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Canonical endomorphism eld on a Lie algebra

Jerzy KOCIK

Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA E-mail: [email protected]

Received Date: March 15, 2010; Revised Date: April 28, 2010

Notation. We will distinguish between purely algebraic and Differential products by using two types of brackets:

Image: Lie algebra product,

[ , ]: Lie commutator of vectorfields, Schouten bracket, Nijenhuis bracket.

The summation convention over repeated indices is adopted throughout the paper.

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Abstract

We show that every Lie algebra is equipped with a natural (1; 1)-variant tensor eld, the \canonical endomorphism eld", determined by the Lie structure, and satisfying a certain Nijenhuis bracket condition. This observation may be considered as complementary to the Kirillov-Kostant-Souriau theorem on symplectic geometry of coadjoint orbits. We show its relevance for classical mechanics, in particular for Lax equations. We show that the space of Lax vector elds is closed under Lie bracket and we introduce a new bracket for vector elds on a Lie algebra. This bracket de nes a new Lie structure on the space of vector elds.

1 Introduction

It is well known that the underlying dual space L* of a Lie algebra L possesses-as a manifold-a canonical Poisson structure in terms of a smooth bivector field Image, which satisfies the Jacobi conditionImage and, when restricted to coadjoint orbits, is nondegenerate and therefore invertible into a symplectic structure [12,16,17]. The existence of these symplectic sheets is the content of the Kirillov-Kostant-Souriau theorem ([3, 9, 15]).

In this paper, we present an overlooked fact that the Lie algebra L itself also possesses-as a manifold-a natural Differential -geometric object, namely, a (1, 1)-type tensor field A ∈ T (1,1) L that we will call the canonical endomorphism field on L. The principal geometric property of A is that it is proportional to its own Nijenhuis derivative (Theorem 2.1).

We discuss the relevance of this object for dynamical systems. It turns out that what Hamilton equations are for the dual space L*, Lax equations are for L. The principal property of A assures that the space of "Lax vector fields" is closed under the Lie commutator and, moreover, it allows one to introduce a new bracket of vector fields on L, which is the analog for Lax equations of the Poisson bracket on Hamiltonian vector fields.

2 The canonical endomorphism field on a Lie algebra

Customarily, one de nes a Lie algebra as a linear space L with a product Image denoted Image (double bracket). The product is bilinear, skew-symmetric (i), and satisfies the Jacobi identity (ii):

Image

In a basis {ei}, the commutator can be represented via "structure constants":

Image (2.1)

Here, we will rather follow [10] and de ne a Lie algebra as a pair {L, c}, where c is a (1; 2)-type tensor that in the above basis is

Image (2.2)

where Image is the dual basis. The algebra product becomes a secondary, derived, concept:

Image. Similarly, the adjoint action of v ∈ L is de ned simply as a (1,1)-tensorImage. Of course, from the structural point of view both de nitions are equivalent, Image.

The point of the present paper is to look at the space L as a at manifold and consider various Differential -geometric objects on it (we will assume that L is real and nite dimensional). The linear structure of this manifold allows one to prolong any tensor T in L to the ("constant") tensor field Image on the manifold L. In particular, the manifold L is equipped with a constant (1, 2)-type tensor field Image

Image (2.3)

where {xi} are coordinates on L associated with the basis {ei}, and where we denote ImageImage. The manifold L is also equipped with a natural vector field, the Liouville vector field, which in a linear coordinate system, is

Image (2.4)

Here is our basic observation.

Theorem 2.1. The manifold of the Lie algebra L possesses a natural field of endomorphisms (i.e., a (1,1)-variant tensor field) Image de ned by

Image (2.5)

Its Nijenhuis derivative [A,A] is a vector-valued biform as follows:

Image (2.6)

Moreover, A acts on the adjoint orbits on L.

We will call A the canonical endomorphism field on L. In the coordinate description, A and its Nijenhuis derivative are

Image (2.7)

The endomorphism field A may be viewed as a family of local transformations that at point x ∈ L can be represented by matrix Image.

Before we give its proof, let us restate the theorem in more standard terms. The natural isomorphism of a tangent space at any x ∈ L with the space L itself will be denoted by Image. Then, Theorem 2.1 states that every Lie algebra L possesses, as a manifold, a unique natural tensor field Image, which at point x ∈ L is de ned as an endomorphism taking a tangent vector v ∈ TxL to

Image (2.8)

or, in a somewhat sloppy notation, Image. Its Nijenhuis derivative [A,A] is a vectorvalued biform, the evaluation of which equals for any v,w ∈ TL:

Image (2.9)

at point x ∈ L, the dependence of which was suppressed in the notation.

Remark 2.2. The canonical endomorphism field A is de ned for an arbitrary algebra and its Differential -geometric properties, including the Nijenhuis bracket [A,A], will re ect the type of this algebra. In the present paper, we restrict to Lie algebras, where the Jacobi identity implies particularly pleasant consequences.

The above theorem may be viewed as a counterpart of the KKS theorem: the essence of which is that the dual space L* is equipped with a bivector field Image(in our languageImage). Instead of the Nijenhuis bracket, we have the Schouten bracketImage. Thus, Ω de nes a Poisson structure, which, moreover, restricts to the coadjoint orbits, on which its inverse Image de nes a symplectic structure, Image=0. Section 8 summarizes these parallels.

3 Lie algebra in pictures

Tensor calculus gains much transparency when expressed in graphical language.

Basic Glyphs. Here are the basic glyphs corresponding to various tensors:

Image

where s is a scalar, v is a vector, α is a covector, A is an endomorphism, and g is a metric or biform. The links with arrows Image and links with circlesImage represent the contravariant and the covariant attributes of a tensor, respectively. You may think of them as contravariant/ covariant (upper/lower) indices in some basis description. Scalars have none.

The "in" and "out" links may go at any direction. Turning and weaving in space do not have any meaning (unlike in some other conventions). For instance,

this representation Image is as good as thisImage

The links may leave the box at any position, but the order of the point of departure is xed: the contravariant indices are ordered clockwise, while the covariant indices counterclockwise. Links may cross without any meaning implied.

Glyphs may be composed into pictograms that represent terms resulting by manipulation with tensors. The tensor contractions are obtained by joining "ins" with "outs". Here are some basic cases.

Evaluation. Here is the evaluation of a covector on a form:

Image

Scalar product. The scalar product of two vectors is a scalar g(v;w), but if only one vector is contracted with g, then the result is a one form:

g(v;w) = Image

Endomorphism. A acting on a vector v or covector results in a vector or covector, respectively:

Image

Trace may be represented by connecting "in" with "out" in a pictogram; if A,B,C ∈ EndL are endomorphisms of some linear space L, then we have

Image

The notable property of trace of a composition of endomorphisms, namely, its invariance under cyclic permutation of the entries, Image becomes in graphical language veri able with a simplicity of a mantra on a japa mala.

Lie algebra in pictures. An algebra is de ned by a (1,2)-variant tensor c, as shown below on the left. Also a product and adjoint representation is shown as follows:

Alg structure = Image

If a single algebra is considered, the letter "c" will be suppressed.

In the case of a Lie algebra, besides skew-symmetry we have the Jacobi identity, which may be written in this way:

Image (3.1)

The labels a and b are only to discern between di erent entries.

Perhaps the simplest derived object is a characteristic one-form Image the value of which on a vectorImage. Its pictograph is

Image

(This one-form vanishes for semisimple algebras.)

The Killing form is de ned as an inner product Image. In the diagrammatic script, it is easy to de ne the corresponding 2-covariant tensor K:

Image

Every Lie algebra possesses a skew-symmetric exterior Lie 3-form Image that for any triple v,w, z ∈ L takes value Image. Using diagrammatic script, we may "draw" the form Image directly-here it is, simpli ed with the use of Jacobi identity (3.1):

Image

where a and b are merely labels to distinguish the covariant entries. If we use the symbol ^ or "alt" inside a loop to denote the signed sum over all permutations of entries of a tensor (skewsymmetrization), then the Lie 3-covariant form is

Image

4 Differential geometry on a Lie algebra

Let us now look at the Differential geometry of Lie algebra viewed as a manifold. In the diagrammatic language, the objects of Theorem 2.1 are

De nition: Image Theorem:Image

Since the contraction with J introduces dependence on poisition (coordinates x), we will use rather notation that will be easier perceptually. Thus, for instance,

Image

Every element (vector) v ∈ L de nes a "constant" vector field Image on manifold L obtained by parallel transport; in coordinates, if v = viei then Image. The canonical endomorphism field A on manifold L applied to such fields de nes a representation of Lie algebra L in terms of vector fields on L, namely, with every algebra element v ∈ L, we associate a vector field:

Image (4.1)

Proposition 4.1. The map Image de nes a homomorphismImage (the in nitesimal representation of L in terms of XL):

Image (4.2)

If the center of L is trivial, the map presents a monomorphism.

Proof. The proposition readily follows from the Jacobi identity.

Corollary 4.2. The following are convenient formulae:

Image

The image of A spans at every point a subspace of the tangent space of L, de ning in this way a distribution:

Image (4.3)

The integral manifolds of this distribution coincide with the adjoint orbits determined by the action of a Lie group on Lie algebra. Note, however, that we may de ne "adjoint orbits" without reference to the Lie group simply as the integral manifolds O of D, satisfying TO = D.

Now we prove the theorem.

Proof of Theorem 2.1. Recall that the Nijenhuis bracket [K,K] of a vector-valued oneform (endomorphism field) K with itself is a vector-valued biform that, evaluated on two fields X and Y , takes the value according to

Image (4.4)

(see, e.g., [13]). Evaluating (half of) the Nijenhuis bracket [A,A] on two constant vectorfields Image and Image and using formulae of Propositions 4.1 and 4.2, one gets the following:

Image

In particular, substitution of Image and Image leads to the coordinate formula (2.7). Now, let us show that A can be restricted to orbits, that is, Image for each point x ∈ O.

First, rewrite (2.8) for X ∈ TxL:

Image

Vector of the vectorfield Xv at point x ∈ L can be expressed as follows:

Image.

Thus,

Image

which was to be proven.

Example 4.3. Consider the 2-dimensional solvable algebra de ned by Image. Then,

Image

The adjoint orbits are lines parallel to e2, and the canonical endomorphism-when restricted to any of them-becomes a dilation.

Example 4.4. The Lie algebra of 3-dimensional rotations, so3, is de ned by relations Image. Thus,

Image

The orbits are spheres de ned by the Killing form. On the unit sphere, tensor A forms an almost complex structure, .

Remark 4.5. In general endomorphismfield A is not integrable. The integrable cases, where the Nijenhuis bracket (2.6) vanishes, include two-step nilpotent algebras such as algebras of type H [7,8]. Note that for vectorfields of in nitesimal representation, the biform (2.6) takes at any point x a vector-value:

Image. (4.5)

Thus, A restricted to an orbit Image is (locally) integrable ifImage for every x ∈ O and every v;w ∈ L. This is true for so(n), n ≤ 4 and for nilpotent algebras of the upper-triangular n × n matrices, n ≤ 5.

5 Other basic properties of the endomorphismfield

The fundamental property of the canonical endomorphismfield (Theorem 2.1) is

Image

Other basic properties of the geometry of a Lie algebra are summarized below.

Corollary 5.1. The endomorphismfield on a Lie algebra satisfies

Image

where O denotes an orbit through x.

Here is a property analogous to the coadjoint representation preserving the Kirillov- Poisson structure on the dual Lie algebra.

Proposition 5.2. The endomorphism A is preserved by the action of the adjoint represen- tation:

Image (5.1)

Proof. Use Leibniz rule to show that Image for everyImageImage

Proposition 5.3. The endomorphismfield on a Lie algebra satisfies

Image

where the objects are as follows: K is the Killing form de ned for two vectors as K(v,w) = Image When evaluated for (J; J), it becomes a quadratic scalar function K(J, J) = Image Similarly,Image is a characteristic form on L de nedImage Prop- erty (iii) states that the endomorphism A is skew-symmetric with respect to the Killing (possibly degenerated) scalar product.

The endomorphism de nes for every k = 1, 2, . . . , a scalar function of the power trace:

Image (5.2)

that will be called Casimir polynomials on L. In the diagrammatical language, they are

Image

etc. Clearly, the second invariant is a quadratic function related to Killing form and will be denoted k = I2 = K(J; J) = k, but the third is obviously not related to the Lie 3-form.

Corollary 5.4. Di erentials of the trace functions are among the annihilators of A:

Image (5.3)

6 The endomorphismfield and dynamical systems

Since the dual Lie algebra L* with its Poisson structure has deep connections with classical mechanics, namely, with Hamiltonian formalism, one may expect that so does a Lie algebra with its endomorphismfield A. The candidate coming to mind rst is Lagrangian mechanics, as suggested by this chain of correspondences:

Image

Duality between tangent bundle TQ over a manifold M, which possesses enough structure so that any ("regular") function L on TQ de nes a dynamical system via Lagrange equations, and the cotangent bundle T*M, with its own symplectic structure w granting a Hamiltonian formalism induced by the Hamiltonian H, suggests that the question mark in the above diagram of analogies should be replaced by some sort of Lagrange formalism. This guess may be supported by the fact that the Lagrange formalism is actually based on the natural endomorphismfield on the tangent ber bundle (see Appendix B).

Yet, it seems that the most direct formalism at the question mark seems-much generalized-Lax equations of motion.

Although Lax equations are typically de ned as matrix equations, the endomorphism A allows one to geometrize it in a new way. In the next sections, we will discuss "Lax vectorfields" on a Lie algebra and will push the analogy with symplectic geometry to see how far it goes.

We show that, quite pleasantly, "Lax vectorfields" form a closed subalgebra under vectorfield commutator. We will also de ne a new "Poisson bracket" in the space of vectorfields on Lie algebra, and prove a homomorphism between Lie algebra of vectorfields with this bracket with the standard Lie algebra of vectorfield.

7 The algebra of Lax vectorfields

Let us start with a general construction. By analogy to symplectic geometry dealing with manifolds equipped with symplectic structure, {M, w}, we may consider a pair {M, A}, where manifold M is equipped with a structure de ned by afield of endomorphisms-(1, 1)-variant tensorfield on M. Exploring further the analogy, we may study dynamical systems described by vectorfields that are de ned by their "potentials"-other vectorfields. Thus, instead of Hamilton equations, we have a map:

Image (7.1)

This contrasts with symplectic geometry, where the potentials of dynamical systems are Differential forms, namely, di erentials of Hamiltonians. It would be natural to require that the set of all such dynamical systems, Image be closed under the Lie bracket of vectorfields. This way it would form a subalgebra of Image. The nal demand would be to have a well-de ned product of vectorfields (potentials), such that the map (7.1) is a homomorphism of the corresponding algebras.

One may ask why one would want to replace one vectorfield by another: one gain may be that in the new form some integrals of motion may be found more easily.

In this section, we show that a Lie algebra with the endomorphismfield de ned in the previous sections forms such a system. In particular, it is equipped with a bracket for potentials that we de ne below.

Consider the underlying linear space L of a Lie algebra Image as a manifold. Any smooth vectorfield B can be viewed as a generator (or "potential") of a dynamical system de ned by vectorfield XB de ned

Image (7.2)

The integral curves of XB satisfy the Lax equations, which in a somewhat imprecise way are expressed as follows:

Image

where the x on the left side is understood as a point in L, while the x inside the bracket on the right side is understood as a vector in L. More accurately,

Image

De nition 7.1. Vectorfields on a Lie algebra L of form (7.2) will be called Lax vectorfields generated by B or Lax dynamical systems. In the diagrammatic representation, the Lax vectorfield is

Image (7.3)

The space of Lax vectorfields will be denoted by Image

A simple and a well-known fact is the existence of Casimir invariants.

Corollary 7.2. The dynamical system de ned by a Lax vectorfield (7.2) leaves Casimir polynomials Ik invariant, XBIk = 0, for any Image.

Proof (graphical). We show the reasoning for I2 = K(J, J) (quadratic polynomials de ned by Killing form):

Image

where rst we used Jacobi identity (3.1) and then skewsymmetry of the resulting w. The right side vanishes as w has two identical entries, x. The argument for the other Casimir invariants is similar.

The geometric meaning of the fundamental Nijenhuis property of the endomorphismfield becomes clear in the current context. Namely, it implies that the space of Lax vectorfields Image is closed under the commutator of vectorfieldsImageA new bracket of vectorfields is implied.

Theorem 7.3. The space of Lax vectorfield forms a subalgebra of the algebra of smooth vectorfields, Image In particular, if XB and XC are two (global) Lax vectorfields, then their commutator is a Lax vectorfield with potential:

Image(7.4)

so that there is an homomorphism between the Lax vectorfields with the regular vectorfield commutator and all vectorfields with { , } product:

Image (7.5)

Proof. This follows from the fact that [A;A] is proportional to A. Rewrite the de nition of the Nijenhuis bracket (4.4) for A and use Theorem 2.1:

Image (7.6)

where in the last part, we see that the endomorphismfield A may be "factored out" thanks to Theorem 2.1. Thus the commutator is of the form (7.2), the formulas in the theorem follow.

Proposition 7.4. The bracket (7.4) can be calculated by the following formula:

Image (7.7)

where Image

Notice that although the two right-most terms are de ned in coordinates, their di erence has a coordinate-free meaning, as it can be de ned by Image.

The bracket { , } turns the space of vectorfields on L into a Lie algebra and can be viewed as a "Differential deformation" of the Lie algebra bracket Image. Due to its involved nature, it may be rather surprising that it de nes a Lie algebra. The Jacobi identity is not a direct consequence and results by intertwined interaction of the Jacobi identities of the Lie algebra L and of the Lie algebra of vectorfields.

Remark 7.5. For two constant vectorfields Image and Image that extend vectors v,w ∈ L, it is Image Thus the bracket formula (7.5) reduces in this case to the in nitesimal representation Image.

Theorem 7.6. The pair Image forms a Lie algebra, that is, the bracket (7.4, 7.8) of vectorfields satisfies the following properties:

Image (7.8)

Proof. If Image are two vectorfields, then we denote Image a vectorfield calculated in linear coordinate system. Thus, formula (7.7) can be written as follows:

Image

where

Image

Now, using the formula Image we get

Image

where the letters (a), (b), (c), and (d) are used to indicate the origin of terms in the second part of the equation. The sum

Image

contains every term of equation (*) in each of the three cyclic permutations of A, B, and C. The sum of such terms marked by any of the numbers (1) to (4) vanishes due to opposite signs. The group of terms marked by (0) and terms marked by (5) both vanish, each due to the Jacobi identity of the Lie algebra product.

Corollary 7.7. There is a Lie algebra homomorphism Image between Lie algebra of vectorfields on L with bracket de ned by (7.4) and the Lie algebra vectorfields on L restricted to Lax vectorfields.

By analogy to Hamiltonian formalism of classical mechanics, we have a property that may be viewed as a counterpart of Poisson theorem.

Corollary 7.8 ("a la Poisson"). If vectorfields B and C are Lax potentials of symmetries of a dynamical system, then {B,C} is a Lax potential of a symmetry as well.

Proof. Use the Jacobi identity for vectorfields:

Image

hence the claim: Image.

Basic examples. What can be used as a Lax potential? The simplest are constant vectorfields, in which case the homomorphism reduces to Proposition 4.1 (see Remark 7.5). Also, a Lax vectorfield may be "reused" as a potential for a new Lax vectorfield. The following formulas for bracket Image may be useful for such dynamical systems:

Image

where v and w are understood as constant vectorfields (the tilde is suppressed for simplicity).

Another class consists of Lax vectorfields generated from linear vectorfields on L. Euler's equations of the motion a rigid body belong to this category. Here is their|somewhat naive- generalization to arbitrary Lie algebra: let R ∈ EndL be a matrix describing the tensor of inertia. If vectorfield Image is used as a "potential", the resulting Lax vectorfield X = Image describes the dynamical system of "rotating body". In the case of the Lie algebra of 3-dimensional orthogonal group Image with the standard coordinates (x, y, z) and for a diagonal matrix R = diag(a, b, c), we get the standard Euler's equations:

Image

Image A more accurate description will be given in a subsequent paper.

8 Analogies and dualities

The analogies between Differential geometry (calculus) on a Lie algebra and on a Lie coalgebra are shown in the following table. Note that the Lie algebra structure c is a (1,2)-variant tensor on the Lie algebra L, but it is a (2,1)-variant tensor on the dual space L*. This results in quite di erent calculus on both spaces treated as manifolds.

Image

On L* as a manifold, λ is (2, 1) variant. In pictures, the canonical Poisson structure on L* and Hamiltonian mechanics may be illustrated as follows:

Image

9 Remark on Lagrange equations

While the cotangent bundle T*Q over a manifold Q possesses a canonical Differential biform Image de ning symplectic structure, the tangent bundle TQ possesses a canonical (1,1)- variant tensorfield Image de ning an endomorphismfield (endomorphisms of T(TQ) and T*(TQ)). In the natural coordinates (xi, vi) on TQ, this tensor can be expressed as Image (sum over i). Its basic property is Ker S = Im S (implying nilpotence S o S = 0). If L is a function on TQ (a Lagrangian), then one de nes a biformImagewhich for a "regular" Lagrangian is nondegenerate and therefore forms a symplectic structure. It is easy to see that Lagrange equations may be written as follows:

Image.

The existence of S and its role in Lagrangian mechanics were noticed rather late [11]; they replace a rather awkward notion of "vertical derivative" that were employed previously to geometrize Euler-Lagrange equations [1].

In a series of papers [4,5], a notion of almost tangent structure on a Differential manifold M has been introduced, as a tensor Image that satisfies

Image (9.1)

where the second condition (ii) is a generalization of the Schouten-Nijenhuis bracket to "vector-valued Differential forms" (see, e.g, [17,18]), which assures (local) integrability of the distribution Ker S. As a result, one obtains all of the structures of the tangent bundle (Ker S gives the bering) except distinguishing the zero-section.

A Lie algebra may provide an example of a generalized version of such an Euler-Lagrange structure, in which the above conditions (9.1) are relaxed.

Image

Whether such potential relationship between Lie algebras and generalized Lagrangian formalism would be fruitful is an interesting question in the context of geometric quantization and representation theory known for coadjoint orbits in the Lie coalgebras.

Acknowledgements

The author would like to thank Zbigniew Oziewicz for discussions that much improved the presentation and Philip Feinsilver for many valuable comments and suggestions.

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