. The aim of this paper is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form $A[y_1;σ_1, δ_1][y_2;σ_2, δ_2]$. We also give partial answers to some questions posed in Zhang and Zhang (2008).
A generalization of down-up algebras was introduced by Cassidy and Shelton (J. Algebra 279 (2004), no. 1), the so-called generalized down-up algebras. We describe the automorphism group of conformal Noetherian generalized down-up algebras L(f,r,s,γ) such that r is not a root of unity, listing explicitly the elements of the group. In the last section we apply these results to Noetherian down-up algebras, thus obtaining a characterization of the automorphism group of Noetherian down-up algebras A(α, β, γ) for which the roots of the polynomial X^2-αX-βare not both roots of unity.