The infinite dimensional Unital 3-Lie Poisson algebra

Chuangchuang Kang; Ruipu Bai; Yingli Wu

arXiv:1904.01005·math.RA·April 3, 2019

The infinite dimensional Unital 3-Lie Poisson algebra

Chuangchuang Kang, Ruipu Bai, Yingli Wu

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TL;DR

This paper constructs an infinite dimensional unital 3-Lie Poisson algebra from a commutative associative algebra, analyzes its structure, and demonstrates embeddings of several important infinite dimensional 3-Lie algebras.

Contribution

It introduces a new infinite dimensional unital 3-Lie Poisson algebra with a minimal generating set and explores its structural properties and embeddings of known 3-Lie algebras.

Findings

01

Existence of a minimal set of six generators for the algebra.

02

The quotient algebra is simple.

03

Embedding of key infinite dimensional 3-Lie algebras into the constructed algebra.

Abstract

From a commutative associative algebra $A$ , the infinite dimensional unital 3-Lie Poisson algebra~ $L$ ~is constructed, which is also a canonical Nambu 3-Lie algebra, and the structure of $L$ is discussed. It is proved that: (1) there is a minimal set of generators $S$ consisting of six vectors; (2) the quotient algebra $L / F L_{0, 0}^{0}$ is a simple 3-Lie Poisson algebra; (3) four important infinite dimensional 3-Lie algebras: 3-Virasoro-Witt algebra $W_{3}$ , $A_{ω}^{δ}$ , $A_{ω}$ and the 3- $W_{\infty}$ algebra can be embedded in $L$ .

Equations177

[[x_{1}, \dots, x_{n}], y_{2}, \dots, y_{n}] = i = 1 \sum n [x_{1}, \dots [x_{i}, y_{2}, \dots, y_{n}], \dots x_{n}] .

[[x_{1}, \dots, x_{n}], y_{2}, \dots, y_{n}] = i = 1 \sum n [x_{1}, \dots [x_{i}, y_{2}, \dots, y_{n}], \dots x_{n}] .

\mbox a d (x_{1}, \dots, x_{n - 1}) (x) = [x_{1}, \dots, x_{n - 1}, x], \forall x \in A,

\mbox a d (x_{1}, \dots, x_{n - 1}) (x) = [x_{1}, \dots, x_{n - 1}, x], \forall x \in A,

[x_{1} \cdot x_{2}, y_{2}, \dots, y_{n}] = [x_{1}, y_{2}, \dots, y_{n}] \cdot x_{2} + x_{1} \cdot [x_{2}, y_{2}, \dots, y_{n}], \forall x_{1}, x_{2}, y_{2}, \dots, y_{n} \in U .

[x_{1} \cdot x_{2}, y_{2}, \dots, y_{n}] = [x_{1}, y_{2}, \dots, y_{n}] \cdot x_{2} + x_{1} \cdot [x_{2}, y_{2}, \dots, y_{n}], \forall x_{1}, x_{2}, y_{2}, \dots, y_{n} \in U .

[x_{1},\cdots,x_{n}]=det\left(\begin{array}[]{ccc}D_{1}(x_{1})&\cdots&D_{1}(x_{n})\\ \cdots&\cdots&\cdots\\ D_{n}(x_{1})&\cdots&D_{n}(x_{n})\\ \end{array}\right),~{}~{}\forall~{}x_{1},x_{2},...,x_{n}\in J,

[x_{1},\cdots,x_{n}]=det\left(\begin{array}[]{ccc}D_{1}(x_{1})&\cdots&D_{1}(x_{n})\\ \cdots&\cdots&\cdots\\ D_{n}(x_{1})&\cdots&D_{n}(x_{n})\\ \end{array}\right),~{}~{}\forall~{}x_{1},x_{2},...,x_{n}\in J,

L_{l_{1}, m_{1}}^{r_{1}} \cdot L_{l_{2}, m_{2}}^{r_{2}} = L_{l_{1} + l_{2}, m_{1} + m_{2}}^{r_{1} + r_{2}}, \forall r_{i}, l_{i}, m_{i} \in Z + \frac{1}{2} Z, 1 \leq i \leq 2.

L_{l_{1}, m_{1}}^{r_{1}} \cdot L_{l_{2}, m_{2}}^{r_{2}} = L_{l_{1} + l_{2}, m_{1} + m_{2}}^{r_{1} + r_{2}}, \forall r_{i}, l_{i}, m_{i} \in Z + \frac{1}{2} Z, 1 \leq i \leq 2.

L_{0, 0}^{0} \cdot L_{l, m}^{r} = L_{l, m}^{r},

L_{0, 0}^{0} \cdot L_{l, m}^{r} = L_{l, m}^{r},

M(\alpha_{1},\alpha_{2},\alpha_{3}):=\left|\begin{array}[]{ccc}r_{1}&r_{2}&r_{3}\\ l_{1}&l_{2}&l_{3}\\ m_{1}&m_{2}&m_{3}\\ \end{array}\right|,~{}~{}\mbox{and }~{}~{}\nu=(1,1,0)\in(\mathbb{Z}+\frac{1}{2}\mathbb{Z})^{3}.

M(\alpha_{1},\alpha_{2},\alpha_{3}):=\left|\begin{array}[]{ccc}r_{1}&r_{2}&r_{3}\\ l_{1}&l_{2}&l_{3}\\ m_{1}&m_{2}&m_{3}\\ \end{array}\right|,~{}~{}\mbox{and }~{}~{}\nu=(1,1,0)\in(\mathbb{Z}+\frac{1}{2}\mathbb{Z})^{3}.

[L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{3}, m_{3}}^{r_{3}}] := M (α_{1}, α_{2}, α_{3}) L_{l_{1} + l_{2} + l_{3} - 1, m_{1} + m_{2} + m_{3} - 1}^{r_{1} + r_{2} + r_{3}} .

[L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{3}, m_{3}}^{r_{3}}] := M (α_{1}, α_{2}, α_{3}) L_{l_{1} + l_{2} + l_{3} - 1, m_{1} + m_{2} + m_{3} - 1}^{r_{1} + r_{2} + r_{3}} .

= = [[L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{3}, m_{3}}^{r_{3}}], L_{l_{4}, m_{4}}^{r_{4}}, L_{l_{5}, m_{5}}^{r_{5}}] [M (α_{1}, α_{2}, α_{3}) L_{l_{1} + l_{2} + l_{3} - 1, m_{1} + m_{2} + m_{3} - 1}^{r_{1} + r_{2} + r_{3}}, L_{l_{4}, m_{4}}^{r_{4}}, L_{l_{5}, m_{5}}^{r_{5}}] M (α_{1}, α_{2}, α_{3}) M (α_{1} + α_{2} + α_{3} - ν, α_{4}, α_{5}) L_{l_{1} + l_{2} + l_{3} + l_{4} + l_{5} - 2, m_{1} + m_{2} + m_{3} + m_{4} + m_{5} - 2}^{r_{1} + r_{2} + r_{3} + r_{4} + r_{5}},

= = [[L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{3}, m_{3}}^{r_{3}}], L_{l_{4}, m_{4}}^{r_{4}}, L_{l_{5}, m_{5}}^{r_{5}}] [M (α_{1}, α_{2}, α_{3}) L_{l_{1} + l_{2} + l_{3} - 1, m_{1} + m_{2} + m_{3} - 1}^{r_{1} + r_{2} + r_{3}}, L_{l_{4}, m_{4}}^{r_{4}}, L_{l_{5}, m_{5}}^{r_{5}}] M (α_{1}, α_{2}, α_{3}) M (α_{1} + α_{2} + α_{3} - ν, α_{4}, α_{5}) L_{l_{1} + l_{2} + l_{3} + l_{4} + l_{5} - 2, m_{1} + m_{2} + m_{3} + m_{4} + m_{5} - 2}^{r_{1} + r_{2} + r_{3} + r_{4} + r_{5}},

+ = + + = + + [[L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{4}, m_{4}}^{r_{4}}, L_{l_{5}, m_{5}}^{r_{5}}], L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{3}, m_{3}}^{r_{3}}] + [L_{l_{1}, m_{1}}^{r_{1}}, [L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{4}, m_{4}}^{r_{4}}, L_{l_{5}, m_{5}}^{r_{5}}], L_{l_{3}, m_{3}}^{r_{3}}] [L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}, [L_{l_{3}, m_{3}}^{r_{3}}, L_{l_{4}, m_{4}}^{r_{4}}, L_{l_{5}, m_{5}}^{r_{5}}]] M (α_{1}, α_{4}, α_{5}) [L_{l_{1} + l_{4} + l_{5} - 1, m_{1} + m_{4} + m_{5} - 1}^{r_{1} + r_{4} + r_{5}}, L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{3}, m_{3}}^{r_{3}}] M (α_{2}, α_{4}, α_{5}) [L_{l_{2} + l_{4} + l_{5} - 1, m_{2} + m_{4} + m_{5} - 1}^{r_{2} + r_{4} + r_{5}}, L_{l_{3}, m_{3}}^{r_{3}}, L_{l_{1}, m_{1}}^{r_{1}}] M (α_{3}, α_{4}, α_{5}) [L_{l_{3} + l_{4} + l_{5} - 1, m_{3} + m_{4} + m_{5} - 1}^{r_{3} + r_{4} + r_{5}}, L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}] M (α_{1}, α_{4}, α_{5}) M (α_{1} + α_{4} + α_{5} - ν, α_{2}, α_{3}) L_{l_{1} + l_{2} + l_{3} + l_{4} + l_{5} - 2, m_{1} + m_{2} + m_{3} + m_{4} + m_{5} - 2}^{r_{1} + r_{2} + r_{3} + r_{4} + r_{5}} M (α_{2}, α_{4}, α_{5}) M (α_{2} + α_{4} + α_{5} - ν, α_{1}, α_{3}) L_{l_{1} + l_{2} + l_{3} + l_{4} + l_{5} - 2, m_{1} + m_{2} + m_{3} + m_{4} + m_{5} - 2}^{r_{1} + r_{2} + r_{3} + r_{4} + r_{5}} M (α_{3}, α_{4}, α_{5}) M (α_{3} + α_{4} + α_{5} - ν, α_{1}, α_{2}) L_{l_{1} + l_{2} + l_{3} + l_{4} + l_{5} - 2, m_{1} + m_{2} + m_{3} + m_{4} + m_{5} - 2}^{r_{1} + r_{2} + r_{3} + r_{4} + r_{5}} .

+ = + + = + + [[L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{4}, m_{4}}^{r_{4}}, L_{l_{5}, m_{5}}^{r_{5}}], L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{3}, m_{3}}^{r_{3}}] + [L_{l_{1}, m_{1}}^{r_{1}}, [L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{4}, m_{4}}^{r_{4}}, L_{l_{5}, m_{5}}^{r_{5}}], L_{l_{3}, m_{3}}^{r_{3}}] [L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}, [L_{l_{3}, m_{3}}^{r_{3}}, L_{l_{4}, m_{4}}^{r_{4}}, L_{l_{5}, m_{5}}^{r_{5}}]] M (α_{1}, α_{4}, α_{5}) [L_{l_{1} + l_{4} + l_{5} - 1, m_{1} + m_{4} + m_{5} - 1}^{r_{1} + r_{4} + r_{5}}, L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{3}, m_{3}}^{r_{3}}] M (α_{2}, α_{4}, α_{5}) [L_{l_{2} + l_{4} + l_{5} - 1, m_{2} + m_{4} + m_{5} - 1}^{r_{2} + r_{4} + r_{5}}, L_{l_{3}, m_{3}}^{r_{3}}, L_{l_{1}, m_{1}}^{r_{1}}] M (α_{3}, α_{4}, α_{5}) [L_{l_{3} + l_{4} + l_{5} - 1, m_{3} + m_{4} + m_{5} - 1}^{r_{3} + r_{4} + r_{5}}, L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}] M (α_{1}, α_{4}, α_{5}) M (α_{1} + α_{4} + α_{5} - ν, α_{2}, α_{3}) L_{l_{1} + l_{2} + l_{3} + l_{4} + l_{5} - 2, m_{1} + m_{2} + m_{3} + m_{4} + m_{5} - 2}^{r_{1} + r_{2} + r_{3} + r_{4} + r_{5}} M (α_{2}, α_{4}, α_{5}) M (α_{2} + α_{4} + α_{5} - ν, α_{1}, α_{3}) L_{l_{1} + l_{2} + l_{3} + l_{4} + l_{5} - 2, m_{1} + m_{2} + m_{3} + m_{4} + m_{5} - 2}^{r_{1} + r_{2} + r_{3} + r_{4} + r_{5}} M (α_{3}, α_{4}, α_{5}) M (α_{3} + α_{4} + α_{5} - ν, α_{1}, α_{2}) L_{l_{1} + l_{2} + l_{3} + l_{4} + l_{5} - 2, m_{1} + m_{2} + m_{3} + m_{4} + m_{5} - 2}^{r_{1} + r_{2} + r_{3} + r_{4} + r_{5}} .

= = + = [L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{3}, m_{3}}^{r_{3}} \cdot L_{l_{4}, m_{4}}^{r_{4}}] = [L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{3} + l_{4}, m_{3} + m_{4}}^{r_{3} + r_{4}}] M (α_{1}, α_{2}, α_{3} + α_{4}) L_{l_{1} + l_{2} + l_{3} + l_{4} - 1, m_{1} + m_{2} + m_{3} + m_{4} - 1}^{r_{1} + r_{2} + r_{3} + r_{4}} M (α_{1}, α_{2}, α_{3}) L_{l_{1} + l_{2} + l_{3} + l_{4} - 1, m_{1} + m_{2} + m_{3} + m_{4} - 1}^{r_{1} + r_{2} + r_{3} + r_{4}} M (α_{1}, α_{2}, α_{4}) L_{l_{1} + l_{2} + l_{3} + l_{4} - 1, m_{1} + m_{2} + m_{3} + m_{4} - 1}^{r_{1} + r_{2} + r_{3} + r_{4}} [L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{3}, m_{3}}^{r_{3}}] \cdot L_{l_{4}, m_{4}}^{r_{4}} + L_{l_{3}, m_{3}}^{r_{3}} \cdot [L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{4}, m_{4}}^{r_{4}}],

= = + = [L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{3}, m_{3}}^{r_{3}} \cdot L_{l_{4}, m_{4}}^{r_{4}}] = [L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{3} + l_{4}, m_{3} + m_{4}}^{r_{3} + r_{4}}] M (α_{1}, α_{2}, α_{3} + α_{4}) L_{l_{1} + l_{2} + l_{3} + l_{4} - 1, m_{1} + m_{2} + m_{3} + m_{4} - 1}^{r_{1} + r_{2} + r_{3} + r_{4}} M (α_{1}, α_{2}, α_{3}) L_{l_{1} + l_{2} + l_{3} + l_{4} - 1, m_{1} + m_{2} + m_{3} + m_{4} - 1}^{r_{1} + r_{2} + r_{3} + r_{4}} M (α_{1}, α_{2}, α_{4}) L_{l_{1} + l_{2} + l_{3} + l_{4} - 1, m_{1} + m_{2} + m_{3} + m_{4} - 1}^{r_{1} + r_{2} + r_{3} + r_{4}} [L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{3}, m_{3}}^{r_{3}}] \cdot L_{l_{4}, m_{4}}^{r_{4}} + L_{l_{3}, m_{3}}^{r_{3}} \cdot [L_{l_{1}, m_{1}}^{r_{1}}, L_{l_{2}, m_{2}}^{r_{2}}, L_{l_{4}, m_{4}}^{r_{4}}],

A_{1} (0, 1; 1, 0), A_{2} (0, 1; 1, 0), A_{3} (0, 1; 1, 0), A_{4} (\frac{1}{2}) \mbox an d A_{5} (\frac{1}{2})

A_{1} (0, 1; 1, 0), A_{2} (0, 1; 1, 0), A_{3} (0, 1; 1, 0), A_{4} (\frac{1}{2}) \mbox an d A_{5} (\frac{1}{2})

S_{1}=\big{\{}-L_{0,1}^{r}+\sqrt{-1}rL_{1,0}^{r},~{}~{}-L_{1,0}^{r}|~{}~{}r\in\mathbb{Z}\big{\}},~{}~{}S_{2}=\big{\{}-L_{0,1}^{r}-\sqrt{-1}rL_{1,0}^{r},~{}~{}L_{1,0}^{r}|~{}~{}r\in\mathbb{Z}\big{\}},

S_{1}=\big{\{}-L_{0,1}^{r}+\sqrt{-1}rL_{1,0}^{r},~{}~{}-L_{1,0}^{r}|~{}~{}r\in\mathbb{Z}\big{\}},~{}~{}S_{2}=\big{\{}-L_{0,1}^{r}-\sqrt{-1}rL_{1,0}^{r},~{}~{}L_{1,0}^{r}|~{}~{}r\in\mathbb{Z}\big{\}},

S_{3}=\big{\{}L_{0,1}^{r},L_{1,0}^{-r}|~{}~{}r\in\mathbb{Z}\big{\}},~{}~{}\quad S_{4}=\big{\{}L_{\frac{1}{2},r}^{(-1)^{r}}|~{}~{}r\in\mathbb{Z}\big{\}},\quad S_{5}=\big{\{}L_{\frac{1}{2},m+\frac{1}{2}}^{r}|~{}~{}r,m\in\mathbb{Z}\big{\}},

S_{3}=\big{\{}L_{0,1}^{r},L_{1,0}^{-r}|~{}~{}r\in\mathbb{Z}\big{\}},~{}~{}\quad S_{4}=\big{\{}L_{\frac{1}{2},r}^{(-1)^{r}}|~{}~{}r\in\mathbb{Z}\big{\}},\quad S_{5}=\big{\{}L_{\frac{1}{2},m+\frac{1}{2}}^{r}|~{}~{}r,m\in\mathbb{Z}\big{\}},

A_{1} (0, 1; 1, 0)

A_{1} (0, 1; 1, 0)

A_{2} (0, 1; 1, 0)

A_{3} (0, 1; 1, 0)

A_{4} (\frac{1}{2})

A_{5} (\frac{1}{2})

{f_{1}, f_{2}, \dots, f_{n}} = \frac{\partial ( f _{1} , f _{2} , \dots , f _{n} )}{\partial ( x _{1} , x _{2} , \dots , x _{n} )} .

{f_{1}, f_{2}, \dots, f_{n}} = \frac{\partial ( f _{1} , f _{2} , \dots , f _{n} )}{\partial ( x _{1} , x _{2} , \dots , x _{n} )} .

X_{l, m}^{r} = y^{l} z^{m} e^{r x}, \forall r, l, m \in Z + \frac{1}{2} Z,

X_{l, m}^{r} = y^{l} z^{m} e^{r x}, \forall r, l, m \in Z + \frac{1}{2} Z,

B = l, m, r \in Z + \frac{1}{2} Z \sum R X_{l, m}^{r} \mbox (or B = l, m, r \in Z + \frac{1}{2} Z \sum C X_{l, m}^{r})

B = l, m, r \in Z + \frac{1}{2} Z \sum R X_{l, m}^{r} \mbox (or B = l, m, r \in Z + \frac{1}{2} Z \sum C X_{l, m}^{r})

X_{l_{1}, m_{1}}^{r_{1}} \cdot X_{l_{2}, m_{2}}^{r_{2}} = y^{l_{1}} z^{m_{1}} e^{r_{1} x} \cdot y^{l_{2}} z^{m_{2}} e^{r_{2} x} = y^{l_{1} + l_{2}} z^{m_{1} + m_{2}} e^{(r_{1} + r_{2}) x} = X_{l_{1} + l_{2}, m_{1} + m_{2}}^{r_{1} + r_{2}} .

X_{l_{1}, m_{1}}^{r_{1}} \cdot X_{l_{2}, m_{2}}^{r_{2}} = y^{l_{1}} z^{m_{1}} e^{r_{1} x} \cdot y^{l_{2}} z^{m_{2}} e^{r_{2} x} = y^{l_{1} + l_{2}} z^{m_{1} + m_{2}} e^{(r_{1} + r_{2}) x} = X_{l_{1} + l_{2}, m_{1} + m_{2}}^{r_{1} + r_{2}} .

[X_{l_{1},m_{1}}^{r_{1}},X_{l_{2},m_{2}}^{r_{2}},X_{l_{3},m_{3}}^{r_{3}}]=\left|\begin{array}[]{ccc}\frac{\partial X_{l_{1},m_{1}}^{r_{1}}}{\partial x}&\frac{\partial X_{l_{2},m_{2}}^{r_{2}}}{\partial x}&\frac{\partial X_{l_{3},m_{3}}^{r_{3}}}{\partial x}\\ \\ \frac{\partial X_{l_{1},m_{1}}^{r_{1}}}{\partial y}&\frac{\partial X_{l_{2},m_{2}}^{r_{2}}}{\partial y}&\frac{\partial X_{l_{3},m_{3}}^{r_{3}}}{\partial y}\\ \\ \frac{\partial X_{l_{1},m_{1}}^{r_{1}}}{\partial z}&\frac{\partial X_{l_{2},m_{2}}^{r_{2}}}{\partial z}&\frac{\partial X_{l_{3},m_{3}}^{r_{3}}}{\partial z}\\ \end{array}\right|.

[X_{l_{1},m_{1}}^{r_{1}},X_{l_{2},m_{2}}^{r_{2}},X_{l_{3},m_{3}}^{r_{3}}]=\left|\begin{array}[]{ccc}\frac{\partial X_{l_{1},m_{1}}^{r_{1}}}{\partial x}&\frac{\partial X_{l_{2},m_{2}}^{r_{2}}}{\partial x}&\frac{\partial X_{l_{3},m_{3}}^{r_{3}}}{\partial x}\\ \\ \frac{\partial X_{l_{1},m_{1}}^{r_{1}}}{\partial y}&\frac{\partial X_{l_{2},m_{2}}^{r_{2}}}{\partial y}&\frac{\partial X_{l_{3},m_{3}}^{r_{3}}}{\partial y}\\ \\ \frac{\partial X_{l_{1},m_{1}}^{r_{1}}}{\partial z}&\frac{\partial X_{l_{2},m_{2}}^{r_{2}}}{\partial z}&\frac{\partial X_{l_{3},m_{3}}^{r_{3}}}{\partial z}\\ \end{array}\right|.

\begin{split}&[X_{l_{1},m_{1}}^{r_{1}},X_{l_{2},m_{2}}^{r_{2}},X_{l_{3},m_{3}}^{r_{3}}]=[y^{l_{1}}z^{m_{1}}e^{r_{1}x},y^{l_{2}}z^{m_{2}}e^{r_{2}x},y^{l_{3}}z^{m_{3}}e^{r_{3}x}]\\ \\ =&\left|\begin{array}[]{ccc}r_{1}y^{l_{1}}z^{m_{1}}e^{r_{1}x}&r_{2}y^{l_{2}}z^{m_{2}}e^{r_{2}x}&r_{3}y^{l_{3}}z^{m_{3}}e^{r_{3}x}\\ l_{1}y^{l_{1}-1}z^{m_{1}}e^{r_{1}x}&l_{2}y^{l_{2}-1}z^{m_{2}}e^{r_{2}x}&l_{3}y^{l_{3}-1}z^{m_{3}}e^{r_{3}x}\\ m_{1}y^{l_{1}}z^{m_{1}-1}e^{r_{1}x}&m_{2}y^{l_{2}}z^{m_{2}-1}e^{r_{2}x}&m_{3}y^{l_{3}}z^{m_{3}-1}e^{r_{3}x}\\ \end{array}\right|\\ =&M(\alpha_{1},\alpha_{2},\alpha_{3})y^{l_{1}+l_{2}+l_{3}-1}z^{m_{1}+m_{2}+m_{3}-1}e^{r_{1}+r_{2}+r_{3}}\\ =&M(\alpha_{1},\alpha_{2},\alpha_{3})X_{l_{1}+l_{2}+l_{3}-1,m_{1}+m_{2}+m_{3}-1}^{r_{1}+r_{2}+r_{3}},\\ \end{split}

\begin{split}&[X_{l_{1},m_{1}}^{r_{1}},X_{l_{2},m_{2}}^{r_{2}},X_{l_{3},m_{3}}^{r_{3}}]=[y^{l_{1}}z^{m_{1}}e^{r_{1}x},y^{l_{2}}z^{m_{2}}e^{r_{2}x},y^{l_{3}}z^{m_{3}}e^{r_{3}x}]\\ \\ =&\left|\begin{array}[]{ccc}r_{1}y^{l_{1}}z^{m_{1}}e^{r_{1}x}&r_{2}y^{l_{2}}z^{m_{2}}e^{r_{2}x}&r_{3}y^{l_{3}}z^{m_{3}}e^{r_{3}x}\\ l_{1}y^{l_{1}-1}z^{m_{1}}e^{r_{1}x}&l_{2}y^{l_{2}-1}z^{m_{2}}e^{r_{2}x}&l_{3}y^{l_{3}-1}z^{m_{3}}e^{r_{3}x}\\ m_{1}y^{l_{1}}z^{m_{1}-1}e^{r_{1}x}&m_{2}y^{l_{2}}z^{m_{2}-1}e^{r_{2}x}&m_{3}y^{l_{3}}z^{m_{3}-1}e^{r_{3}x}\\ \end{array}\right|\\ =&M(\alpha_{1},\alpha_{2},\alpha_{3})y^{l_{1}+l_{2}+l_{3}-1}z^{m_{1}+m_{2}+m_{3}-1}e^{r_{1}+r_{2}+r_{3}}\\ =&M(\alpha_{1},\alpha_{2},\alpha_{3})X_{l_{1}+l_{2}+l_{3}-1,m_{1}+m_{2}+m_{3}-1}^{r_{1}+r_{2}+r_{3}},\\ \end{split}

(- \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}} + \frac{1}{2} m ω^{2} x^{2}) ψ (x) = E ψ (x) .

(- \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}} + \frac{1}{2} m ω^{2} x^{2}) ψ (x) = E ψ (x) .

x = \frac{ℏ}{mω} q,

x = \frac{ℏ}{mω} q,

\frac{ℏ ω}{2} (- \frac{d ^{2}}{d q ^{2}} + q^{2}) ψ (q) = E ψ (q),

\frac{ℏ ω}{2} (- \frac{d ^{2}}{d q ^{2}} + q^{2}) ψ (q) = E ψ (q),

- \frac{d ^{2}}{d q ^{2}} + q^{2} = (- \frac{d}{d q} + q) (\frac{d}{d q} + q) + \frac{d}{d q} q - q \frac{d}{d q} .

- \frac{d ^{2}}{d q ^{2}} + q^{2} = (- \frac{d}{d q} + q) (\frac{d}{d q} + q) + \frac{d}{d q} q - q \frac{d}{d q} .

(\frac{d}{d q} q - q \frac{d}{d q}) f (q) = \frac{d}{d q} (q f (q)) - q \frac{df ( q )}{d q} = f (q),

(\frac{d}{d q} q - q \frac{d}{d q}) f (q) = \frac{d}{d q} (q f (q)) - q \frac{df ( q )}{d q} = f (q),

\frac{d}{d q} q - q \frac{d}{d q} = 1.

\frac{d}{d q} q - q \frac{d}{d q} = 1.

- \frac{d ^{2}}{d q ^{2}} + q^{2} = (- \frac{d}{d q} + q) (\frac{d}{d q} + q) + 1.

- \frac{d ^{2}}{d q ^{2}} + q^{2} = (- \frac{d}{d q} + q) (\frac{d}{d q} + q) + 1.

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TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology

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00footnotetext: Corresponding author: Ruipu Bai, E-mail: [email protected].

The infinite dimensional Unital 3-Lie Poisson algebra

Chuangchuang Kang

College of Mathematics and Information Science, Hebei University, Baoding 071002, P.R. China

[email protected]

,

Ruipu Bai

College of Mathematics and Information Science, Hebei University, Key Laboratory of Machine Learning and Computational Intelligence of Hebei Province, Baoding 071002, P.R. China

[email protected]

and

Yingli Wu

College of Mathematics and Information Science, Hebei University, Baoding 071002, P.R. China

[email protected]

Abstract.

From a commutative associative algebra $A$ , the infinite dimensional unital 3-Lie Poisson algebra $\mathfrak{L}$ is constructed, which is also a canonical Nambu 3-Lie algebra, and the structure of $\mathfrak{L}$ is discussed. It is proved that: (1) there is a minimal set of generators $S$ consisting of six vectors; (2) the quotient algebra $\mathfrak{L}/\mathbb{F}L_{0,0}^{0}$ is a simple 3-Lie Poisson algebra; (3) four important infinite dimensional 3-Lie algebras: 3-Virasoro-Witt algebra $\mathcal{W}_{3}$ , $A_{\omega}^{\delta}$ , $A_{\omega}$ and the 3- $W_{\infty}$ algebra can be embedded in $\mathfrak{L}$ .

Key words and phrases:

3-Lie algebra, 3-Lie Poisson algebra, canonical Nambu 3-Lie algebra

1. Introduction

The role of $n$ -Lie Poisson algebra in string theory has received more and more attention in recent years([1, 2, 3, 4]). For example, the applications of $n$ -Lie Poisson algebra in noncommutative geometry and the quantum geometry of branes in M-theory have been considered in [5]. The theory of $n$ -Lie Poisson algebras provides an useful method to describe multiple M2 branes ([6, 7, 8]), and algebraical and geometrical structures of Nambu-Poisson manifold ([9, 10, 11]).

The discovery of 3-bracket in 1973 triggered a huge amount of innovative scientific inquiry. In [12], N. Nambu first proposed the notion of 3-bracket for constructing the generalized Hamiltonian dynamics. In [13], Takhtajan studied the algebraic structures in Nambu mechanics, and indicated the relation between Nambu mechanics and $n$ -Lie algebras. In 1985, Filippov introduced $n$ -Lie algebras ([14]), and then the structures are studied in [15, 16, 17, 18, 19, 20]. These earlier studies demonstrate a strong and consistent association between $n$ -Lie algebras and $n$ -Lie Poisson algebras. The $n$ -Lie Poisson algebra comes from the concept of generalized Poisson ( $n$ -Poisson or Nambu-Poisson) structure which is naturally defined for $n$ -brackets with an even number of entries, and parallels the properties of higher order generalized Lie algebras (or $n$ -Lie algebras)([21]).

An $n$ -Lie Poisson algebra is an associative commutative algebra with a totally antisymmetric $n$ -Lie bracket satisfying the generalized Leibniz rule and the fundamental identity (FI) given in [14]. Actually, removing the property of Leibniz rule requirement for the $n$ -Lie Poisson algebras, while keeping the fundamental identity, we get $n$ -Lie algebras. Further, if the $n$ -bracket need not be anticommutative, we can get another useful algebras which are called $n$ -Leibniz algebras (or $n$ -Loday algebras) ([22, 23]). Physically, the fundamental identity is a consistency condition for the time evolution, which is given in terms of $(n-1)$ Hamiltonian functions that determines derivations of $n$ -Lie algebras([24]).

However, the main challenge faced by researchers is how to find $n$ -Lie Poisson algebras. Since the multiple multiplication and Leibniz rule should be satisfied simultaneously, the structure of $n$ -Lie Poisson algebras is more complicated than that of $n$ -Lie algebras. So the construction of 3-Lie Poisson algebras as the special case of $n$ -Lie Poisson algebras is very important.

In [25], Curtright constructed a 3-Virasoro-Witt algebra through the use of $su(1,1)$ enveloping algebra techniques. In [26], Bai provided two methods for constructing infinite dimensional 3-Lie algebras $A_{\omega}$ and $A_{\omega}^{\delta}$ from group algebras. In [27], Chakrabortty obtained $W_{\infty}$ 3-algebras by using “lone-star” product of generators in $W_{\infty}$ -algebras as well as their commutation relations, and appropriate double scaling limits. Connected strongly to physics, the 3-Lie algebras constructed as above are very important.

The purpose of this paper is to construct an infinite dimensional 3-Lie Poisson algebra $\mathfrak{L}$ which contains 3-Virasoro-Witt algebra $\mathcal{W}_{3}$ , 3-Lie algebras $A_{\omega}$ , $A_{\omega}^{\delta}$ , and $W_{\infty}$ 3-algebras simultaneously.

In Section 2, we introduce some basic notions. In Section 3, we provide an infinite dimensional unital 3-Lie Poisson algebra $\mathfrak{L}$ and realize it by canonical Nambu $3$ -Lie algebras. Section 4 we pay close attention to the Lie structure of unital 3-Lie Poisson algebra $\mathfrak{L}$ and study four types of 3-Lie subalgebras. Section 5 is devoted to derivations of unital 3-Lie Poisson algebra.

Unless otherwise stated, algebras and vector spaces are over a field $\mathbb{F}$ of characteristic zero, $\mathbb{Z}$ is the set of integers, $\mathbb{Z}^{+}$ is the set of positive integers, and $\mathbb{R}$ is the real field. For any algebra $A$ , and $S\subseteq A$ , we use $\langle S\rangle$ to denote the subalgebra of $A$ generated by $S$ .

2. Preliminary

Definition 2.1.

[14]** An $n$ -Lie algebra $A$ * is a vector space over $\mathbb{F}$ endowed with an $n$ -ary multi-linear skew-symmetric multiplication $[,\cdots,]$ satisfying Fundamental Identity (FI), for all $x_{1},x_{2},\cdots,x_{n},y_{2},\cdots,y_{n}\in A$ ,*

[TABLE]

Let $A$ be an $n$ -Lie algebra. For any $x_{1},\cdots,x_{n-1}\in A$ , the linear mapping $\mbox{ad}(x_{1},\cdots,x_{n-1}):A\rightarrow A$ defined by

[TABLE]

is called the left multiplication determined by $x_{1},\cdots,x_{n-1}$ . Thanks to (2.1), left multiplications are derivations.

Definition 2.2.

[28]** An $n$ -Lie Poisson algebra* (or Nambu-Poisson algebra) $(U,\cdot,[,\cdots,])$ over a field $\mathbb{F}$ is a linear vector space $U$ with $\mathbb{F}$ -linear multiplications $\cdot:U\otimes U\rightarrow U$ , and $[,\cdots,]:\wedge^{n}U\rightarrow U$ satisfying*

•

$(U,\cdot)$ * is an associative commutative algebra;*

•

$(U,[,\cdots,])$ * is an n-Lie algebra;*

•

the following Leibniz rule holds:

[TABLE]

If there is an unit element in $(A,\cdot)$ , then $A=(U,\cdot,[,\cdots,])$ is called an unital $n$ -Lie Poisson algebra.

An $n$ -Lie Poisson algebra $A=(U,\cdot,[,\cdots,])$ is usually denoted by $A=(U,[,\cdots,])$ or $A$ , for all $u,v\in U$ , $u\cdot v$ is denoted by $uv.$

Recall Jacobian algebras given in [26]. Let $J$ be a commutative associative algebra, $D_{1},\cdots,D_{n}$ be commutative derivations of $J$ , then the multiplication on $J$ defined by

[TABLE]

satisfies Definition 2.2. Therefore, $(J,[,\cdots,])$ is an $n$ -Lie Poisson algebra, and it is called the Jacobian algebra defined by commutative derivations $\{D_{1},\cdots,D_{n}\}$ .

3. Unital 3-Lie Poisson algebra

Let $A$ be a commutative associative algebra with a basis $\{L_{l,m}^{r}~{}|~{}l,m,r\in\mathbb{Z}+\frac{1}{2}\mathbb{Z}\}$ , and the multiplication in the basis be as follows

[TABLE]

Since for all $r,l,m\in\mathbb{Z}+\frac{1}{2}\mathbb{Z}$ ,

[TABLE]

$L_{0,0}^{0}$ is the only unit element of $A$ .

For convenience, for all $\alpha_{1}=(l_{1},m_{1},r_{1}),~{}\alpha_{2}=(l_{2},m_{2},r_{2}),~{}\alpha_{3}=(l_{3},m_{3},r_{3})\in(\mathbb{Z}+\frac{1}{2}\mathbb{Z})^{3}$ , $M(\alpha_{1},\alpha_{2},\alpha_{3})$ denotes the determinant

[TABLE]

Define the 3-ary multiplication $[,,]:A^{\wedge 3}\rightarrow A$ as follows, $\forall~{}~{}\alpha_{1}=(l_{1},m_{1},r_{1})$ , $\alpha_{2}=(l_{2},m_{2},r_{2})$ , $\alpha_{3}=(l_{3},m_{3},r_{3})\in(\mathbb{Z}+\frac{1}{2}\mathbb{Z})^{3}$ ,

[TABLE]

By the above notations, we have the following result.

Theorem 3.1.

$(A,\cdot,[,,])$ * is an unital 3-Lie Poisson algebra.*

Proof.

We only need to prove that (2.1) and (2.3) hold. By (3.4), for all $\alpha_{i}=(l_{i},m_{i},r_{i})\in(\mathbb{Z}+\frac{1}{2}\mathbb{Z})^{3},~{}1\leq i\leq 5$ ,

[TABLE]

Thanks to the properties of determinant, (2.1) holds. Since

[TABLE]

(2.3) holds. Therefore, $(A,\cdot,[,,])$ is an unital 3-Lie Poisson algebra. ∎

In the following, $\mathfrak{L}$ ** denotes the unital 3-Lie Poisson algebra $(A,\cdot,[,,])$ in Theorem 3.1, and $\mathfrak{A}$ denotes the $3$ -Lie algebra structure of $\mathfrak{L}$ .**

Now we give some symbols. Let

[TABLE]

be 3-Lie subalgebras of 3-Lie algebra $\mathfrak{A}$ generated by the subsets

[TABLE]

respectively. Then

[TABLE]

In [13], authors gave the canonical Nambu 3-Lie algebra on $\mathbb{R}$ . Let $V=C^{\infty}(\mathbb{R}^{n})$ (or $V=C^{\infty}(\mathbb{C}^{n})$ ). Then $(V,\{,\cdots,\})$ is a canonical Nambu $n$ -Lie algebra, where for $\forall f_{1},f_{2},\cdots,f_{n}\in V$ with $n$ coordinates $x_{1},x_{2},\cdots,x_{n}\in\mathbb{R}$ ( or $x_{1},x_{2},\cdots,x_{n}\in\mathbb{C}$ ),

[TABLE]

Theorem 3.2.

Let $\mathfrak{L}$ be the 3-Lie Poisson algebra in Theorem 3.1 over the real field $\mathbb{R}$ (or in the complex field $\mathbb{C}$ ), then $\mathfrak{L}$ can be realized by a canonical Nambu 3-Lie algebra.

Proof.

Let

[TABLE]

where $y,z\in\mathbb{R}^{+}=[0,+\infty],x\in\mathbb{R}$ (or $x,y,z\in\mathbb{C}$ ). Then

[TABLE]

is an associative commutative algebra with the multiplication

[TABLE]

Define the 3-Lie multiplication

[TABLE]

Thanks to Eq (2.4), $(B,\cdot,[,,])$ is a 3-Lie Poisson algebra, and

[TABLE]

where $\alpha_{i}=(l_{i},m_{i},r_{i})\in(\mathbb{Z}+\frac{1}{2}\mathbb{Z})^{\otimes 3},1\leq i\leq 3$ . Therefore, $\chi:\mathfrak{L}\rightarrow B$ defined by, $\chi(L_{l,m}^{r})=X_{l,m}^{r}$ , for all $l,m,r\in\mathbb{Z}+\frac{1}{2}\mathbb{Z}$ , is an algebra isomorphism between $\mathfrak{L}$ and $(B,\cdot,[,,])$ . ∎

4. Applications of unital 3-Lie Poisson algebra $\mathfrak{L}$

In this section, we study some applications of $\mathfrak{L}$ . We will prove that four important 3-Lie algebras: 3-Virasoro-Witt algebra $\mathcal{W}_{3}$ in [25], $A_{\omega}^{\delta}$ in [29], $A_{\omega}$ in [26] and 3- $W_{\infty}$ algebra in [27] can be embedded in $\mathfrak{L}$ .

4.1. 3-Virasoro-Witt algebra

Witt algebra (centerless Virasoro algebra) is an important complex Lie algebra in two-dimensional conformal field theory and string theory. It has been generalized to higher arties as 3-Virasoro-Witt algebra in [25]. First we show that how to construct 3-Virasoro-Witt algebra by creation and annihilation operators $a^{\dagger}$ and $a$ .

We know that $a^{\dagger}$ and $a$ have wide applications in quantum mechanics. Consider the Schrödinger equation for the one-dimensional time independent quantum harmonic oscillator

[TABLE]

Set

[TABLE]

a direct computation yields that

[TABLE]

moreover

[TABLE]

For arbitrary differentiable function $f(q)$ , since

[TABLE]

we have

[TABLE]

Therefore, (4.2) can be reduced to

[TABLE]

So the Schrödinger equation (4.1) becomes

[TABLE]

Define the creation operator and annihilation operator as

[TABLE]

then the Schrödinger equation reduces to

[TABLE]

It is a simplification of the Schrödinger equation. Furthermore, set $p=-i{\frac{d}{dq}}$ , then $[q,p]=i$ , and

[TABLE]

Thanks to (4.3) and (4.4),

[TABLE]

Set

[TABLE]

where $\gamma,\beta$ are parameters. Using the Nambu commutator

[TABLE]

and Eq (4.5), we have

[TABLE]

Replacing the basis in Eq (4.8) with

[TABLE]

and taking limit $\beta\rightarrow\infty$ , we get the 3-Virasoro-Witt algebra.

Lemma 4.1.

[25]** Let $V$ be a commutative associative algebra with a basis $\{Q_{n},R_{n}\}_{n\in\mathbb{Z}}$ , and define 3-ary linear skew-symmetric multiplication $[,,]$ on $V$ as follows

[TABLE]

where z is a parameter. Then the multiplication (4.9) does not satisfy Eq (2.1), except when $z=\pm 2\sqrt{-1}$ . In that cases, $(V,[,,])$ is a 3-Lie algebra, which is called 3-Virasoro-Witt algebra, and is denoted by $\mathcal{W}_{3}$ .

Theorem 4.2.

3-Virasoro-Witt algebras $\mathcal{W}_{3}({z=2\sqrt{-1}})$ and $\mathcal{W}_{3}({z=-2\sqrt{-1}})$ ) can be embedded in $\mathfrak{L}$ , and which are isomorphic to $\mathfrak{A}_{1}~{}(0,1~{}~{};~{}~{}1,0)$ and $\mathfrak{A}_{2}~{}(0,1~{}~{};1,0~{}~{})$ , respectively.

Proof.

From (3.4) and (3.5), we have

[TABLE]

Thanks to (4.9), $\Gamma_{1}:$ $\mathfrak{A}_{1}(0,1~{}~{};~{}~{}1,0)\rightarrow$ $\mathcal{W}_{3}(z={2\sqrt{-1}})$ defined by

[TABLE]

is a 3-Lie algebra isomorphism. By a similar discussion to the above, $\Gamma_{2}:$ $\mathfrak{A}_{2}(0,1~{}~{};~{}~{}1,0)\rightarrow$ $\mathcal{W}_{3}(z={-2\sqrt{-1}})$ defined as the above is a 3-Lie algebra isomorphism. Therefore, 3-Virasoro-Witt algebras $\mathcal{W}_{3}(z={2\sqrt{-1}})$ and $\mathcal{W}_{3}(z={-2\sqrt{-1}})$ can be embedded in $\mathfrak{L}$ . ∎

4.2. The 3-Lie algebra $A_{\omega}^{\delta}$ constructed by an involution and a derivation

In this subsection, we study an infinite-dimensional 3-Lie algebra $A_{\omega}^{\delta}$ which is constructed by a commutative associative algebra, an involution and a derivation ([29]).

Lemma 4.3.

[29]** Let $E$ be a commutative associative algebra with a basis $\{S_{r},T_{r}\}_{r\in\mathbb{Z}}.$ Then $A_{\omega}^{\delta}:=(E,[,,])$ is a simple 3-Lie algebra, where

[TABLE]

Theorem 4.4.

The linear mapping $\rho:A_{\omega}^{\delta}\rightarrow\mathfrak{A}_{3}~{}(0,1~{}~{};~{}~{}1,0)$ defined by

[TABLE]

is a 3-Lie algebra isomorphism, therefore, 3-Lie algebra $A_{\omega}^{\delta}$ can be embedded in $\mathfrak{L}$ .

Proof.

By Eqs (3.4) and (3.7), and a direct computation, we have

[TABLE]

Thanks to Lemma 4.3,

[TABLE]

Then we have

[TABLE]

Therefore, 3-Lie algebra $A_{\omega}^{\delta}$ can be embedded in $\mathfrak{L}$ . ∎

4.3. 3-Lie algebra $A_{\omega}$ constructed by Laurent polynomials

Now we study the infinite dimensional 3-Lie algebra $A_{\omega}$ which is constructed by Laurent polynomials in [26]

Lemma 4.5.

[26]** Let U be a vector space with a basis $\{U_{n}\}_{n\in\mathbb{Z}}$ over $\mathbb{F}$ . Then $U$ is a simple 3-Lie algebra with the multiplication

[TABLE]

The 3-Lie algebra $(U,[,,])$ is denoted by $A_{\omega}$ .

Theorem 4.6.

The linear mapping $\tau:A_{\omega}\rightarrow\mathfrak{A}_{4}~{}(\frac{1}{2}),$ defined by

[TABLE]

is a 3-Lie algebra isomorphism, therefore, the 3-Lie algebra $A_{\omega}$ can be embedded in $\mathfrak{L}$ .

Proof.

Thanks to Eqs (3.4) and (3.7),

[TABLE]

By Lemma 4.5,

[TABLE]

It follows

[TABLE]

$\tau$ is a 3-Lie algebra isomorphism, therefore, the 3-Lie algebra $A_{\omega}$ can be embedded in $\mathfrak{L}$ . ∎

4.4. 3- $W_{\infty}$ algebra

The algebra $W_{\infty}$ is a higher-spin extension of the Virasoro algebra ([30]). In [27], authors obtained a 3- $W_{\infty}$ algebra by using “lone-star” product and commutative relations of generators in $W_{\infty}$ and appropriate double scaling limits on the generators.

Lemma 4.7.

[27]** Let $W$ be a commutative associative algebra with a basis $\{W_{m}^{r}\}_{r,m\in\mathbb{Z}}.$ Then $(W,[,,])$ is a 3-Lie algebra with the multiplication

[TABLE]

which is called 3- $W_{\infty}$ algebra, and is simply denoted by 3- $W_{\infty}$ .

Theorem 4.8.

The linear mapping $\phi:3\textrm{-}W_{\infty}\rightarrow\mathfrak{A}_{5}~{}(\frac{1}{2})$ defined by

[TABLE]

is a 3-Lie algebra isomorphism. Therefore, 3- $W_{\infty}$ can be embedded in $\mathfrak{L}$ .

Proof.

From (3.4) and (3.9), and Lemma.4.7, for $\forall m_{1},m_{2},m_{3},r_{1},r_{2},r_{3}\in\mathbb{Z},$

[TABLE]

therefore, the 3- $W_{\infty}$ algebra is isomorphic to $\mathfrak{A}_{5}~{}(\frac{1}{2})$ and 3- $W_{\infty}$ algebra can be embedded in $\mathfrak{L}$ . ∎

5. Structure of unital 3-Lie Poisson algebra $\mathfrak{L}$

From (3.2) and (3.4), $L_{0,0}^{0}$ is the center of $\mathfrak{L}$ , that is, $[L_{0,0}^{0},\mathfrak{L},\mathfrak{L}]=0$ and $L_{0,0}^{0}\cdot L_{l,m}^{r}=L_{l,m}^{r}\cdot L_{0,0}^{0}$ for all $l,m,r\in\mathbb{Z}+\frac{1}{2}\mathbb{Z}$ . Denotes $C_{0}=\mathbb{F}L_{0,0}^{0}$ . Thanks to (3.2), and $[L_{0,0}^{0},\mathfrak{A},\mathfrak{A}]=0$ , $C_{0}$ is an ideal of $\mathfrak{L}$ .

Theorem 5.1.

The quotient 3-Lie algebra $\mathfrak{A}~{}/C_{0}$ is an infinite dimensional simple 3-Lie algebra.

Proof.

Let $I$ be a nonzero ideal of the quotient algebra $\mathfrak{A}~{}/C_{0}$ , and $u\in I_{\neq C_{0}}$ . Then we can suppose

[TABLE]

If $n=1$ , then $a_{1}\neq 0$ , and for all $L_{l,m}^{r}+C_{0}\neq C_{0}$ satisfying $(l+1,m+1,r)\neq k(l_{1},m_{1},r_{1})$ ( $\forall k\in\mathbb{Z}+\frac{1}{2}\mathbb{Z}$ ), there exist $a,s,t\in\mathbb{Z}+\frac{1}{2}\mathbb{Z}$ , such that $\left|\begin{array}[]{ccc}r_{1}&a&r\\ l_{1}&s&l+1\\ m_{1}&t&m+1\\ \end{array}\right|\neq 0.$ Then

[TABLE]

therefore, for all $L_{l,m}^{r}+C_{0}\neq C_{0}$ satisfying $(l+1,m+1,r)\neq k(l_{1},m_{1},r_{1})$ , $k\in\mathbb{Z}+\frac{1}{2}\mathbb{Z}$ , we have $L_{l,m}^{r}+C_{0}\in I$ .

For $L_{l,m}^{r}+C_{0}\neq C_{0}$ satisfying $(l+1,m+1,r)=k(l_{1},m_{1},r_{1})$ , $k\in\mathbb{Z}+\frac{1}{2}\mathbb{Z}$ , there are $l^{\prime},m^{\prime},r^{\prime}\in\mathbb{Z}+\frac{1}{2}\mathbb{Z}$ , such that $(l+1,m+1,r)\neq k(l^{\prime},m^{\prime},r^{\prime})$ for all $k\in\mathbb{Z}+\frac{1}{2}\mathbb{Z}$ . By the above discussion, we have $L_{l^{\prime},m^{\prime}}^{r^{\prime}}+C_{0}\in I_{\neq C_{0}}$ , therefore, $L_{l,m}^{r}+C_{0}\in I.$ It follows $I=$$\mathfrak{A}~{}/C_{0}$ .

Now assume $n=t\in\mathbb{Z}^{+}$ is true. We will discuss the case $n=t+1$ . Thanks to (3.4), there are $r_{0}\in\mathbb{Z}+\frac{1}{2}\mathbb{Z}$ such that $r_{0}\neq 0$ and $r_{i}+r_{t+1}+r_{0}\neq 0$ for all $1\leq i\leq t$ . Then

[TABLE]

Therefore, we have $I=$$\mathfrak{A}~{}/C_{0}$ . The proof is complete. ∎

Theorem 5.2.

1) The subspace $H$ spanned by

[TABLE]

is a subalgebra of $\mathfrak{L}$ and satisfies $[H,H,H]=0$ . Furthermore, for any $L^{r}_{l,m}\in\mathfrak{L}$ , $[L^{r}_{l,m},H,H]\subseteq H$ if and only if $L^{r}_{l,m}\in H$ , and $L_{l,m}^{r}\cdot L_{i,i}^{t}\in H$ if and only if $L^{r}_{l,m}\in H$ , that is, $l=m$ .

2) For any $i,t\in\mathbb{Z}+\frac{1}{2}\mathbb{Z}$ , the left multiplication $\mbox{ad}(L_{i,i}^{t},L_{-i+1,-i+1}^{-t}):\mathfrak{L}\rightarrow\mathfrak{L}$ is semi-simple, that is,

[TABLE]

where $\mathfrak{L}(k,t)$ is the subspace of $\mathfrak{L}$ spanned by

[TABLE]

Furthermore, if $t\neq 0$ , then $H=\mathfrak{L}(0,t)$ . If $t=0$ , then $\mathfrak{L}(0,0)=\mathfrak{L}$ .

Proof.

Thanks to (3.4), for all $L_{i,i}^{s},L_{j,j}^{t},L_{h,h}^{r}\in H$

[TABLE]

therefore, $H$ is a subalgebra of $\mathfrak{L}$ and $[H,H,H]=0.$

For any $L_{l,m}^{r}\in\mathfrak{L}$ , and $\forall i,j,s,t\in\mathbb{Z}+\frac{1}{2}\mathbb{Z}$ , by (3.4),

$[L_{l,m}^{r},L_{i,i}^{s},L_{j,j}^{t}]=\left|\begin{array}[]{ccc}r&s&t\\ l&i&j\\ m&i&j\\ \end{array}\right|L_{i+j+l-1,i+j+m-1}^{r+s+t}\in H$ if and only if $i+j+l-1=i+j+m-1$ , that is, $L^{r}_{l,m}\in H$ . We get 1).

Apply (3.4) and a direct computation, for all $L_{l,m}^{r}\in\mathfrak{L},$

[TABLE]

Therefore, $L_{l,m}^{r}\in\mathfrak{L}(t(m-l),t)$ , and $\mathfrak{L}=\sum\limits_{k\in\mathbb{Z}+\frac{1}{2}\mathbb{Z}}\mathfrak{L}(k,t).$

Thanks to (5.2), if $t\neq 0$ , then $H=\mathfrak{L}(0,t)$ , and if $t=0$ , then $\mathfrak{L}(0,0)=\mathfrak{L}.$ The result 2) follows. ∎

Definition 5.3.

Let $\delta$ be a linear mapping of $\mathfrak{L}$ . If $\delta$ satisfies that $\delta\in Der(A,\cdot)\cap Der(A,[,,])$ , that is, for all $l_{i},m_{i},r_{i}\in\mathbb{Z}+\frac{1}{2}\mathbb{Z},1\leq i\leq 3,$

[TABLE]

then $\delta$ is called a derivation of $\mathfrak{L}$ .

Theorem 5.4.

The left multiplications $ad(L_{l_{1},m_{1}}^{r_{1}},L_{l_{2},m_{2}}^{r_{2}}):\mathfrak{L}\rightarrow\mathfrak{L}$ , for all $\alpha_{i}=(l_{i},m_{i},r_{i})\in(\mathbb{Z}+\frac{1}{2}\mathbb{Z})^{3},1\leq i\leq 3$ are derivations of $\mathfrak{L}$ .

Proof.

Apply Eqs (2.1), (2.2) and (3.1). ∎

Lemma 5.5.

$\mathfrak{L}$ * has the minimal set of generators*

[TABLE]

Proof.

By (3.1), $(L_{l,m}^{r})^{j}=\underbrace{L_{l,m}^{r}\cdot L_{l,m}^{r}\cdot...\cdot L_{l,m}^{r}}_{j}=L_{jl,jm}^{jr}$ , $\forall r,l,m\in\mathbb{Z}+\frac{1}{2}\mathbb{Z},~{}j\in\mathbb{Z}_{\geq 0}$ , then

[TABLE]

Thanks to $L_{l,m}^{r}=L_{l,0}^{0}\cdot L_{0,m}^{0}\cdot L_{0,0}^{r}$ , we get $S=\{L_{0,0}^{-1},L_{-1,0}^{0},L_{0,-1}^{0},L_{0,0}^{\frac{1}{2}},L_{\frac{1}{2},0}^{0},L_{0,\frac{1}{2}}^{0}\}$ is a minimal set of generators of $\mathfrak{L}$ . ∎

Theorem 5.6.

Derivation algebra $Der(A,\cdot)$ of commutative associative algebra $(A,\cdot)$ is spanned by

[TABLE]

Proof.

By (5.3),

[TABLE]

and

[TABLE]

Then we have

[TABLE]

Similarly, we have

[TABLE]

Thanks to $\delta\in Der(A,\cdot)$ , we have

[TABLE]

It follows the result.∎

By Lemma 5.5 and Theorem 5.6, for discussing derivations of $\mathfrak{L}$ , we need to study the properties of $\delta(L_{0,0}^{\frac{1}{2}})$ , $\delta(L_{\frac{1}{2},0}^{0})$ and $\delta(L_{0,\frac{1}{2}}^{0})$ , where $\delta\in Der(\mathfrak{L})$ . So, for any $\delta\in Der(\mathfrak{L})$ , suppose

[TABLE]

Theorem 5.7.

If $\delta\in Der(A,\cdot)$ , then $\delta$ is a derivation of $\mathfrak{L}$ if and only if

[TABLE]

satisfies the following identities

[TABLE]

Proof.

For any $\delta\in Der(A,\cdot)$ , thanks to Theorem 5.6, $\delta$ is a derivation of $\mathfrak{L}$ if and only if $\delta$ satisfies

[TABLE]

We get that Eq (5.10) holds if and only if

[TABLE]

Therefore, if $\left|\begin{array}[]{cc}l_{2}&l_{3}\\ m_{2}&m_{3}\\ \end{array}\right|=0$ , (5.10) holds. If $\left|\begin{array}[]{cc}l_{2}&l_{3}\\ m_{2}&m_{3}\\ \end{array}\right|\neq 0$ , we have

[TABLE]

By the similar discussion to the above, Eqs (5.11) and (5.12) hold if and only if

[TABLE]

hold, respectively. Thanks to $L_{l+l_{2}+l_{3}-1,m+m_{2}+m_{3}-1}^{r+r_{2}+r_{3}}\neq 0$ , and Eqs (5.14), (5.15) and (5.16), we get that $\delta$ is a derivation of $\mathfrak{L}$ if and only if Eq (5.9) holds. ∎

Bibliography30

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] Axenides M. and Floratos E., Nambu-Lie 3-algebras on fuzzy 3-manifolds, J. High Energy Phys . 2009, 2009(2):39-39.
2[2] Sochichiu C., On Nambu-Lie 3-algebra representations, ar Xiv preprint ar Xiv:0806.3520, 2008.
3[3] de Azc rraga J. A. and Izquierdo J. M., n-ary algebras: a review with applications, J. Math. Phys . 2010, 43.29:293001.
4[4] Papadopoulos G., M 2-branes, 3-Lie algebras and Pl cker relations, J. High Energy Phys . 2008.05 (2008): 054.
5[5] De Bellis J., Saemann C.and Szabo R. J., Quantized Nambu Poisson manifolds and n-Lie algebras, J. Math. Phys . 2010, 51(12): 122303.
6[6] Bagger J. and Lambert N., Gauge symmetry and supersymmetry of multiple M 2-branes, Phys.rev.d . 2008, 77(6):215-240.
7[7] Bagger J. and Lambert N., Modeling Multiple M 2’s, Phys.rev.d . 2006, 75.
8[8] Gustavsson A., Algebraic structures on parallel M 2-branes, Nuclear Phys. B . 2009, 811(1):66-76.

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The infinite dimensional Unital 3-Lie Poisson algebra

Abstract.

Key words and phrases:

1. Introduction

2. Preliminary

Definition 2.1**.**

Definition 2.2**.**

3. Unital 3-Lie Poisson algebra

Theorem 3.1**.**

Proof.

Theorem 3.2**.**

Proof.

4. Applications of unital 3-Lie Poisson algebra L\mathfrak{L}L

4.1. 3-Virasoro-Witt algebra

Lemma 4.1**.**

Theorem 4.2**.**

Proof.

4.2. The 3-Lie algebra AωδA_{\omega}^{\delta}Aωδ​ constructed by an involution and a derivation

Lemma 4.3**.**

Theorem 4.4**.**

Proof.

4.3. 3-Lie algebra AωA_{\omega}Aω​ constructed by Laurent polynomials

Lemma 4.5**.**

Theorem 4.6**.**

Proof.

4.4. 3-W∞W_{\infty}W∞​ algebra

Lemma 4.7**.**

Theorem 4.8**.**

Proof.

5. Structure of unital 3-Lie Poisson algebra L\mathfrak{L}L

Theorem 5.1**.**

Proof.

Theorem 5.2**.**

Proof.

Definition 5.3**.**

Theorem 5.4**.**

Proof.

Lemma 5.5**.**

Proof.

Theorem 5.6**.**

Proof.

Theorem 5.7**.**

Proof.

Definition 2.1.

Definition 2.2.

Theorem 3.1.

Theorem 3.2.

4. Applications of unital 3-Lie Poisson algebra $\mathfrak{L}$

Lemma 4.1.

Theorem 4.2.

4.2. The 3-Lie algebra $A_{\omega}^{\delta}$ constructed by an involution and a derivation

Lemma 4.3.

Theorem 4.4.

4.3. 3-Lie algebra $A_{\omega}$ constructed by Laurent polynomials

Lemma 4.5.

Theorem 4.6.

4.4. 3- $W_{\infty}$ algebra

Lemma 4.7.

Theorem 4.8.

5. Structure of unital 3-Lie Poisson algebra $\mathfrak{L}$

Theorem 5.1.

Theorem 5.2.

Definition 5.3.

Theorem 5.4.

Lemma 5.5.

Theorem 5.6.

Theorem 5.7.