2,161 research outputs found
Classification of factorial generalized down-up algebras
We determine when a generalized down-up algebra is a Noetherian unique
factorisation domain or a Noetherian unique factorisation ring
A multiparameter family of irreducible representations of the quantum plane and of the quantum Weyl algebra
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.Comment: 12 pages, Section 2 has been reorganized, new material added in a new
Section
A Parametric Family of Subalgebras of the Weyl Algebra II. Irreducible Modules
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.Comment: 30 pages, a few of the sections have been placed in a different order
at the suggestion of the refere
Non-Noetherian generalized Heisenberg algebras
In this note, we classify the non-Noetherian generalized Heisenberg algebras H(f) introduced in [R. Lu and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl. 475 (2015) 276-291]. In case deg f > 1, we determine all locally finite and also all locally nilpotent derivations of H(f) and describe the automorphism group of these algebras
Primitive ideals of U-q(Sl(n)(+))
Let U-q(g(+)) be the quantized enveloping algebra of the nilpotent Lie algebra g(+) = sl(n+1)(+) which occurs as the positive part in the triangular decomposition of the simple Lie algebra sl(n+1) of type A(n). Assuming the base field K is algebraically closed and of characteristic 0, and that the parameter q is an element of K* is not a root of unity, we define and study certain quotients of U-q(g(+)) which coincide with the Weyl-Hayashi algebra when n = 2 (see Alev and Dumas, 1996, Hayashi, 1990; Kirkman and Small, 1993). We show that these are simple Noetherian domains, with a trivial center and even Gelfand-Kirillov dimension. Hence, they play a role analogous to that played by the Weyl algebras in the classical case. In the remainder of the article, we study the primitive spectrum of U-q(sl(4)(+)) in detail, somewhat in the spirit of Launois (to appear). We determine all primitive ideals of U-q(sl(4)(+)), find a set of generators for each one, compute their heights and find a simple U-q(sl(4)(+))-module corresponding to each primitive ideal of U-q(sl(4)(+))
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