# Algebraic Inverses on Lie Algebra Comultiplications

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- $A:=\langle {a}_{1},{a}_{2},\dots ,{a}_{n}\rangle $ and $B:=\langle {b}_{1},{b}_{2},\dots ,{b}_{m}\rangle $ are graded vector spaces over a field $\mathbb{F}$ of characteristic 0.
- We denote the degree of generators ${a}_{s},s=1,2,\dots ,n$ by $|{a}_{s}|={d}_{s}$ with ${d}_{1}\le {d}_{2}\le \dots \le {d}_{n}$, and similarly, for ${b}_{t},t=1,2,\dots ,m$.
- $L\left(A\right)$ and $L\left(B\right)$ are the free graded Lie algebras with the Lie brackets $[\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}]$ generated by A and B, respectively.

**Example**

**1.**

- Ω is the loop functor;
- Σ is the suspension functor;
- deg$\left({a}_{i}\right)=2i$ for $i=1,2,\dots ,n$;
- deg$\left({b}_{j}\right)=4j$ for $j=1,2,\dots ,m$.

**Definition**

**1.**

**Example**

**2.**

- ${P}_{1}=0$;
- ${P}_{s}={\sum}_{i,j}({a}_{i}{a}_{j}^{\prime}+{a}_{i}^{\prime}{a}_{j})$ for $s=2,3,\dots ,n$ and $i,j\in \{1,2,\dots ,n-1\}$ with ${d}_{i}+{d}_{j}={d}_{s}$.

**Definition**

**2.**

## 3. Algebraic Loops and Inverses

**Definition**

**3.**

**Remark**

**1.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Definition**

**4.**

**Definition**

**5.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 4. Conclusions and Further Prospects

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Lee, D.-W.
Algebraic Inverses on Lie Algebra Comultiplications. *Symmetry* **2020**, *12*, 565.
https://doi.org/10.3390/sym12040565

**AMA Style**

Lee D-W.
Algebraic Inverses on Lie Algebra Comultiplications. *Symmetry*. 2020; 12(4):565.
https://doi.org/10.3390/sym12040565

**Chicago/Turabian Style**

Lee, Dae-Woong.
2020. "Algebraic Inverses on Lie Algebra Comultiplications" *Symmetry* 12, no. 4: 565.
https://doi.org/10.3390/sym12040565