[Paper Review] Center of skew PBW extensions

José Oswaldo Lezama Serrano, Helbert Javier Venegas Ramírez|arXiv (Cornell University)|Apr 15, 2018

Advanced Topics in Algebra11 references3 citations

TL;DR

This paper computes the center of a broad class of noncommutative algebras via skew PBW extensions, showing under natural parameter conditions that the center is either trivial or polynomial-type. It applies these results to prove new examples of cancellative noncommutative algebras, advancing the Zariski cancellation problem in noncommutative algebraic geometry.

ABSTRACT

In this paper we compute the center of many noncommutative algebras that can be interpreted as skew $PBW$ extensions. We show that, under some natural assumptions on the parameters that define the extension, either the center is trivial, or, it is of polynomial type. As an application, we provided new examples of noncommutative algebras that are cancellative.

Motivation & Objective

• To compute the center of a wide class of noncommutative algebras that are skew PBW extensions over a field K with char(K) = 0.
• To determine conditions on the defining parameters of skew PBW extensions under which the center is trivial or polynomial-type.
• To apply center computations to the Zariski cancellation problem, identifying new cancellative noncommutative algebras.
• To extend existing results on center computations in quantum algebras and noncommutative rings using a unified framework.
• To provide a systematic method for analyzing central elements and subalgebras in skew PBW extensions via algebraic constraints derived from commutation relations.

Proposed method

• Interprets noncommutative algebras as skew PBW extensions using the definition involving injective endomorphisms σ_i and σ_i-derivations δ_i.
• Uses the fundamental equation x_i f = f x_i for f in the center to derive constraints on the parameters and structure of f.
• Applies the defining relations of skew PBW extensions—specifically x_i r = σ_i(r)x_i + δ_i(r) and x_j x_i = c_{i,j}x_i x_j + lower terms—to analyze central elements.
• Groups algebras into subclasses based on parameter conditions (e.g., bijective σ_i, invertible c_{i,j}, roots of unity), enabling systematic center computation.
• Employs known center results for specific algebras (e.g., Weyl algebra, quantum plane) as benchmarks and references.
• Applies the criterion that if Z(A) = K, then A is universally cancellative, to deduce cancellativity of new algebras.

Experimental results

Research questions

• RQ1 Under what conditions on the parameters of a skew PBW extension is the center trivial or of polynomial type?
• RQ2 Which noncommutative algebras covered by skew PBW extensions admit a computable center using the proposed method?
• RQ3 How can center computations be leveraged to determine cancellativity in noncommutative algebras?
• RQ4 What new classes of cancellative algebras can be identified using the center structure of skew PBW extensions?
• RQ5 In which cases does the center of a skew PBW extension coincide with a polynomial ring over the base field K?

Key findings

• The center of a skew PBW extension is either trivial or isomorphic to a polynomial ring in finitely many variables under natural assumptions on the defining parameters.
• For bijective skew PBW extensions with parameters q that are roots of unity, the center contains polynomial subalgebras such as K[x_i^l] or K[xy - quv], depending on the algebra.
• The paper proves that any skew PBW extension with center equal to K is universally cancellative, hence cancellative.
• New cancellative algebras are identified, including the Weyl algebra A_n(K), extended Weyl algebra B_n(K), and various quantum and differential operator algebras.
• Specific central subalgebras are computed for algebras like the q-Heisenberg algebra (with center generated by C_i = (q^2-1)x_i y_i z_i - y_i^2) and the Jordan plane.
• The results apply to 24 specific algebras, including quantum symplectic spaces, quantum enveloping algebras, and diffusion-type algebras, all shown to be cancellative when their center is K.