Discussion on the Complex Structure of Nilpotent Lie Groups G_k

Caiyu Du, Yu Wang

College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, China.

DOI: 10.4236/ojapps.2024.146092                

Abstract

Consider the real, simply-connected, connected, s-step nilpotent Lie group G endowed with a left-invariant, integrable almost complex structure J, which is nilpotent. Consider the simply-connected, connected nilpotent Lie group G_k, defined by the nilpotent Lie algebra g/a_k, where g is the Lie algebra of G, and a_k is an ideal of g. Then, J gives rise to an almost complex structure J_k on G_k. The main conclusion obtained is as follows: if the almost complex structure J of a nilpotent Lie group G is nilpotent, then J can give rise to a left-invariant integrable almost complex structure J_k on the nilpotent Lie group G_k, and J_k is also nilpotent.

Keywords

Almost Complex Structure, Nilpotent Lie Group, Nilpotent Lie Algebra

1. Introduction

In the year 2000, Cordero and others [1] conducted research on nilpotent complex structures on connected simply connected real even-dimensional nilpotent Lie groups G with left-invariant integrable almost complex structures. They provided definitions for an ascending sequence $\left\{a_k,k\ge 0\right\}$ compatible with the integrable almost complex structure J of G, as well as the definition of nilpotent complex structure. Building upon Cordero et al.’s research on nilpotent complex structures, this paper demonstrates that if the left-invariant integrable almost complex structure J on the Lie group G is nilpotent, then J can induce a left-invariant integrable almost complex structure J_k on G_k, and J_k is also nilpotent. The study of nilpotent complex structures on the nilpotent Lie group G_k can further investigate topics such as spectral sequences, Dolbeault cohomology groups, and minimal models of compact nilpotent manifolds discussed in references [2] [3] [4] .

The aim of this paper is to investigate the scenario of a connected simply-connected s-step nilpotent Lie group G with a left-invariant integrable almost complex structure J, where J is nilpotent. Through the examination of the connected simply-connected nilpotent Lie group G_k defined by the nilpotent Lie algebra g/a_k, the objective is to ascertain whether J can induce an almost complex structure J_k on G_k, and further demonstrate that J_k is also nilpotent.

In addressing this issue, the paper is divided into two parts. The first part serves as background knowledge, introducing fundamental concepts related to connected simply connected s-step nilpotent Lie groups G with left-invariant integrable almost complex structures. The second part provides evidence that if the left-invariant integrable almost complex structure J is nilpotent, then J can induce a left-invariant integrable almost complex structure J_k on G_k, and J_k is also nilpotent.

2. Background Knowledge

2.1. Integrable Complex Structure

Let V connected simply connected 2n-dimensional real vector space. The so-called complex structure J on V is a linear transformation $J:V\to V$ , satisfying:

$J^2=-\text{id}:V\to V$ .

Let M be a 2n-dimensional smooth manifold, and J be a smooth (1,1)-type tensor field on M. For each point $x\in M$ , J_k a linear transformation from the tangent space $T_{x}M$ to itself. If each $J_k\left(x\right)\left(x\in M\right)$ is a complex structure on the tangent space $T_{x}M$ , then the tensor field J is called a almost complex structure on M. The smoothness of the tensor field J implies that if X is a smooth tangent vector field on M, then JX is also a smooth tangent vector field on M.

Let G be a Lie group with a left-invariant almost complex structure, $g=LieG$ . Then, we can define a linear map $J:g\to g$ and $J^2=-\text{id}$ . J is called a complex structure on g. If J satisfies:

$\left[JX,JY\right]=\left[X,Y\right]+J\left[JX,Y\right]+J\left[X,JY\right]$ for any $\left(X,Y\in g\right)$ , (1)

then J is integrable. Without distinction, the left-invariant integrable almost complex structure on G and the integrable complex structure on g are both denoted by J.

2.2. On Sequences of Nilpotent Lie Algebras

Let g be a Lie algebra. Suppose

$g^0=g,g^1=\left[g^0,g\right],\cdots ,g^l=\left[g^{l-1},g\right],\cdots$ (2)

It can be easily proven that $g^i$ is an ideal of g, and $g^i\subseteq g^{i-1}$ . The sequence $\left\{g^k,k\ge 0\right\}$ is called the descending central series of g. If there exists an $s\in N$ such that $g^s=\left\{0\right\}$ and $g^{s-1}\ne \left\{0\right\}$ , then g is called an s-step nilpotent Lie algebra [5] [6] .

Let G be a 2n-dimensional real nilpotent Lie group with a left-invariant integrable almost complex structure, $g=LieG$ , and $g^{\text{*}}$ be the dual space of g. Let $\left\{w_1,w_2,\cdots ,w_n\right\}$ denote a complex basis, and $\left\{w_1,{\bar{w}}_1,w_2,{\bar{w}}_2,\cdots ,w_n,{\bar{w}}_n\right\}$ denote a corresponding real basis. Therefore,

$d{w}_i={\displaystyle \underset{j<k}{\sum }{A}_{ijk}{w}_j\wedge w_k}+{\displaystyle \underset{j,k}{\sum }{B}_{ijk}{w}_j\wedge {\bar{w}}_k}+{\displaystyle \underset{j<k}{\sum }{C}_{ijk}{\bar{w}}_j\wedge {\bar{w}}_k}\left(1\le i\le n\right)$ , (3)

because $d\alpha \left(X,Y\right)=-\alpha \left(\left[X,Y\right]\right)$ $\left(X,Y\in g^C,\alpha \in {\left(g^C\right)}^{*}\right)$ , we use the exterior derivative on $g^{\text{*}}$ to describe the Lie bracket on g.

Property 1 [7] . Let G be a real nilpotent Lie group, $g=LieG$ . G has a left-invariant integrable almost complex structure if and only if $C_{ijk}=0$ , i.e.

$d{w}_i={\displaystyle \underset{j<k\le n}{\sum }{A}_{ijk}{w}_j\wedge w_k}+{\displaystyle \underset{j,k\le n}{\sum }{B}_{ijk}{w}_j\wedge {\bar{w}}_k}$ ( $1\le i\le n$ ). (4)

The structure equation can define connected and simply connected nilpotent Lie groups left-invariant integrable almost complex structure, so we can study some properties of Lie groups through this structure equation.

Definition 1 [8] . Let G be a connected simply connected s-step nilpotent Lie group, $g=LieG$ . Define a sequence in g as

$g_0=0,g_1=\left\{X\in g|\left[X,g\right]\subseteq g_0\right\},\cdots ,g_k=\left\{X\in g|\left[X,g\right]\subseteq g_{k-1}\right\},\cdots .$ (5)

Then g₁ is the center of g, where $g_1\ne \left\{0\right\}$ . $g_s=g$ , $g_k\subseteq g_{k+1}$ , and for any $k\left(0\le k\le s-1\right)$ chosen such that $\dim g_k<\dim g_{k+1}$ . Thus, there exists an ascending central series

$g_0=0\subset g_1\subset g_2\subset \cdots \subset g_{s-1}\subset g_s=g.$ (6)

Property 2.If the sequence $\left\{g_k,k\ge 0\right\}$ satisfies Equation (5), then:

a) g₁ is the center of g and $g_1\ne 0$ ;

b) For any $k\ge 0$ , $g_k\subseteq g_{k+1}$ ;

c) For any $k\left(0\le k\le s-1\right)$ such that $\dim g_k<\dim g_{k+1}$ $\left(0\le k\le s-1\right)$ . We have $g_0=0\subset g_1\subset g_2\subset \cdots \subset g_{s-1}\subset g_s=g$ .

Proof. a) Since $g_0=\left\{0\right\}$ , then $g_1=\left\{X\in g|\left[X,g\right]=0\right\}=C\left(g\right)$ . Moreover, since g is a nilpotent Lie algebra, $C\left(g\right)\ne 0$ , hence g₁ is the center of g and $g_0\subset g_1$ .

b) We use induction to prove $g_k\subseteq g_{k+1}$ $\left(k\ge 0\right)$ .

For $k=0$ , by a), we have $g_0\subset g_1$ . Assume that when $k=i$ holds, $g_i\subseteq g_{i+1}$ . We’ll prove that for $k=i+1$ , $g_{i+1}\subseteq g_{i+2}$ . For any $x_1\in g_{i+1}$ and $x\in g$ , we have $\left[x_1,x\right]\in g_i\subseteq g_{i+1}$ . According to Equation (5), $x_1\in g_{i+2}$ , thus $g_{i+1}\subseteq g_{i+2}$ , hence $g_k\subseteq g_{k+1}$ $\left(k\ge 0\right)$ .

c) First, we’ll prove that there exists an integer s such that $g_s=g$ .

Since g is an s-step nilpotent Lie algebra, there exists a descending central series

$g^0=g\supset g^1=\left[g,g^0\right]\supset \cdots \supset g^{k+1}=\left[g,g^k\right]\supset \cdots g^s=\left\{0\right\}$ .

Next, we’ll use induction to prove $g^{s-i}\subseteq g_i$ . When $k=0$ , we have $g^s=\left\{0\right\}=g_0$ . Assuming $k=i$ holds, $g^{s-i}\subseteq g_i$ , we’ll prove that for $k=i+1$ , $g^{s-i-1}\subseteq g_{i+1}$ . According to Equation (2), we have $g^{s-i}=\left[g,g^{s-i-1}\right]$ . Also, since $g^{s-i}\subseteq g_i$ , then $g^{s-i-1}\subseteq g_{i+1}$ . Thus, for any $i\left(0\le i\le s\right)$ such that $g^{s-i}\subseteq g_i$ , we have $g_s\supseteq g^0=g$ , and since $g_s\subseteq g$ , we conclude that $g_s=g$ .

Next, we prove that $g_0=0\subset g_1\subset g_2\subset \cdots \subset g_{s-1}\subset g_s=g$ , namely, for any $k\left(0\le k\le s-1\right)$ such that $\dim g_k<\dim g_{k+1}$ , then $g_k\subset g_{k+1}$ is strict.

When $k=0$ , by conclusion (a), we have $g_0\subset g_1$ . Assuming for $k=i$ holds, $g_i\subset g_{i+1}$ , we’ll prove that for $k=i+1$ , $g_{i+1}\subset g_{i+2}$ . According to Equation (5), we have $g_{i+1}=\left\{X\in g|\left[X,g\right]\subseteq g_i\right\}$ , $g_{i+2}=\left\{X\in g|\left[X,g\right]\subseteq g_{i+1}\right\}$ , thus $g_{i+1}\subset g_{i+2}$ . Also, since $g_s=g$ , then for any $k\left(0\le k\le s-1\right)$ such that $\dim g_k<\dim g_{k+1}$ , we have $g_k\subset g_{k+1}$ is strict. Thus, we have

$g_0=0\subset g_1\subset g_2\subset \cdots \subset g_{s-1}\subset g_s=g$

holds.

Before introducing the nilpotent complex structure on G, let’s first discuss under what conditions g is a complex Lie algebra. Let $g^C$ denote the complexification of g, and let J be the complex structure on the Lie algebra g. Then we

have $g^C=g^{-}\oplus g^{+}$ , where $\pm i$ are the eigenvalues of J, and $g^{-}=\left\{X+iJX|X\in g\right\}$ and $g^{+}=\left\{X-iJX|X\in g\right\}$ are the eigenspaces of J.

Property 3 [9] . The eigenspaces $g^{\pm }$ of J are ideals of $g^C$ .

Theorem 1. Let J be the integrable complex structure on the Lie algebra g. If $J\left[X,Y\right]=\left[JX,Y\right]$ for all $X,Y\in g$ , then g is a complex Lie algebra.

Proof: Let $g^{+}=\left\{X-iJX|X\in g\right\}$ . According to Property 3, we have $\left[X,Y\right]\in g^{+}$ for all $X,Y\in g^{+}$ .

Let $g^{-}=\left\{X+iJX|X\in g\right\}$ . According to Property 3, we have $\left[X,Y\right]\in g^{-}$ for all $X,Y\in g^{-}$ .

Since $g\subset g^C=g^{-}\oplus g^{+}$ , any $X,Y\in g$ satisfies $X=X_1+X_2$ ( $X_1\in g^{-}$ , $X_2\in g^{+}$ ), $Y=Y_1+Y_2$ ( $Y_1\in g^{-}$ , $Y_2\in g^{+}$ ).

According to Property 3, we have $\left[X_1,Y_2\right]\in g^{+}$ and $\left[X_1,Y_2\right]\in g^{-}$ . Since $g^{+}\cap g^{-}=\left\{0\right\}$ , it follows that $\left[X_1,Y_2\right]=0$ and $\left[X_2,Y_1\right]=0$ .

To prove that g is a complex Lie algebra, we need to show that the Lie bracket of g is C-linear, i.e., $\left[\left(a+ib\right)X,Y\right]=\left(a+ib\right)\left[X,Y\right]$ ( $X,Y\in g$ ).

$\begin{array}{c}\left[\left(a+ib\right)X,Y\right]=\left[\left(a+ib\right)\left(X_1+X_2\right),Y_1+Y_2\right]\\ =\left[a\left(X_1+X_2\right),Y_1+Y_2\right]+\left[ib\left(X_1+X_2\right),Y_1+Y_2\right]\\ =a\left[X_1+X_2,Y_1+Y_2\right]+b\left[i\left(X_1+X_2\right),Y_1+Y_2\right]\\ =a\left[X_1+X_2,Y_1+Y_2\right]+b\left[J{X}_1-J{X}_2,Y_1+Y_2\right]\\ =a\left[X_1+X_2,Y_1+Y_2\right]+b\left[J{X}_1,Y_1+Y_2\right]-b\left[J{X}_2,Y_1+Y_2\right]\end{array}$

$\begin{array}{c}=a\left[X_1+X_2,Y_1+Y_2\right]+bJ\left[X_1,Y_1+Y_2\right]-bJ\left[X_2,Y_1+Y_2\right]\\ =a\left[X_1+X_2,Y_1+Y_2\right]+bJ\left[X_1,Y_1\right]-bJ\left[X_2,Y_2\right]\\ =a\left[X_1+X_2,Y_1+Y_2\right]+ib\left[X_1,Y_1\right]+ib\left[X_2,Y_2\right]\\ =\left(a+ib\right)\left[X_1+X_2,Y_1+Y_2\right]\end{array}$

Therefore, the Lie algebra is C-linear, proving that g is a complex Lie algebra.

Suppose G is a connected simply connected nilpotent Lie group, $g=LieG$ , and g has a complex structure J. Then g is a complex vector space, but generally not a complex Lie algebra. According to Theorem 1, if the complex structure satisfies $J\left[X,Y\right]=\left[JX,Y\right]$ for all $X,Y\in g$ , then g is a complex Lie algebra. To study the nilpotent complex structure of the Lie group G, we introduce the ascending sequence $\left\{a_k,k\ge 0\right\}$ related to the nilpotent Lie algebra g.

Definition 2 [10] . Let G be a connected simply connected s-step nilpotent Lie group, and suppose it has a left-invariant integrable almost complex structure. $g=LieG$ and J is the complex structure on the Lie algebra g. g has the sequence

$a_0=\left\{0\right\},\cdots ,a_k=\left\{X\in g|\left[X,g\right]\subseteq a_{k-1},\left[JX,g\right]\subseteq a_{k-1}\right\},\cdots ,$ (7)

called the compatible with the integrable almost complex structure J of G.

Property 4. If $\left\{a_k,k\ge 0\right\}$ is the compatible with the integrable almost complex structure J of G, then a_k is an ideal of g, $a_k\subseteq a_{k+1}$ , and $a_k\subseteq g_k$ $\left(k\ge 0\right)$ .

Proof: We use induction to prove $a_k\subseteq a_{k+1}$ .

When $k=0$ , because $\left[a_1,g\right]\subseteq a_0=\left\{0\right\}$ , then $\left[a_1,g\right]\subseteq a_0\subseteq a_1$ . Suppose that when $k=i$ , $a_i\subseteq a_{i+1}$ holds, we need to prove that when $k=i+1$ , $a_{i+1}\subseteq a_{i+2}$ . According to Equation (7), we have $a_{i+2}=\left\{X\in g|\left[X,g\right]\subseteq a_{i+1},\left[JX,g\right]\subseteq a_{i+1}\right\}$ , $a_{i+1}=\left\{X\in g|\left[X,g\right]\subseteq a_i,\left[JX,g\right]\subseteq a_i\right\}$ . Since $a_i\subseteq a_{i+1}$ holds, $a_{i+1}\subseteq a_{i+2}$ , hence $a_k\subseteq a_{k+1}$ $\left(k\ge 0\right)$ .

To prove that a_k is an ideal of g, because $\left[a_k,g\right]\subseteq a_{k-1}\subseteq a_k$ , a_k is an ideal of g. By induction, we prove $a_k\subseteq g_k$ $\left(k\ge 0\right)$ .

When $k=0$ , since $a_0=0$ , $g_0=0$ , then $a_0\subseteq g_0$ . Suppose that when $k=i$ , $a_i\subseteq g_i$ holds, we need to prove that when $k=i+1$ , $a_{i+1}\subseteq g_{i+1}$ .

According to Equations (5) and (7), we have

$a_{i+1}=\left\{X\in g|\left[X,g\right]\subseteq a_i,\left[JX,g\right]\subseteq a_i\right\}$ , $g_{i+1}=\left\{X\in g|\left[X,g\right]\subseteq g_i\right\}$ .

Since $a_i\subseteq g_i$ holds, $a_{i+1}\subseteq g_{i+1}$ , which completes the proof.

Lemma 1 [1] . If there exists $k\ge 0$ such that $a_k=a_{k+1}$ , then for any $r\ge k$ , $a_r=a_k$ .

Now let’s look at some properties between the ascending sequence $\left\{a_k,k\ge 0\right\}$ and the ascending central sequence $\left\{g_k,k\ge 0\right\}$ of g.

Lemma 2 [1] . If $\left\{a_k,k\ge 0\right\}$ and $\left\{g_k,k\ge 0\right\}$ are respectively the ascending sequences of g, then the following three conclusions hold.

i) If there exists $k\ge 0$ such that $a_k=g_k$ , then $g_k=J\left(g_k\right)$ ;

ii) If there exists $k>0$ such that $a_{k-1}=g_{k-1}$ , then $a_k=g_k$ if and only if $g_k=J\left(g_k\right)$ ;

iii) If $a_{k-1}=g_{k-1}$ , then a_k is the largest J-invariant subspace of g_k.

Under the conditions of Lemma 2 and according to Equations (5), (7), we know $a_0=g_0=0$ , then

a) If there exists $k\ge 0$ such that $J\left(g_k\right)\not\subset g_k$ , then $a_k\subset g_k$ is strict;

b) If there exists an integer s such that $a_{s-1}=g_{s-1}$ , then $a_s=g_s=g$ ;

c) a₁ is the largest J-invariant subspace of g₁;

d) If any term g_k of $\left\{g_k,k\ge 0\right\}$ is J-invariant, then for any $k\ge 0$ , $a_k=g_k$ and $a_s=g_s=g$ .

Next, consider some related properties between the ascending sequence $\left\{a_k,k\ge 0\right\}$ and the descending central sequence $\left\{g^l,l\ge 0\right\}$ of g. Under the conditions of Lemma 2, suppose $\left\{g^l,l\ge 0\right\}$ is the descending central sequence of g. Then we have

① If for some $k\ge 0$ and some $l\ge 0$ , $g^l\subset a_k$ and $J\left(g^{l-1}\right)=g^{l-1}$ , then $g^{l-1}\subset a_{k+1}$ ;

② If for some $k\ge 0$ , $\left[g,g\right]\subseteq a_k$ , then $a_{k+1}=g$ ;

③ If any $g^l\in \left\{g^l,l\ge 0\right\}$ is J-invariant, then $a_s=g_s=g$ [1] .

2.3. Nilpotent Complex Structure

Definition 3 [11] . Let G be a connected simply connected s-step nilpotent Lie group, and suppose it has a left-invariant integrable almost complex structure. $g=LieG$ and J is the complex structure on the Lie algebra g. If there exists $t>0$ such that $a_t=g$ , then the left-invariant integrable almost complex structure J is called a nilpotent left-invariant complex structure.

Lemma 3 [1] . Let $\left\{w_i,1\le i\le n\right\}$ be a (1,0)-type left-invariant form complex basis for $g^{*}$ , satisfying the structural equation

${\displaystyle \underset{j<k<i}{\sum }{A}_{ijk}{w}_j\wedge w_k}+{\displaystyle \underset{j,k<i}{\sum }{B}_{ijk}{w}_j\wedge {\bar{w}}_k}$ ( $1\le i\le n$ ).

If $\left\{Z_i,{\bar{Z}}_i,1\le i\le n\right\}$ is a basis for g and dual to the basis $\left\{w_i,{\bar{w}}_i,1\le i\le n\right\}$ , let $X_i=\mathrm{Re}\left(Z_i\right)$ , $Y_i=\mathrm{Im}\left(Z_i\right)$ $\left(1\le i\le n\right)$ , then any term $a_l$ $\left(1\le l\le n\right)$ in the ascending sequence $\left\{a_l,l\ge 0\right\}$ contains at least generators $X_{n-l+1},Y_{n-l+1},\cdots ,X_n,Y_n$ .

Proposition 1. Under the conditions of Lemm3, we have

(i) If a_k is a member of the sequence $\left\{a_k,k\ge 0\right\}$ , and $a_k\ne g$ , then $\dim a_{k+1}\ge 2+\dim a_k$ ;

(ii) If $\left\{g_k,k\ge 0\right\}$ is the ascending central sequence of g, then $\dim g_k\ge \dim a_k\ge 2k$ $\left(1\le k\le n\right)$ ;

(iii) There exists a unique integer t such that $\dim a_{t-1}<\dim a_t$ and $a_t=g$ $\left(s\le t\le n\right)$ .

Theorem 2 [1] . Let $\left\{a_k,k\ge 0\right\}$ be the compatible with the integrable almost complex structure J of G. If there exists a(1,0)-type left-invariant form complex basis $\left\{w_i,1\le i\le n\right\}$ for $g^{*}$ such that the basis satisfies the structural equation

$d{w}_i={\displaystyle \underset{j<k<i}{\sum }{A}_{ijk}{w}_j\wedge w_k}+{\displaystyle \underset{j,k<i}{\sum }{B}_{ijk}{w}_j\wedge {\bar{w}}_k}$ ( $1\le i\le n$ ),

then the left-invariant integrable almost complex structure is nilpotent if and only if it is almost nilpotent.

Raghunathan [12] concludes: let G be a connected simply connected nilpotent Lie group with Lie algebra $g=LieG$ . G has a lattice D if and only if g admits a basis with rational structure constants. By applying Theorem 2, Theorem 3 can be obtained.

Theorem 3 [1] . Given the structure equations of Theorem 2:

$d{w}_i={\displaystyle \underset{j<k<i}{\sum }{A}_{ijk}{w}_j\wedge w_k}+{\displaystyle \underset{j,k<i}{\sum }{B}_{ijk}{w}_j\wedge {\bar{w}}_k}$ ( $1\le i\le n$ ),

a connected simply connected nilpotent Lie group G with a left-invariant almost complex structure that is nilpotent can be defined, and its left-invariant complex structure is nilpotent. Then a complex structure, which is also nilpotent, can be defined on the compact homogeneous nilpotent manifold G/D. Conversely, if the left-invariant integrable almost complex structure of a connected simply connected nilpotent Lie group G is nilpotent, with structure equations

$d{w}_i={\displaystyle \underset{j<k<i}{\sum }{A}_{ijk}{w}_j\wedge w_k}+{\displaystyle \underset{j,k<i}{\sum }{B}_{ijk}{w}_j\wedge {\bar{w}}_k}$ ( $1\le i\le n$ ),

then G has a left-invariant complex structure that is nilpotent.

3. Exploring Complex Structures on Nilpotent Lie Group G_k

This section mainly discusses that if the left-invariant integrable almost complex structure J on a Lie group G is nilpotent, then the nilpotent Lie group G_k has a left-invariant integrable almost complex structure J_k, and J_k is nilpotent (where $k<t$ , and t is the smallest integer such that $a_t=g$ ). By Property 4, a_k is an ideal of g, and since g is a nilpotent Lie algebra, $g/a_k$ is also a nilpotent Lie algebra. Let G_k be the connected simply connected nilpotent Lie group defined by the nilpotent Lie algebra $g/a_k$ .

3.1. Complex Structure of Nilpotent Lie Group G_k

Definition 4. Let G be a connected simply connected nilpotent Lie group with a left-invariant integrable almost complex structure, $g=LieG$ , and J be the complex structure on g. Define the mapping on $g/a_k$ :

$J_k:g/a_k\to g/a_k$

$\bar{x}\mapsto \tilde{J}\left(\bar{x}\right)=J\left(x\right)+a_k$ , $x\in g,\bar{x}\in g/a_k$ .

Lemme 5. Suppose G is a connected simply connected nilpotent Lie group with a left-invariant integrable almost complex structure that is nilpotent, then G_k has a left-invariant integrable almost complex structure.

Proof: First, we prove that J_k is a complex structure. Since J is linear, and according to Definition 4,

$J_k\left(\bar{x}\right)=J\left(x\right)+a_k$ , $x\in g,\bar{x}\in g/a_k$ ,

J_k is linear. Next, we prove that J_k is a complex structure. Since

$\begin{array}{c}{J}_k\left(J_k\left(\bar{x}\right)\right)=J_k\left(Jx+a_k\right)\\ =J\left(Jx+a_k\right)+a_k\\ =J^{2}x+a_k=-\bar{x}\end{array}$

thus $J_k^2=-id:g/a_k\to g/a_k$ , so J_k is a complex structure on $g/a_k$ .

For any $\bar{X},\bar{Y}\in g/a_k$ , we have $\bar{X}=X+a_k$ , $\bar{Y}=Y+a_k$ ( $X,Y\in g$ ),

$\begin{array}{c}\left[J_k\left(\bar{X}\right),J_k\left(\bar{Y}\right)\right]=\left[JX+a_k,JY+a_k\right]\\ =\left[JX,JY\right]+a_k\\ =\left[\bar{X},\bar{Y}\right]+J\left[JX,Y\right]+J\left[X,JY\right]+a_k\\ =\left[\bar{X},\bar{Y}\right]+J_k\left(\left[JX,Y\right]+a_k\right)+J_k\left(\left[X,JY\right]+a_k\right)\\ =\left[\bar{X},\bar{Y}\right]+J_k\left(\left[J_k\left(\bar{X}\right),\bar{Y}\right]\right)+J_k\left(\left[\bar{X},J_k\left(\bar{Y}\right)\right]\right)\end{array}$

so G_k has a left-invariant integrable almost complex structure. □

We know that the left-invariant integrable almost complex structure J of the Lie group G induces a left-invariant integrable almost complex structure J_k on the Lie group G_k. Next, we first give a sequence $\left\{b_k,k\ge 0\right\}$ on $g/a_k$ , and then use this sequence to prove that if J is nilpotent, then J_k is also nilpotent.

Definition 5. Let G be a connected simply connected nilpotent Lie group with a left-invariant integrable almost complex structure, $g=LieG$ , J be the complex structure on g, and J_k be the complex structure on $g/a_k$ . $g/a_k$ has a sequence

$b_0=\left\{0\right\},\cdots ,b_k=\left\{\bar{X}\in g/a_k|\left[\bar{X},g/a_k\right]\subseteq b_{k-1},\left[\tilde{J}\left(\bar{X}\right),g/a_k\right]\subseteq b_{k-1}\right\},\cdots$ . (8)

3.2. Properties of the Complex Structure of Nilpotent Lie Group G_k

Property 5. $g/a_k$ has a sequence $\left\{b_k,k\ge 0\right\}$ satisfying Equation (8), which implies that

(1) For any $k\ge 0$ , $b_k\subseteq b_{k+1}$ ;

(2) If there exists $k\ge 0$ such that $b_k=b_{k+1}$ , then for any $r\ge k$ , $b_r=b_k$ .

Proof: We use induction to prove $b_k\subseteq b_{k+1}$ .

When $k=0$ , according to Equation (8), we have $b_0=\left\{0\right\}$ and $b_0\subseteq b_1$ .

Assume that when $k=i$ , $b_i\subseteq b_{i+1}$ holds. When $k=i+1$ , $b_{i+1}\subseteq b_{i+2}$ holds. According to Equation(8), we have

$b_{i+2}=\left\{\bar{X}\in g/a_k|\left[\bar{X},g/a_k\right]\subseteq b_{i+1},\left[J_k\left(\bar{X}\right),g/a_k\right]\subseteq b_{i+1}\right\}$ ,

$b_{i+1}=\left\{\bar{X}\in g/a_k|\left[\bar{X},g/a_k\right]\subseteq b_i,\left[J_k\left(\bar{X}\right),g/a_k\right]\subseteq b_i\right\}$ ,

and since $b_i\subseteq b_{i+1}$ holds, $b_{i+1}\subseteq b_{i+2}$ holds. Thus, $b_k\subseteq b_{k+1}$ ( $k\ge 0$ ).

Next, we prove property (2). According to Equation (8), we have

$\begin{array}{c}{b}_{i+2}=\left\{\bar{X}\in g/a_k|\left[\bar{X},g/a_k\right]\subseteq b_{i+1},\left[J_k\left(\bar{X}\right),g/a_k\right]\subseteq b_{i+1}\right\}\\ =\left\{\bar{X}\in g/a_k|\left[\bar{X},g/a_k\right]\subseteq b_i,\left[J_k\left(\bar{X}\right),g/a_k\right]\subseteq b_i\right\}\\ =b_{k+1}\end{array}$

which means that for any $r\ge k$ , $b_r=b_{k+1}=b_k$ . □

Definition 6. Let G be a connected simply connected nilpotent Lie group with a left-invariant integrable almost complex structure, $g=LieG$ , J_k be the complex structure on $g/a_k$ , and $\left\{b_k,k\ge 0\right\}$ be an ascending sequence on $g/a_k$ . If there exists $t>0$ such that $b_t=g/a_k$ , then the left-invariant integrable almost complex structure of G_k is called nilpotent left-invariant complex structure.

Theorem 4. Suppose a connected simply connected s-step real nilpotent Lie group G has a left-invariant integrable almost complex structure J, and the sequence $\left\{a_k,k\ge 0\right\}$ is a ascending sequence of g. If the left-invariant integrable almost complex structure J is nilpotent, then $b_n=a_{n+k}/a_k$ and the left-invariant integrable almost complex structure of G_k is nilpotent.

Proof: To prove that the left-invariant integrable almost complex structure of G_k is nilpotent, we only need to prove that there exists t such that $b_t=g/a_k$ . We will prove by induction that there exists n such that $b_n=g/a_k$ .

$b_0=a_k/a_k=\left\{0\right\}$ , Next, we prove $b_1=a_{k+1}/a_k$ .

For any $\bar{X}\in a_{k+1}/a_k$ , we have $\bar{X}=X+a_k\left(X\in a_{k+1}\right)$ , then $\left[\bar{X},g/a_k\right]=\left[X,g\right]+a_k$ and $\left[J_k\left(\bar{X}\right),g/a_k\right]=\left[JX,g\right]+a_k$ According to Equation (7), we know that $\left[X,g\right]\subseteq a_k$ and $\left[J\left(X\right),g\right]\subseteq a_k$ , so

$\left[J_k\left(\bar{X}\right),g/a_k\right]=0$ and $\left[\bar{X},g/a_k\right]=0$ .

Therefore $\bar{X}\in b_1$ , implying $b_1\supseteq a_{k+1}/a_k$ .

For any $\bar{X}\in b_1$ , we have $\bar{X}=X+a_k\left(X\in g\right)$ , according to Equation (8),

$b_1=\left\{\bar{X}\in g/a_k|\left[\bar{X},g/a_k\right]\subseteq b_0=\left\{0\right\},\left[J_k\left(\bar{X}\right),g/a_k\right]\subseteq b_0=\left\{0\right\}\right\}$ ,

which means $\left[X,g\right]+a_k\subseteq \left\{0\right\}$ and $\left[JX,g\right]+a_k\subseteq \left\{0\right\}$ , so $\left[X,g\right]\subseteq a_k$ and $\left[JX,g\right]\subseteq a_k$ , hence $X\in a_{k+1}$ , so $\bar{X}\in a_{k+1}/a_k$ , which means $b_1\subseteq a_{k+1}/a_k$ .

So $b_1=a_{k+1}/a_k$ .

Assume that when $n=i$ , $b_i=a_{k+i}/a_k$ holds.

Next, we prove that when $n=i+1$ , $b_{i+1}=a_{k+i+1}/a_k$ holds.

For any $\bar{X}\in b_{i+1}$ we have $\bar{X}=X+a_k\left(X\in a_{k+i+1}\right)$ , then $\left[\bar{X},g/a_k\right]=\left[X,g\right]+a_k$ and $\left[J_k\left(\bar{X}\right),g/a_k\right]=\left[JX,g\right]+a_k$ . according to Equation (8), we know that $\left[X,g\right]\subseteq a_{k+i}$ and $\left[JX,g\right]\subseteq a_{k+i}$ . thus $\left[J_k\left(\bar{X}\right),g/a_k\right]\subseteq a_{k+i}/a_k=b_i$ and $\left[\bar{X},g/a_k\right]\subseteq a_{k+i}/a_k=b_i$ . Therefore $\bar{X}\in b_{i+1}$ , implying $b_{i+1}\supseteq a_{k+i+1}/a_k$ .

For any $\bar{X}\in b_{i+1}$ , we have $\bar{X}=X+a_k\left(X\in g\right)$ , according to Equation (8), $b_{i+1}=\left\{\bar{X}\in g/a_k|\left[\bar{X},g/a_k\right]\subseteq b_i=a_{k+i}/a_k,\left[J_k\left(\bar{X}\right),g/a_k\right]\subseteq b_i=a_{k+i}/a_k\right\}$ , which means $\left[X,g\right]+a_k\subseteq a_{k+i}/a_k$ and $\left[JX,g\right]+a_k\subseteq a_{k+i}/a_k$ , so $\left[X,g\right]\subseteq a_{k+i}$ and $\left[JX,g\right]\subseteq a_{k+i}$ , hence $X\in a_{k+i+1}$ , so $\bar{X}\in a_{k+i+1}/a_k$ , which means $b_{i+1}\subseteq a_{k+i+1}/a_k$ .

So $b_{i+1}=a_{k+i+1}/a_k$ .

Next, we prove that the complex structure J_k on g/a_k is nilpotent. Since the complex structure J on the Lie group G is nilpotent, there exists $t>0$ such that $a_t=g$ , then $b_{t-k}=a_t/a_k=g/a_k$ , thus J_k is nilpotent.

According to Theorem4, if a connected simply connected s-step real nilpotent Lie group G has a left-invariant integrable almost complex structure J that is nilpotent, then J can induce a nilpotent left-invariant integrable almost complex structure J_k on the Lie group G_k.

4. Summary

Let g be a Lie algebra. If $g^{*}$ has a (1,0)-type left-invariant complex structure with complex basis $\left\{w_1,1\le i\le n\right\}$ , satisfying the structural equation

$d{w}_i={\displaystyle \underset{j<k<i}{\sum }{A}_{ijk}{w}_j\wedge w_k}+{\displaystyle \underset{j,k<i}{\sum }{B}_{ijk}{w}_j\wedge {\bar{w}}_k}$ ( $1\le i\le n$ ),

then we can define a connected simply connected nilpotent Lie group G. Its left-invariant integrable almost complex structure J on G is nilpotent, and J induces a left-invariant integrable almost complex structure J_k on the nilpotent Lie group G_k, and J_k is nilpotent.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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