Symplectic structures on almost abelian Lie algebras
Luis Pedro Castellanos Moscoso
Hiroshima University, 1 Chome-3-2 Kagamiyama, Higashihiroshima, Hiroshima
739-0046.
e-mail : luis90@hiroshima-u.ac.jp

Abstract. We are interested in the classification or finding conditions for the existence of leftinvariant symplectic structures on Lie groups. In this paper, we study this problem for the case ofalmost abelian Lie algebras, which are Lie algebras that contain a codimension 1 abelian subalgebra. We show that, for this case, the problem is reduced to studying a certain matrix congruence
relation. We obtain a classification of symplectic structures for all almost abelian Lie algebras andstate necessary and sufficient conditions for the existence of such structures.

1

Introduction

In this section, we recall some basic notions on left-invariant symplectic forms
on Lie groups. Then, we present some of the basic facts about almost abelian Liealgebras.
1.1

Left-invariant symplectic 2-forms

Let G be a simply connected Lie group with dimension 2n and g its corresponding Lie algebra. We are interested in the space of left-invariant symplectic forms onG:^2∗nω(·,·) ∈T G | ω ̸= 0, left-invariant, dω = 0 .
This is a subset of the set of all nondegenerate left-invariant 2-forms, denoted by^2Ω (G) := ω(·,·) ∈T ∗ G | ω n ̸= 0, left-invariant .
It is well known that the space Ω (G) can be identified with the space of nondegenerate 2-forms on g, denoted by^2∗nΩ(g) := ω (·,·) ∈g | ω ̸= 0 .
We are interested in left-invariant symplectic structures, so we consider the subsetof closed 2 forms:Ω(g) ⊃ Ωc (g) := {ω ∈ Ω(g) | dω = 0}.
There is a natural equivalence relation in these sets.

Definition. Let ω1 , ω2 ∈ Ω(g). Then, (g, ω1 ) and (g, ω2 ) are said to be equivalent
up to automorphism and scale if there exist ϕ ∈ Aut(g) and a constant c ̸= 0 suchthat c · (ϕ)∗ ω1 = ω2 .
We have the corresponding equivalence classesΩ(g)/“up to symplectomorphism and scale”∪Ωc (g)/“up to symplectomorphism and scale”.
1.2The moduli space

In this section, we define the moduli space of left-invariant nondegenerate 2forms on a Lie group. From the theory of homogeneous spaces we also have theidentificationΩ(g) ∼= GL(2n, R)/Spn (R).
2n∼Here we identify g = R , and then the general linear group GL(2n, R) acts transitively on Ω (g) byg.ω(·,·) = ω(g −1 (·),g −1 (·)) ∀g ∈ GL(2n, R).
Here Spn (R) denotes the symplectic group.
Define the set R× := R \ 0. Then we can consider the set
R× Aut(g) := ϕ ∈ GL(2n, R) | ϕ ∈ Aut(g), c ∈ R× ,which is a subgroup of GL(2n, R). Hence, it naturally acts on Ω (g). We can thenconsider the orbit space of this action.
Definition. The orbit space of the action of R× Aut(g) on Ω (g) will be called themoduli space of left-invariant nondegenerate 2-forms and will be denoted by
PΩ(g) := R× Aut(g) \ Ω(g) := R× Aut(g).ω | ω ∈ Ω(g) .
Similarly, we define the setPΩc (g) := R× Aut(g) \ Ωc (g) := R× Aut(g).ω | ω ∈ Ωc (g) .
One can easily see that, if ω1 , ω2 ∈ Ω(g) are in the same R× Aut(g)-orbit, then they
are equivalent up to automorphism and scale. Therefore, there is a surjection fromthe moduli space PΩ(g) onto the quotient space. We have the following diagram
PΩ(g)

↠∪
PΩc (g)Ω(g)/“up to symplecto-morphism and scale”∪↠Ωc (g)/“up to symplecto-morphism and scale”

1.3

Almost abelian Lie algebras
Here we define a special type of Lie algebras.

Definition. A non-abelian Lie algebra g is called almost abelian if it contains acodimension 1 abelian subalgebra.
The following facts about almost abelian Lie algebras can be found in [3].
Proposition 1.1. An almost abelian Lie algebra g has a codimension 1 abelian
ideal L, and is therefore isomorphic to the semidirect productg = Re1 ⋉ L.
The Lie algebra structure of an almost abelian Lie algebra g = Re1 ⋉ L iscompletely determined by ade1 :[e1 , v] = ade1 v, v ∈ L.
Therefore, any pair (V, T ), where V is vector space and T a nonzero linear map
determines an almost abelian Lie algebra g and vice versa, but different maps canyield isomorphic Lie algebras.
Definition. Two pairs (V1 , T1 ) and (V2 , T2 ) are said to be similar if there exists aninvertible map ϕ such that ϕT1 ϕ−1 = T2 . In this case we write (V1 , T1 ) ∼ (V2 , T2 ).
Isomorphism classes of almost abelian Lie algebras correspond to the similarity
classes of linear operators on vector spaces up to scaling.
Theorem 1.2. Two almost abelian Lie algebras g = Re1 ⋉ L and g′ = Re′1 ⋉ L′
are isomorphic if and only if (L, ade1 ) ∼ (L′ , λade′1 ) for some constant λ ̸= 0.
This theorem allows us to consider, without loss of generality, only almost
abelian Lie algebras g such that the map ade1 is given in Jordan Normal form.
2A matrix problem
Consider the following matrix problem.

Let JN ∈ M (N × N, R), N odd, be a matrix in real Jordan normal form. Define
the following two sets:
TJN := {A ∈ GL(N, R) | AJN − JN A = 0} ,()tCJN + JN
C = 0,HJN := C ∈ Skew(N × N, R)
Rank(C) = N − 1.

We want to study the following quotient space:
HJN /“congruence over TJN ”.

(2.1)

The next proposition gives us canonical representatives for this quotient space.
Proposition 2.1 (Canonical representatives). Let C ∈ HJN . There exist S ∈ TJN
and a permutation matrix P such that¯
S t CS = P t JP.
The matrix J¯ being defined by0−I N +123
I N +12= −101J¯.
Main results

The next proposition states the connection between the previous matrix problem
and the study of symplectic structures on almost abelian Lie algebras.
Proposition 3.1. Let g = Re1 ⋉ L be a 2n-dimensional almost abelian Lie algebrawith ade1 = JN , then
HJN /“congruence over TJN ” ↠ PΩc (g)Therefore, to study symplectic forms on almost abelian Lie algebras we need tostudy the quotient space 2.1. For example, if for a given almost abelian Lie algebra
g there is only one equivalence class in the corresponding quotient space, then there
is only one element in the moduli space and therefore only one symplectic form ong. If the number of equivalence classes is finite, then we get an upper limit in thenumber of possible symplectic forms.
Proposition 2.1 and Proposition 3.1 combine to gives us our main result: a classification of symplectic forms on all almost abelian Lie algebras and conditions forthe existence of such structures. There have been some recent result for some special cases (see [1],[2]) . Before stating the main result lets introduce some notation.
For a matrix A, let NA (m, λ) denote be number of Jordan blocks of size m witheigenvalue λ ∈ C of the corresponding Jordan normal form of A.
Theorem 3.2. Let g = Re1 ⋉ L be an almost abelian Lie algebra.

1. There exists a symplectic form ω if and only if the following conditions hold(a) Nade1 (m, λ) = 0 (mod 2) for all λ ∈ iR, m ∈ N+ .
(b) Nade1 (m, λ) = Nade1 (m, −λ) for all λ ∈ C \ (R ∪ iR), m ∈ N+ .
(c) One of the following:i. Nade1 (m, λ) = Nade1 (m, −λ) for all λ ∈ R \ {0}, m ∈ N+ and there
exists only one l ∈ N+ such that Nade1 (l, 0) is odd.
ii. Nade1 (m, 0) is even for all odd m ∈ N+ . One of the following:
A. There exists a unique α ∈ R \ {0} such that Nade1 (1, α) −Nade1 (1, α) = 1.

Nade1 (m, λ) = Nade1 (m, −λ) for all λ ∈R \ {α, 0}, m ∈ N+ .
B. There exist a unique l ∈ N+ and a unique α ∈ R \ {0} such thatNade1 (l, α) − Nade1 (l, −α) = 1 and Nade1 (l + 1, −α) − Nade1 (l +1, α) = 1. Nade1 (m, λ) = Nade1 (m, −λ) whenever λ ∈ R \ {α, 0}or m ∈ N+ \ {l}.
2. If ade1 is given in Jordan normal form and ω is a symplectic form, then there
exist a permutation P such that ω ∼= P.ω0 . ω0 is the canonical symplectic
form in the chosen basis.

References[1] Giovanni Bazzoni, Marco Freibert, Adela Latorre, and Nicoletta Tardini, Complex symplectic
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10.1016/j.laa.2023.08.006. MR4634959[2] Luis Pedro Castellanos Moscoso, Left-invariant symplectic structures on diagonal almost
abelian Lie groups, Hiroshima Math. J. 52 (2022), no. 3, 357–378, DOI 10.32917/h2021055.
MR4515688[3] Zhirayr Avetisyan, The structure of almost Abelian Lie algebras, Internat. J. Math. 33 (2022),
no. 8, Paper No. 2250057, 26, DOI 10.1142/S0129167X22500574. MR4462443