# (q, σ, τ)-Differential Graded Algebras

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. First Order $(\mathit{\sigma},\mathit{\tau})$-Differential Calculus with Right Partial Derivatives

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

## 3. $(\mathit{q},\mathit{\sigma},\mathit{\tau})$-Differential Graded Algebra

**Definition**

**2.**

- (a)
- d commutes with endomorphisms $\sigma ,\tau $, i.e., $\sigma \circ d=d\circ \sigma ,\tau \circ d=d\circ \tau ,$
- (b)
- d satisfies the $(q,\sigma ,\tau )$-Leibniz rule:$$d(uv)=d(u)\phantom{\rule{0.166667em}{0ex}}\tau (v)+{q}^{\left|u\right|}\phantom{\rule{0.166667em}{0ex}}\sigma (u)\phantom{\rule{0.166667em}{0ex}}d(v),$$
- (c)
- ${d}^{N}(u)=0$ for any element $u\in \mathcal{A}$.

## 4. Construction of $(\mathit{q},\mathit{\sigma},\mathit{\tau})$-Differential Graded Algebra by Means of the Graded $\mathit{q}$-Commutator

**Theorem**

**1.**

- (a)
- q is a primitive ${N}^{\mathrm{th}}$ root of unity,
- (b)
- ${\xi}^{N}=\lambda \phantom{\rule{0.166667em}{0ex}}\mathbf{1}$, where λ is a non-zero complex number,
- (c)
- $\sigma (\xi )=\tau (\xi )=\xi ,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\sigma \circ \tau =\tau \circ \sigma ,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\sigma}^{N}={\tau}^{N}$,

**Proof.**

## 5. Construction of $(\mathit{q},\mathit{\sigma},\mathit{\tau})$-Differential Graded Algebra by Means of $(\mathit{\sigma},\mathit{\tau})$-Pre- Cosimplicial Algebra

**Definition**

**3.**

- (1)
- $\mathfrak{A}={\oplus}_{n\ge 0}{\mathfrak{A}}^{n}$ is a graded algebra,
- (2)
- $\sigma ,\tau $ are degree zero endomorphisms of a graded algebra $\mathfrak{A}$,
- (3)
- for any homogeneous elements $u,v\in \mathfrak{A}$ and any integer $i\in \{0,1,\dots ,|u|+|v|+1\}$, we have:$${\mathfrak{f}}_{i}(uv)=\left\{\begin{array}{c}{\mathfrak{f}}_{i}(u)\phantom{\rule{0.166667em}{0ex}}\tau (v),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left|u\right|\ge i,\hfill \\ \sigma (u)\phantom{\rule{0.166667em}{0ex}}{\mathfrak{f}}_{i-\left|u\right|}(v),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0\le \left|u\right|<i,\hfill \end{array}\right.\phantom{\rule{1.em}{0ex}}{\mathfrak{f}}_{\left|u\right|+1}(u)\tau (v)=\sigma (u)\phantom{\rule{0.166667em}{0ex}}{\mathfrak{f}}_{0}(v).$$

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Abramov, V.; Liivapuu, O.; Makhlouf, A. (*q*, *σ*, *τ*)-Differential Graded Algebras. *Universe* **2018**, *4*, 138.
https://doi.org/10.3390/universe4120138

**AMA Style**

Abramov V, Liivapuu O, Makhlouf A. (*q*, *σ*, *τ*)-Differential Graded Algebras. *Universe*. 2018; 4(12):138.
https://doi.org/10.3390/universe4120138

**Chicago/Turabian Style**

Abramov, Viktor, Olga Liivapuu, and Abdenacer Makhlouf. 2018. "(*q*, *σ*, *τ*)-Differential Graded Algebras" *Universe* 4, no. 12: 138.
https://doi.org/10.3390/universe4120138