Joint Invariants of Symplectic and Contact Lie Algebra Actions

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Department of Mathematics and Statistics

Joint Invariants of Symplectic and Contact Lie Algebra Actions

—

Fredrik Andreassen

Master’s thesis in Mathematics June 2020

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Abstract

By restricting generating functions of infinitesimal symmetries of symplectic and contact vector spaces to quadratic forms, we obtain a finite-dimensional Lie sub- algebrag, consisting of vector fields isomorphic to the linear symplectic or confor- mal symplectic algebra. This allows us to look for joint invariants of the diagonal action ofg on product manifolds M^×m. We find an explicit recipe for creating a transcendence basis for the field ofm-fold rational joint invariants overR, starting from a base space M of any dimension n≥2.

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Acknowledgments

I’d like to express my sincere gratitude to Boris Kruglikov for providing me with a great topic for my master’s thesis, and for the many hours put into guiding me through the writing process.

I would also like to thank Jørn Olav Jensen for lots of good discussions and feedback.

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Our main objective is to obtain a complete description of the field of joint rational invariants of an extended Lie algebra action defined on symplectic and contact manifolds of varying dimensions. Along the way, we aim to illustrate how to describe the algebra of joint polynomial invariants using minimal free resolutions.

We will also briefly consider how our results can be used to generate symmetric invariants. Before diving into the computations, we introduce the necessary tools and definitions needed to make sense of them.

Chapter 2 opens with a section on classical invariant theory, using binary forms to illustrate the key ideas. Those ideas will be used to motivate a more general notion of an invariant, to be described in the following section. We proceed to describe a small variety of computational strategies for finding invariants of a given group action, all of which will be put to use in the central part of the thesis. Finally, we consider spaces of invariants consisting of polynomials as well as of rational functions.

In Chapter 3 we start looking for joint invariants on symplectic manifolds. Most of the groundwork will be laid in the 2-dimensional case, where the limited com- plexity of the problem admits a direct approach using infinitesimal methods. As we obtain polynomial generators for our space of invariants, we start by describing it as an algebra for the first few product spaces. We then turn to the task of describing the field of invariants for arbitrarily large products. Our results will be of great use in the equivalent endeavor in 4 dimensions and beyond.

We will compute joint invariants on contact manifolds in Chapter 4. Here the infinitesimal approach fails, but we are able to find invariants by obtaining an explicit description of the group action. On contact manifolds, generators for the space of invariants are found to be rational functions. After finding a description for the field of joint invariants in 3 and 5 dimensions, we proceed to consider spaces of higher dimensions.

The thesis is concluded in chapter 5 with a few computations illustrating how to find symmetric joint invariants on products of manifolds of low dimensions. By averaging the action of the symmetric group on the space of invariants, we can find generators for the field and algebra of symmetric invariants. However this time the structure of the algebra of invariants, even the generating set, is quite complicated, and so we describe only some particular examples.

Having computed the joint invariants we can solve the equivalence problem for the finite collection of points (ordered or symmetric) with respect to our group Sp(2n,R) or CSp(2n,R). For this the method of joint invariant signatures can be applied. We refer to [Olv01] for details.

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2 A brief overview of Invariant Theory

This chapter will provide an introduction to the main elements of invariant theory.

2.1 The classical perspective

Classical invariant theory is concerned with those properties of mathematical objects which are intrinsic to the objects themselves (i.e. not an artifact of the underlying coordinates in which they are represented). This section will outline this theory using homogeneous polynomials of two variables, also known as ”binary forms” in the literature, as well as their inhomogeneous counterparts.

For more on classical invariant theory, see [Olv99].

2.1.1 Homogeneous polynomials

Formally, ahomogeneous polynomialQ(x, y) in two variables is an expression of the following form:

Q(x, y) =

n

X

i=0

a_ixⁱyⁿ⁻ⁱ =

n

X

i=0

n i

˜

a_ixⁱyⁿ⁻ⁱ, a_i ∈R(or C)

It is natural to interpret the above as the local coordinate expression of some function Qon a 2-dimensional manifoldM, given in the local coordinates (x, y).

In principle, M could be any such manifold. However, we will require M to be either R² orC², and our coordinates (x, y) to be global.

Since we are interested in finding properties of homogeneous polynomials which do not depend on the coordinates of the space they are defined on, a good place to start is by examining how the expression for Qchanges under a linear change of coordinates.

If we let M =R², we can identify the points (1,0),(0,1) with the vectors e₁, e₂, and in this sense regard our space as avector space with basis (e₁, e₂). Under this identification, a point (v¹, v²)∈R² can be regarded as the vectorv =v¹e₁+v²e₂. From this point of view, changing the coordinates of our space corresponds to changing to some new basis (¯e₁,¯e₂).

Denoting the same point after a change of coordinates by ¯v = ¯v¹e¯₁+ ¯v²e¯₂, we have the following relationship between the old and the new coordinates:

v¯¹

¯ v²

=

α β γ δ

v¹ v²

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This can be written more concisely as ¯v = Av, where A is the invertible matrix above. There is a known isomorphism between the space of homogeneous polynomials and the space of symmetric tensors:

P⁽ⁿ⁾(R²)' S⁽ⁿ⁾(R²)^∗

For the case wheren= 1, we can regardQas a member ofS⁽¹⁾(R²)^∗ = (R²)^∗with (x, y) being the dual basis of (e₁, e₂). Expanding in this basis,Q(x, y) =a₀x+a₁y with the action on v given by Q(v) := hQ, vi = v¹a₀+v²a₁.

Let (¯x,y) be the dual basis of (¯¯ e₁,e¯₂). Then we can also express Q in this basis as ¯Q(¯x,y) = ¯¯ a₀x¯+ ¯a₁y. We would like to find a relation between the old and¯ new coefficients. Using the well known relationship between the components of a covector under a change of coordinates, we can quickly conclude that: ¯a=A^−Ta.

In matrix notation:

a₀ a₁

=

α γ β δ

¯ a₀

¯ a₁

Generalizing to the case where n=k, we can regard Qas a member of S^k(R²)^∗, with the action on v given by Q(v) := hQ, v⊗...⊗v

| {z }

kentries

i

The relationship between the old and new coefficients then becomes:

¯

a=A^−T ...A^−T

| {z }

kentries

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Notice that from this point of view, R² as amanifold stays fixed under a change of coordinates. We are merely changing the labels assigned to each point. It is also possible to interpret the invertible matrixAas an automorphism ofM. From this second perspective, the coordinates (x, y) and (¯x,y) are both the standard¯ coordinate functions on R². However, the points they are labeling are differently arranged. Here ¯Q can be interpreted as a transformed version of Q, defined by the relation ¯Q(¯x,y) =¯ Q(x, y). Of course, the transformation properties of its coefficients will be exactly the same in either interpretation.

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2.1.2 Inhomogeneous polynomials

We define an inhomogeneous polynomial Q(p) of one variable as a formal expression of the the following form:

Q(p) =

n

X

i=0

b_ipⁱ

As was the case with homogeneous polynomials of two variables, we can interpret such an expressions as representing some function Qdefined on a 1-dimensional manifold N in the coordinate p.

Here, we will take N to be the projective lineP(R²), where points corresponds to linear subspaces of R². To every homogeneous polynomial Q(x, y) : R² → R, there is a corresponding function Q:P(R²)→R. In affine coordinates onP(R²), this function will be represented as an inhomogeneous polynomial.

Ex: Q(x, y) =x²+ 3xy+ 2y², let x=p, y = 1 Q(p) =p²+ 3p+ 2

As before, we are interested in how this expression for Q(p) changes as we change to different coordinates ¯p.

A linear transformation:

(x¯=αx+βy

¯

y=γx+δy of R², induces a M¨obius transformation p¯= αp+β

γp+δ of P(R²).

In the special case where ¯y =y, the linear transformation induces an affine transformation. Starting with the relationship between the homogeneous polynomials Qand ¯Q, we can deduce the relationship between the inhomogeneous polynomials after a change of coordinates:

Q(x, y) = ¯Q(¯x,y)¯ yⁿQ

x y

= ¯yⁿQ¯ x¯

¯ y

, let x=p, y = 1 Q(p) = ¯yⁿQ¯

x¯

¯ y

Q(p) = (γp+δ)ⁿQ¯

αp+β γp+δ

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This relationship warrants a couple of remarks:

(i) The ”naive degree” of Q(p) is not always preserved under M¨obius transformations.

Ex: A= 0 1

1 1

=⇒ p¯= 1 p+ 1

Q(p) = p²−1 =⇒ Q(¯¯ p) = −2¯p+ 1

In fact, we will define the degree of an inhomogeneous polynomial to be the degree of its homogeneous counterpart.

(ii) Roots of Q(p) are mapped to roots of ¯Q(¯p).

That is,Q(p₀) = 0 iff ¯Q(¯p₀) = 0. This implies that the number of distinct roots of a polynomial does not depend on the coordinates in which it is being represented.

It is well known that p₀ is a root of Q(p) of multiplicity k iff: Q(p0) =Q⁰(p0) = ...=Q^(k−1)(p0) = 0

The resultant gives us a way to determine whether or not two polynomials have any common roots:

LetP(x) = a_mx^m+am−1x^m−1y+...+a₀y^m Q(x) = b_nxⁿ+bn−1xⁿ⁻¹y+...+b₀yⁿ

Then theresultantof P and Q is defined as the determiant of the (m+n)×(m+n) Sylvester matrix:

Res[P, Q] =

am 0 · · · 0 bn 0 · · · 0 am−1 a_m · · · ... bn−1 b_n · · · ... ... am−1 . .. ... ... bn−1 . .. · · · ... ... ... · · · a_m ... ... · · · . .. ... ... ... · · · am−1 b₀ ... · · · b_n a₀ ... · · · ... 0 b₀ · · · b_n−1

0 a0 · · · ... ... ... . .. · · · ... ... ... . .. ... ... ... · · · . .. ... 0 0 · · · a₀ 0 0 · · · b₀

| {z }

n columns | {z }

m columns P(x) andQ(x) have a common root iff Res[P, Q] = 0.

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We identify Q⁰(p) with Q_x(x, y) and define thediscriminant of Qas:

∆[Q] = Res[Q, Q_x] nⁿ˜an

2.1.3 Classical invariants

Now we come to the main definitions of this section. A classical invariant of weightk of a binary formQ(x, y) of degreen, is a functionI(a₀, ... , a_n) satisfying the following equation under linear transformations:

I(a₀, ... , a_n) = (αδ−βγ)^kI(¯a₀, ... ,¯a_n)

Ex: Let n =2. ∆[Q] is an invariant of weight 2:

∆ = (αδ−βγ)²∆¯

ac−b² = (αδ−βγ)²(¯a¯c−¯b²)

A classical covariant of weight k of a binary form Q(x, y) of degree n is a function J(a₀, ... , a_n, x, y) satisfying the following equation under linear transformations:

J(a₀, ... , a_n, x, y) = (αδ−βγ)^kJ(¯a₀, ... ,¯a_n,x,¯ y)¯

Ex1: Q itself is a covariant of weight 0 under the relationQ(x, y) = ¯Q(¯x,y)¯

Ex2: The Hessian H =Q_xxQ_yy−Q²_xy is a covariant of weight 2.

A classical joint covariant of weight k of a system Q₁(x, y), ... , Q_m(x, y) is a function J(a¹₀, ... , a¹_n, ... , a^m₀ , ... , a^m_n, x, y) satisfying the following equation under linear transformations:

J(a¹₀, ... , a¹_n, ... , a^m₀ , ... , a^m_n, x, y) = (αδ−βγ)^kJ(¯a¹₀, ... ,¯a¹_n, ... ,¯a^m₀ , ... ,¯a^m_n,x,¯ y)¯

A joint covariant not depending on (x, y) is called a (classical) joint invariant.

Ex: The resultantRes[Q, P] is a joint invariant of weight mn+mk+nj, where m =deg[Q], n=deg[P], k=weight[Q], j =weight[P]

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Using covariants and invariants we can completely classify all quadratic and cubic polynomials under M¨obius transformations:

Canonical forms for complex quadratics: Q(p) =ap²+ 2bp+c

distinct roots I p ∆6= 0

double root II 1 ∆ = 0, Q6≡0

identically zero III 0 Q≡0

Canonical forms for complex cubics: Q(p) =ap³+ 3bp²+ 3cp+d

distinct roots I p² −1 ∆6= 0

double root II p ∆ = 0, H 6≡0

triple root III 1 H ≡0, Q6≡0

identically zero IV 0 Q≡0

For quadratics, ∆ and Q are the only covariants. Cubics have one more (T, the Jacobian of Q and H). A quartic has 5 covariants (2 invariants). We can construct new covariants by multiplying any two covariants, whose weight will equal the product of their respective weights, or by adding two of the same weight.

We are however, mostly interested in independent covariants.

2.2 A broader picture

In the previous section we considered coordinate changes on R² and C² from two different perspectives. From the first perspective, the spaces stayed fixed, as we were merely transforming the coordinates used to label their points. From the second perspective, a coordinate change is associated with an automorphism of the underlying space. As it turns out, adopting the second point of view allows us to significantly broaden our notion of an invariant and the spaces they inhabit.

The key observation will be that the set of automorphisms of any space forms a group. We can thus consider a change of coordinates to be a particular instance of an action by some element of a transformation group. In this broader picture, we will consider any function defined on the space in question that is unaffected by the action an invariant. The goal of this section is to make all of this precise.

2.2.1 Lie group actions and representations

To start off with, we would like to restrict ourselves to those automorphisms that preserve the smooth structure of a smooth manifold M, i.e. diffeomorphisms.

The set of all diffeomorphisms of M is denoted by Diff(M) and forms a group.

All transformation groups we consider will be Lie groups.

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A (left) smooth Lie group action is a smooth map Φ : G×M → M where (g, p)7→g·psatisfies:

g1·(g2·p) = (g1g2)·p,∀gi ∈G, p∈M e·p=p

An action of a Lie group Ginduces a homomorphism σ:G→ Diff(M). A group action is faithful if σ(G) ' G. A group action is regular if all orbits have the same dimension, and there exists neighborhoods of every point on M such that the intersection with each orbit is connected (see [Olv95], p.41). Unless otherwise stated, all group actions considered are assumed to be regular and faithful.

Ex1: Let M = R². The group of rotations SO(2) is a one-dimensional group which rotates the plane around a fixed point. Group elements correspond to the angle of rotation, and orbits are circles of constant radii around the fixed point, as well as the point itself.

Ex2: LetM =Rⁿ. The Euclidean groupE(n) is the group consisting of translations, rotations and reflections preserving the euclidean metric onRⁿ. This group acts transitively, which means that there is only one orbit consisting of the entire space.

There is one particular kind of group action which has especially desirable properties: a Lie Group representation is a smooth group homomorphism

Π :G→GL(V), where GL(V) is the general linear group consisting of all linear automorphisms of some vector space V.

In general, group actions will be non-linear. However, given any group action Φ, there is a way to induce a linear action via a representation on an associated space. Given a manifold M, its function space F(M), consisting of all functions F :M →R is a vector space.

We define the induced representation ΠΦ : G → GL(F(M)) with the action given by g·F = ¯F, where we define ¯F(¯x) := F(g⁻¹·x).¯

It is often the case that we are only interested in subrepresentations of ΠΦ. For instance, in the setting of smooth manifolds it is natural to consider only the induced representations of smooth functions, in the case of algebraic manifolds of rational functions, and in the case of projective varieties of homogeneous polynomials. If our manifold M is a vector space, with the group action being the standard action of GL(n,R), we will find that in all the above mentioned cases, restricting to our desired subspace ofF(M) yields a subrepresentation. However, this will not always be the case. Let M = P(R²), and let the action on M be the linear fractional action of GL(2,R). Here the space of polynomials is not a subrepresentation of ΠΦ. To see why, consider the effect of the induced action on the coordinate representations of Q(p) as defined in section 2.1:

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Q(¯¯ p) =Q(g⁻¹·p)¯ Q¯

αp+β γp+δ

=Q(p)

The left hand side of the above is in general not a polynomial expression. If we generalize our notion of an induced representation slightly, we can fix this problem. We start with a preliminary definition:

A multiplier for a Lie group action defined on the space M is a map µ:G×M →C\ {0}satisfying:

µ(g₁·g₂, x) = µ(g₁, g₂·x)µ(g₂, x),∀g_i ∈G, x∈M µ(e, x) = 1

A multiplier representation is a homomorphism ΠΦ,µ :G→GL(F(M)) with an action given by g·F = ¯F, where we define ¯F(¯x) := F(g⁻¹·x) =¯ µ(g, x)F(x).

For the GL(2,R)-action on the functions of two variables, if we let our multiplier be µ(g, x) = (γp+δ)⁻ⁿ, the space of polynomials corresponding to our previous example above will now be a subrepresentation of ΠΦ,µ:

Q(p) = (γp+δ)ⁿQ¯

αp+β γp+δ

2.2.2 Invariants

Let G be a Lie Group with an action defined on the space M. An invariant is a function I : M → R satisfying I(g · x) = I(x), ∀g ∈ G. Equivalently, an invariant I(x) is a fixed point on the function space under the action of the induced representation Π_Φ.

Proposition: LetI denote a real-valued function on a manifold M. The following conditions are equivalent:

i) I is a G-invariant function ii) I is constant on the orbits ofG.

iii) All level sets{I(x) = c} are G-invariant subsets of M.

(See [Olv99], p.73). It immediately follows from iii) that constant functions are always invariant. It follows from ii) that these are the only invariants under a transitive group action.

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Ex1: Let M = R², G = SO(2). If we let 0 be our fixed point under the action of G, any function of the form I(x² +y²) will be an invariant.

Ex2: LetM =R²,G=E(n). The Euclidean group here acts transitively onM, thus the only invariants will be constant functions.

Let G be a Lie Group with actions defined on the spaces M₁, ... , M_m. We can then define an inducedCartesian Product action on the spaceM₁×...×M_m given by g·(x1, ... , xm) := (g·x1, ... , g·xm), ∀g ∈G, xi ∈Mi.

A joint invariant is a function J :M₁×...×M_m →Rsatisfying J(g ·x₁, ... , g·x_m) =J(x₁, ... , x_m),∀g ∈G.

Usually, we are most interested in the case where the M_i’s are all copies of the same space. A Joint invariant on the cartesian product of m copies of M, under the induced Cartesian product action, is referred to as an m-fold joint invariant. In this case, we refer to the induced action as the extended action of G.

Ex: The Euclidean group acting on M = R², failed to yield any non-trivial ordinary invariants. However, consider M ×M with the extended action of G= E(n). Any function of the form I(d((x₁, y₁),(x₂, y₂)), where d is the Euclidean distance function, will be a 2-fold joint invariant.

A symmetric m-fold joint invariant is an invariant of the extended group G×S^m, with an action given by σ :G×S^m×M^×m →M^×m.

Ex: The joint invariant from the previous example is also a symmetric 2-fold joint invariant.

Letµ be a multiplier for a Lie Group action defined on the spaceX. Arelative invariant is a function R : X → R satisfying R(g·x) = µ(g, x)R(x), ∀g ∈ G.

If the group acting is GL(n,R), it can be shown that the multiplier will always be a determinantal factor raised to the power k (see [Olv99]). In this case we define the weight of a relative invariant to be the value of k. The product of two relative invariants of weight k andm, is a new relative invariant of weightk+m.

In particular, multiplying an invariant of weight k with an absolute invariant of weight −k, yields an absolute invariant of weight 0.

We can fit our classical notions of invariants into this broader picture as follows:

Let the G=GL(2), with the induced action on F(C²).

A classical invariant is a relative invariant I :P⁽ⁿ⁾ →C

A classical covariant is a joint relative invariant J :P⁽ⁿ⁾×C² →C

A classical joint covariant is a joint relative invariant I :P⁽ⁿ⁾×...× P^(m)×C² →C

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2.3 Computational methods

This section will be discussing different computational strategies to finding invariants of a given Lie group acting on a space.

Recall that an invariant is a function I(x) ∈ F(M) satisfying I(g ·x) = I(x),

∀g ∈σ(G). Here every element g is a diffeomorphism of M, which suggests that it might be possible to rephrase our defining equation for an invariant slightly.

Consider the following diagram:

M M

R

g

(I◦g)(x)

I(x)

It should be clear that I(x) is an invariant if and only if the above diagram commutes. Recognizing (I◦g)(x) as the pullback of I(x) underg, we can write:

g^∗I(x) =I(x), ∀g ∈σ(G)

We can recognize the form of this equation as a symmetry equation. However, when computing symmetries of a geometric object, the unknown element is the group acting. In our case, the group is known, and the unknown elements are the functions its actions preserve.

2.3.1 The method of moving frames

In general there is no one way to solve the system described above, which generically will be highly nonlinear. If, under certain conditions, we have an explicit local expression for the action ofGonM, one approach is given byThe method of moving frames.

Let (x¹, ... , xⁿ) be local coordinates around a point p in some n-dimensional manifold M, and let Φ :G×M →M be a Lie group action. A local expression for Φ is given by:

ϕ(g,(x¹, ... , xⁿ)) = (ϕ₁(g,(x¹, ... , xⁿ)), ..., ϕ_n(g,(x¹, ... , xⁿ)))

As Gis also a smooth manifold, we can pick local coordinates (y¹, ... , y^r) around the identitye∈G. The idea will be to eliminate thergroup parameters by equat- ing the first r component functions ofϕto a set of constants ˜x¹, ... ,x˜^r. Consider the following system of equations, called the normalization equations:

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ϕ₁((y¹, ... , y^r),(x¹, ... , xⁿ)) = ˜x¹ ... ϕ_r((y¹, ... , y^r),(x¹, ... , xⁿ)) = ˜x^r

After solving this system for the r group parameters (y¹, ... , y^r), we define a moving frame as a map γ :M →Ggiven by substituting its solution back into ϕ₁, ... , ϕ_r.

Proposition: Assume that the action is free and regular. Then the functions I₁, ..., In−r defined below form a complete system ofn−rfunctionally independent invariants for the action of G:

I₁(x¹, ... , xⁿ) = ϕ_r+1(γ(x¹, ... , xⁿ),(x¹, ... , xⁿ)) ...

In−r(x¹, ... , xⁿ) = ϕn(γ(x¹, ... , xⁿ),(x¹, ... , xⁿ)) (See [Olv99], thm 8.25, p.164)

2.3.2 Inﬁnitesimal methods

Another, often more computationally straightforward approach to finding invariants of a Lie group action, can be taken by making use of the corresponding Lie algebra. Given a Lie group G, there is a natural action ofGon itself given by left translations. Let l_g1 :G→G denote a left translation by g₁, which we define by l_g1(g₂) = g₁g₂. A vector field X ∈ D(G) is said to be left invariant if it satisfies (l_g)∗X =X, ∀g ∈ G. We denote the space of all left invariant vector fields of G by Lie(G), and call it the Lie algebraof G. It has some very useful properties:

• Lie(G) is closed under the Lie bracket operation (hence the name).

• Every vector field in Lie(G) is complete.

• It is isomorphic to the tangent space at the identity of G.

There is a natural map Exp: Lie(G)→ G from the algebra to the group known as the Exponential map. It is given by Exp(X) =γ(1), where γ is the unique integral curve of X starting at the identity. Taking the linear span of X yields a curve γ(t) = Exp(Xt), corresponding to a one-parameter subgroup in G.

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In general, not every element of G can be reached via the exponential map of some X ∈Lie(G). The image of Lie(G) under Exp is some neighborhood around the identity of G. Generically, the map also fails to be injective. Only in some cases, like when G is compact or nilpotent and simply connected, do we have that Exp(Lie(G))'G. For more information on this topic, see [Hal15].

A Lie group action Φ : G×M → M, will induce a Lie algebra homomorphism Φ : Lie(G)ˆ → D(M). We can associate a global flow ϕ_X : R×M → M, where ϕX(t, p) =γX(t)·p to each elementX ∈Lie(G). Then we can define ˆΦ(X) = ˆX, such that ˆXp = dϕ_X(t, p)

dt t=0

.

Lie(G) g

G G

Φˆ

Exp ϕXˆ

σ

We denote the image of Gunder σ byG ⊂ Diff(M). Similarly, letg⊂ D(M) be the image of Lie(G) under ˆΦ.

The map ˆΦ is sometimes referred to as an infinitesimal generator of group actions (see [Lee13], p.526). Our strategy will be to reach for the connected component of G via the flow of vector fields ing.

Recall our defining equation for an invariant function: g^∗I(x) = I(x), ∀g ∈ G.

Given that this holds for every g ∈G, it will in particular hold for any ϕ_X_ˆ(t, p)∈G, where ˆX ∈g. Thus, we can write:

ϕXˆ(t, p)^∗I(p) = I(p) dϕXˆ(t, p)^∗

dt t=0

I(p) = dI(p) dt

t=0

LXˆI(p) = 0, ∀Xˆ ∈g

We recognize the last expression as the Lie equation. As before, when the vector fields are the unknowns, this is an (infinitesimal) symmetry equation. Even though there is some loss of information in going from the full picture to the infinitesimal one, the Lie equation has the advantage of always being a linear system of PDE’s. In practice, solutions to this system will often yield all invariants of a given group action. Another useful fact is that orbits of the action of G are integral submanifolds of the flow of the Lie algebra (see [Lee13]).

It is not a given that we have a full group acting on a space to begin with. Any Lie algebra homomorphism, defines aLie algebra action. Finding invariants of

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a Lie algebra action is the task we will be doing in the latter half of this thesis.

Generically there is a slight complication when considering Lie algebra actions due to the fact that there is no guarantee that every ˆX ∈ g will be a complete vector field. All we can say is that locally, the flow ϕXˆ(t, p) will correspond to a family of diffeomorphisms which we identify with elements in some G. For our purposes, this turns out to be good enough.

When we are looking for m-fold joint invariants of a group action, the general way to go about it remains very much the same. Let G be a Lie group with an action Φ defined onM. Recall that we defined theextended group action Φ^×k on M^×k to be the map given by Φ^×k :G×M^×k →M^×k such that:

g·(p₁, ..., p_k) = (Φ(g, p₁), ...,Φ(g, p_k)). We can consider a joint invariant of Φ an ordinary invariant under the action of Φ^×k onM^×k, and proceed as before. This action induces a new homomorphsm σ^×k :G→ Diff(M^×k). We label the image of G under σ^×k byG^×k.

Similarly, given a Lie algebra g ∈ D(M), we define the extended Lie algebra g^×k ∈ D(M^×k) as the image of the induced Lie algebra homomorphism ˆΦ^×k. In local coordinates: If ˆX =f(x¹, ..., xⁿ)ⁱ∂_xⁱ is an element of g,

Xˆ^×k =f(x¹₁, ..., xⁿ₁)ⁱ∂_xⁱ

1+...+f(x¹_k, ..., xⁿ_k)ⁱ∂_xⁱ

k is the extended vector field ing^×k.

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2.4 The space of invariants

Having spent some time defining invariants and looking at different ways to find them, we now turn our attention to the space of invariants itself. Given a Lie group G acting on a manifold M, we will denote the space of all functional invariants by I^G.

Let I₁, I₂ ∈ I^G. Then we have allready seen that the sum I₁+I₂ as well as the product I₁I₂ yields another invariant. This makes the space I^G into a ring. If our invariants are real functions, then multiplication of invariants by elements r ∈ R turns out to yield another invariant as well. We can thus consider the space I^G ⊂ F(M) as a sub-algebra.

2.4.1 Generating sets

The first question we would like to answer, is whether or not a subset a⊂ I^G is a generating set for the space of invariants as anR-algebra.

Recall that orbits of a regular group action are immersed submanifolds of M, which are integral manifolds of Π, the distribution defined by the Lie algebra g ⊂ D(M). The dimension of the orbits is the dimension of these submanifolds, which is equal to the rank of Π (proposition 9.26 in [Olv99], p.209). Given that our group acts regularly on M, the following theorem completely determines the number of functionally independent invariants of G:

Theorem

LetGbe a Lie group with a regular action Φ defined on ann-dimensional manifold M. If the orbits of Φ are of dimension s, then there exists m−s functionally independent local invariants I₁, ..., I_s∈ F(M).

(See [Olv95], p.46). We conclude that ais a generating set ofI^G iff it containss functionally independent elements. From this we can also infer that when Π has reached maximal rank, s= dim(M)−dim(G).

2.4.2 The algebra of polynomial invariants

In the special case where the groupG is semi-simple, and the manifold M is any affine space, there exists a generating set a⊂ I^G consisting of polynomials. This is result is known as Hilbert’s theorem. For proof and further reading, see [Hil93] and [MFK94].

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In particular, if M = Rⁿ, we can consider our generating set a as a subset of R[x], wherex= (x¹, ..., xⁿ). We define the algebra of polynomial invariants as the subalgebra generated by a⊂R[x] and label itI^G.

We can learn more about I^G by a deeper examination of its set of generators.

How do we know that a given generating set is a minimal one? Are the generators independent or is there some relation between them? The answers to these questions are to be found within the framework of syzygy-modules.

Given a generating set of cardinality m, a={a1, a2, ... , am|ai ∈I^G}, we denote by F the free commutative R-algebra generated by a. It should be clear that F 'R[a₁, ... , a_m]. We can also consider F a free module over itself. In fact, we can consider bothF and I^G to be R[a₁, ... , a_m]-modules. There is a natural map φ : F → I^G given by φ( ˜a_i) = a_i. Denoting the kernel of φ by S₁, we get the following exact sequence:

0→S₁ →F →I^G→0

The R[a1, ... , am]-moduleS1 is called the 1st module of syzygies of I^G. By a syzygywe mean an element ofS₁. In other words, a syzygy is a relation between the generators of I^G, of the form:

r_i1a_j1+...+ra_ika_jk = 0,

whereri1, ..., rik ∈F, aj1, ..., ajk ∈I^G, k ≤m. IfS1 is a free module, there is noth- ing more to be done. However, there might be relations between its generators as well. In that case, we can proceed as before. Let b = {b₁, b₂, ... , b_l|b_i ∈ S₁} be a generating set of S1. ThenF1 'R[b1, ... , bl] is the free algebra generated by b. We get another exact seqence:

0→S₂ →F₁ →S₁ →0,

where S₂ is the 2nd module of syzygies of I^G. Equivalently, we can define a mapφ1 :F1 →F such thatφ1(F1) = S1. Continuing this way we get a long exact sequence, called a free resolution of I^G:

...−→^φ³ F₂ −→^φ² F₁ −→^φ¹ F −→^φ I^G→0

All modules F_i are free and each map is a surjection onto the kernel of the next.

It is natural to ask whether or not the free resolution of I^G is a finite sequence.

The following theorem answers the question:

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Hilbert’s syzygy theorem

IfM is a finitely generatedR[a₁, ..., a_m]-module, then them-th module of syzygies S_m of M is free.

(For proof and more information on free resolutions, see [Eis06]). It follows immediately thatF_i = 0 fori > m. By Hilbert’s basis theorem (see [Hil93]), each S_i is finitely generated. If we at each point choose a minimal generating set, we get a minimal free resolution of I^G, which is unique up to isomorphism and finite with length at most m.

By homological methods, it is possible to show that the alternating sum of the dimensions of the free modules F_i equals 0 for a minimal free resolution. These considerations are outside the scope of this text, but for further information on this topic, see [Eis13].

The notation F_i emphasizes that the modules in question arefree. To emphasize their generating setsa,b,c, .., we will adopt the following convention for depicting free resolutions:

R[x]⊃I^G←R[a]←R[b]←R[c]←...←0

2.4.3 The ﬁeld of rational invariants

In general, the Lie group G acting on our manifold M will not be semi-simple, and so we can’t consider our space of invariants as a polynomial subalgebra even if M = Rⁿ. However, if the center of the group acts by semi-simple elements, there exists a generating set a ⊂ I^G consisting of rational functions. This is a consequence of the following more general theorem:

Theorem (Rosenlicht)

If the action ofGisalgebraic, andM is any affine or projective space, a finite set of rational invariants separates the orbits.

(For proof and further reading, see [Ros56] and [KL16]). In light of the above, we can define the field of rational invariants as the subfield generated by a ⊂ R(x), labeled J^G. As the kernel of any field homomorphism is either 0 or the the whole field, the notions of syzygy-modules and free resolutions are inapplicable in this context. Instead, we will make use of concepts from field theory to further describe J^G.

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Given a field extension L|K, we define its Transcendence degree to be the largest cardinality of an algebraically independent subset S of L over K. If in addition, L is an algebraic extension of the field K(S), we refer to S as a Transcendence basis. For more on field extensions, see [DF04].

In particular, the field R(x) is a field extension of transcendence degree n over R. If a generating set ¯a⊆a of J^G of cardinality d is a transcendence basis, then d is also the transcendence degree of J^G over R. We denote this as follows:

R(x)⊃J^G 'R(¯a)⊃^d R

Of course, even in the cases when our space of invariants has a polynomial generating set, we can still consider the space as a subspace of the field of rational functions. From this point of view, we look for a rationally independent generating set to describe the space J^G as a field extension over R.

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3 Computations on even-dimensional symplectic manifolds

In this chapter we will compute ordered joint invariants on symplectic manifolds of dimensions 2 and 4, concluding with a discussion on how our results generalize to higher dimensions. We begin with a preliminary exposition of symplectic geometry.

A symplectic manifold (M, ω) is an even-dimensional manifold M equipped with a closed, alternating, non-degenerate 2-formω, called asymplectic form. In contrast to Riemannian geometry, there are no notions of lengths or angles in symplectic geometry. However, ω does provide a canonical volume form on M, given by ωⁿ = ω ∧...∧ω (n entries, where dimM = 2n), which means there is still a notion of 2n-dimensional hypervolume. Restricting to subspaces S on which ω|_S is non-degenerate, we can define even-dimensional volumes of lower dimension as well. Such a subspace is called a symplectic subspace. A volume form provides an orientation, which means that a symplectic manifold isoriented.

Unlike in the Riemannian case, symplectic manifolds are locally similar, by the following theorem:

Theorem(Darboux)

Let (M, ω) be a 2n-dimensional symplectic manifold. For any pointq ∈M, there exists local coordinates (x¹, ..., xⁿ, p¹, ..., pⁿ) centered at q, in which ω has the following form:

ω =

n

X

i=1

dxⁱ∧dpⁱ

(For a proof, see [Lee13]). We can interpret the action of ω_q on a pair of tangent vectors X_q, Y_q ∈ T_qM as a sum of areas of parallelograms defined by the projections of Xq, Yq to the symplectic subspacesR²(xⁱ,pⁱ) of TqM.

The symplectic form establishes a canonical isomorphism between T M andT^∗M given by ω:T M →T^∗M, where X 7→ι_Xω. Given any H ∈ C^∞(M), we can use this isomorphism to associate a Hamiltonian vector field X_H to H, defined by the relation ι_X_Hω =dH.

In local Darboux coordinates, the Hamiltonian vector field corresponding to the function H is given by:

X_H =

n

X

i=1

∂H

∂pⁱ

∂

∂xⁱ −∂H

∂xⁱ

∂

∂pⁱ

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It is straightforward to deduce that Hamiltonian vector fields correspond to the infinitesimal symmetries of ω, i.e. L_Xω = 0, iff X =X_H for some H ∈ C^∞(M).

By theinfinitesimal Stoke’s theorem L_X_Hω =d(ι_X_Hω) +ι_X_Hdω. Sinceωis closed, the second term vanishes. Using the definition of X_H, we see that d(ι_X_Hω) = d(dH) = 0 as well, which verifies the claim.

Thus, there is a correspondence between elements of C^∞(M) and sym(ω). We can use the Hamiltonian vector fields to induce an operation on C^∞(M), turning the space into a Lie algebra. Let F, H ∈ C^∞(M). Then we define the Poisson bracket {, }:C^∞(M)× C^∞(M)→ C^∞(M) by {F, H}:=X_H(F).

In local Darboux coordinates:

{F, H}=

n

X

i=1

∂H

∂pⁱ

∂F

∂xⁱ − ∂H

∂xⁱ

∂F

∂pⁱ

It is follows that X{F,H} = [XF, XH], and so the map Φ :C^∞(M) →sym(ω) defined by Φ(H) = X_H is a Lie algebra homomorphism, where the kernel of Φ con- sists of constant functions. Hence, sym(ω) is an infinite-dimensional Lie algebra.

We would like to consider a finite-dimensional subalgebra g. Let M be a linear symplectic manifold. It should be clear that the subspace P⁽²⁾(M)⊂ C^∞(M) is closed under the Poisson bracket.

This allows us to define g=Im(Φ|_P(2)(M)(C^∞(M))) ⊂sym(ω).

3.1 2-dimensional M

We start with the case where M =R²(x, p) is our base space endowed with the standard symplectic form ω=dx∧dp. Its infinitesimal symmetries are given by:

sym(ω) ={X_f =f_p∂_x−f_x∂_p|f ∈ C^∞(M)}

By restricting the generating functions f to consist of quadratic functions of the form f =a₀x²+a₁xy+a₂y², we get a 3-dimensional subalgebra:

g= h−x∂_p, x∂_x−p∂_p, p∂_xi We observe that g'sp(2) =sl(2).

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The action of g on M has two orbits: the fixed point {0} and the open orbit R²\ {0}. We regard a generic point as a point contained in the latter orbit, and say that the algebra acts transitively on generic points of M. Thus, there will be no invariants on the base space. Solving the system L_Xf = 0,∀X ∈ g gives us the trivial solution where f =const, which confirms that this is the case. We conclude that I^G=R.

3.1.1 M ×M

We now move onto the space

M

×

M =

_R⁴

(x

₁

, x

₂

, p

₁

, p

₂

)

. To look for joint invariants, we start by extending the algebra by applying the recipe from section

2.3

. It gives us the following:

_x₂i The rank of the distribution defined by the vector fields in g^×2 is 3, which is the maximal rank. This implies that there will be one independent invariant. More- over, as the algebra acting is semi-simple, we know that it will be a polynomial.

Solving the Lie equation gives us:

a

₁₂

= x

₁

p

₂ −

x

₂

p

)

. We denote the algebra of 2-fold joint invariants by

I

^G×2. It is generated by one element, with the following minimal free resolution:

R

[x

₁

, x

₂

, p

₁

, p

₂

]

⊃

I

^G×2 ← _R

[a

₁₂

]

←

0

This implies that

I

^G×2 ' _R

[a

₁₂

]

[a

₁₂

, a

₁₃

, a

₂₃

]

.

in the Lie equation, we find that all 6

a

, p

_i

)

is fixed only the stabilizer

G

I

0

So far we have considered our generators for the algebra of joint invariants as polynomials. However, it is also possible to consider them rational functions.

₁₂

Hence, we can drop

a

₃₄ from the set of generators of the field of rational joint invariants, which we will denote

J

^G×4. From this perspective,

J

^G×4 is a field extension over R of transcendence degree 5:

≤

i < j < k < l

≤

5.

To shorten our expressions, we hereby adopt a shorthand notation for the set of generators for the free algebras we will be constructing. Let a denote the set of all

a

_ij’s. Likewise, let b denote the set of all

b

_ijkl’s.

Let us find out if

S

₁ is a free module, or if there are relations between its generators. We proceed as before, by constructing the free algebra

F

₁ ' _R

[b]

,

Joint Invariants of Symplectic and Contact Lie Algebra Actions