#
Reduced Pre-Lie Algebraic Structures, the Weak and Weakly Deformed Balinsky–Novikov Type Symmetry Algebras and Related Hamiltonian Operators^{ †}

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. General Setting

#### 2.1. Pre-Lie Algebraic Structures and Related Hamiltonian Operators

**Proposition**

**1.**

**Theorem**

**2.**

**Remark**

**3.**

**Proposition**

**4.**

**Sketch**

**of**

**a**

**proof.**

**Proposition**

**5.**

#### 2.2. Lie–Poisson Brackets, Skew-Symmetric Derivations and Balinsky–Novikov Type Algebraic Structures

**Example**

**6.**

**Example**

**7.**

**Example**

**8.**

**Example**

**9.**

**Remark**

**10.**

## 3. Weak and Weakly Deformed Balinsky–Novikov Type Algebras

#### 3.1. A Weak Balinsky–Novikov Type Symmetry Algebra

**Remark**

**11.**

**Theorem**

**12.**

**Lemma**

**13.**

**Proof.**

**Remark**

**14.**

**Proposition**

**15.**

#### 3.2. A Weakly Deformed Balinsky–Novikov Type Symmetry Algebra

**Remark**

**16.**

**Proposition**

**17.**

## 4. The Riemann Type Reduced Pre-Lie Algebra Isomorphism and Related Algebraic Properties

**Theorem**

**18.**

**Lemma**

**19.**

**Proof.**

**Lemma**

**20.**

**Proof.**

#### A General Riemann Type Pre-Lie Algebra Structure

**Lemma**

**21.**

- (i)
- A is associative and $(A,[A,A])=0$,
- (ii)
- ${a}^{2}\in Z(A)$,
- (iii)
- $(A,A)\subseteq Z(A)$ (in particular, if $A=(A,A)$, then A is commutative),
- (iv)
- if A has unity, then $2[A,A]=0$ (in particular, in $\mathbb{F}\ne 2$, then A is commutative),
- (v)
- if $\mathbb{F}\ne 2$, then ${u}^{2}=0$ for any $u\in [A,A]$.

**Proof.**

**Proposition**

**22.**

- (i)
- if $(A,A)=0$, then ${a}^{2}=0$ for any $a\in A$ (in particular, if A is finite-dimensional, then A is nilpotent),
- (ii)
- if $(A,A)$ is nonzero proper in A, then A is nilpotent with NI $\le 3$ (and so A is at most 2-step Lie nilpotent).

**Proof.**

## 5. The Balinsky–Novikov Algebra and Its Fermionic Modification

**Theorem**

**23.**

- (1)
- Suppose that N is non-commutative and $char\mathbb{F}\ne 2$. If N is simple (respectively prime), then it contains a commutative Lie ideal A that contains every commutative Lie ideal of N and ${N}^{L}/A$ is a simple (respectively prime) Lie algebra.
- (2)
- If ${N}^{L}$ is a simple (respectively prime or semiprime) Lie algebra, then N is a simple (respectively prime or semiprime) BNA.

**Proposition**

**24.**

- (i)
- the Lie algebra $\mathcal{L}(N)=L(N)+{X}_{N}$ is a sum of a subalgebra $L(N)$ and an ideal ${X}_{N}:=R(N)+R(N)R(N)$, where $L(N)$ and ${N}^{L}/lannN$ are isomorphic, where $lannN:=\{x\in N\mid xN=0\}$ and ${X}_{N}$ is at most 2-step Lie nilpotent,
- (ii)
- if $L(N)\subseteq Der({N}^{L})$, then $[N,N]\subseteq lannN$ and $\mathcal{L}(N)=L(N)+R(N)\subseteq Der({N}^{L})$.

## 6. Elementary Properties of Fermionic BNA’s

**Lemma**

**25.**

**Proof.**

**Lemma**

**26.**

**Proof.**

**Lemma**

**27.**

**Proof.**

**Corollary**

**28.**

**Proof.**

**Lemma**

**29.**

**Proof.**

**Lemma**

**30.**

**Proof.**

**Proposition**

**31.**

**Proof.**

**Proof**

**of**

**Proposition**

**24**

## 7. Lie Structure of Semiprime BNA’s

**Lemma**

**32.**

**Proof.**

**Lemma**

**33.**

- (i)
- the left annihilator $lannA:=\{n\in N\mid nA=0\}$ of A in N is an ideal of N,
- (ii)
- if $char\mathbb{F}\ne 2$, then ${I}_{N}(U):=\{u\in U\mid uN+Nu\subseteq U\}$ is an ideal of N and ${I}_{N}(U)\subseteq U$,
- (iii)
- if $char\mathbb{F}\ne 2$, then $[U,U]=0$ or U contains a non-central ideal of N,
- (iv)
- if $z\in Z(N)$, then $zN:=\{zn\mid n\in N\}$ is an ideal of N,
- (v)
- $T(U):=\{x\in N\mid [x,N]\subseteq U\}$ is a Lie ideal of N and $U\subseteq T(U)$,
- (vi)
- $Z(U)$ is a Lie ideal of N,
- (vii)
- $Z(A)$ is an ideal of N,
- (viii)
- the centralizer ${C}_{N}(A):=\{z\in N\mid za=az\phantom{\rule{4pt}{0ex}}forany\phantom{\rule{4pt}{0ex}}a\in A\}$ of A is an ideal of N,
- (ix)
- ${C}_{N}(U)$ is a Lie ideal of N,
- (x)
- if N is prime, then $Z(N)=0$ or it is an associative and commutative domain.

**Proof.**

**Lemma**

**34.**

**Proof.**

**Lemma**

**35.**

**Proof.**

**Proposition**

**36.**

**Proof.**

**Proof**

**of**

**Theorem**

**23**

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Artemovych, O.D.; Balinsky, A.A.; Blackmore, D.; Prykarpatski, A.K.
Reduced Pre-Lie Algebraic Structures, the Weak and Weakly Deformed Balinsky–Novikov Type Symmetry Algebras and Related Hamiltonian Operators. *Symmetry* **2018**, *10*, 601.
https://doi.org/10.3390/sym10110601

**AMA Style**

Artemovych OD, Balinsky AA, Blackmore D, Prykarpatski AK.
Reduced Pre-Lie Algebraic Structures, the Weak and Weakly Deformed Balinsky–Novikov Type Symmetry Algebras and Related Hamiltonian Operators. *Symmetry*. 2018; 10(11):601.
https://doi.org/10.3390/sym10110601

**Chicago/Turabian Style**

Artemovych, Orest D., Alexander A. Balinsky, Denis Blackmore, and Anatolij K. Prykarpatski.
2018. "Reduced Pre-Lie Algebraic Structures, the Weak and Weakly Deformed Balinsky–Novikov Type Symmetry Algebras and Related Hamiltonian Operators" *Symmetry* 10, no. 11: 601.
https://doi.org/10.3390/sym10110601