Ideals on the Quantum Plane’s Jet Space

The goal of this paper is to introduce some rings that play the role of the jet spaces of the quantum plane and unlike the quantum plane itself possess interesting nontrivial prime ideals. We will prove some results (Theorems 1–4) about the prime spectrum of these rings.

(q,σ,τ)-Differential Graded Algebras

We propose the notion of ( q , σ , τ ) -differential graded algebra, which generalizes the notions of ( σ , τ ) -differential graded algebra and q-differential graded algebra. We construct two examples of ( q , σ , τ ) -differential graded algebra, where the first one is constructed by means of the generalized Clifford algebra with two generators (reduced quantum plane), where we use a ( σ , τ ) -twisted graded q-commutator. In order to construct the second example, we introduce the notion of ( σ , τ ) -pre-cosimplicial algebra.

$(q,\sigma,\tau)$-Differential Graded Algebras

We propose a notion of $(q,\sigma,\tau)$-differential graded algebra, which generalizes the notions of $(\sigma,\tau)$-differential graded algebra and $q$-differential graded algebra. We construct two examples of $(q,\sigma,\tau)$-differential graded algebra, where the first one is constructed by means of generalized Clifford algebra with two generators (reduced quantum plane), where we use a $(\sigma,\tau)$-twisted graded $q$-commutator. In order to construct the second example, we introduce a notion of $(\sigma,\tau)$-pre-cosimplicial algebra.