Canonical endomorphism field on a Lie algebra

We show that every Lie algebra is equipped with a natural $(1,1)$-variant tensor field, the"canonical endomorphism field", naturally 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 fields is closed under Lie bracket and we introduce a new bracket for vector fields on a Lie algebra. This bracket defines a new Lie structure on the space of vector fields.


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 bi-vector field Ω ∈ ∧ 2 T L * , which satisfies the Jacobi condition [Ω, Ω] = 0, and, when restricted to coadjoint orbits, is nondegenerate and therefore invertible into a symplectic structure [16,17,12]. 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 shall 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.

The canonical endomorphism field on a Lie algebra
Customarily one defines a Lie algebra as a linear space L with a product L × L → L denoted [[v, w]] (double bracket). The product is bilinear, skew-symmetric (i), and satisfies the Jacobi identity (ii): In a basis {e i }, the commutator can be represented via "structure constants": [[e i , e j ]] = c k ij e k Here we shall rather follow [10] and define a Lie algebra as a pair {L, c} where c is a (1, 2)-type tensor that in the above basis is where {ε i } is the dual basis. The algebra product becomes a secondary, de- The point of the present paper is to look at the space L as a flat manifold and consider various differential-geometric objects on it. (We shall assume that L is a real and finite dimensional.) The linear structure of this manifold allow one to prolong any tensor T in L to the ("constant") tensor field T on the manifold L. In particular, the manifold L is equipped with a constant (1, 2)-type tensor field λ = c: where {x i } are coordinates on L associated with the basis {e i } and where we denote ∂ i ≡ ∂/∂x i . The manifold L is also equipped with a natural vector field, the Liouville vector field, which in a linear coordinate system is Here is our basic observation: The manifold of the Lie algebra L possesses a natural field of endomorphisms (i.e., a (1,1)-variant tensor field) A ∈ T (1,1) L defined by Moreover A acts on the adjoint orbits on L.
We shall call A the canonical endomorphism field on L. In the coordinate description, A and its Nijenhuis derivative are The endomorphism field A may be viewed as a family of local transformations that at point x ∈ L can be represented by matrix A k j (x) = x i c k ij . 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 µ x : T x L → L. Then Theorem 2.1 states that every Lie algebra L possesses, as a manifold, a unique natural tensor field A ∈ T (1,1) L, which at point x ∈ L is defined as an endomorphism taking a tangent vector or, in a somewhat sloppy notation, is a vector-valued biform the evaluation of which equals for any v, w ∈ T L: at point x ∈ L, the dependence of which was suppressed in the notation.

Remark 2.2.
The canonical endomorphism field A is defined for an arbitrary algebra and its differential-geometric properties, including the Nijenhuis bracket [A, A], will reflect 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 bi-vector field Ω = x k c k ij ∂ i ∧ ∂ j (in our language Ω = J λ). Instead of the Nijenhuis bracket we have the Schouten bracket [Ω, Ω] Sch = 0. Thus Ω defines a Poisson structure, which, moreover, restricts to the coadjoint orbits, on which its inverse ω defines a symplectic structure, ω = 0. Section 8 summarizes these parallels.

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: where s is a scalar, v is a vector, α is a covector, A is an endomorphism, g is a metric or biform. The links with arrows and links with circles 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 any direction. Turning and weaving in space does not have any meaning (unlike in some other convections). For instance: this representation T is as good as this T The links may leave the box at any position, but the order of the point of departure is fixed: 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:

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: Endomorphism A acting on a vector v or covector α results in a vector or covector, respectively: Trace may be represented by connecting "in' with "out" in a pictogram; If A, B, C ∈ End L are endomorphism of some linear space L, then we have: The notable property of trace of a composition of endomorphisms, namely its invariance under cyclic permutation of the entries, Tr , becomes in graphical language verifiable with a simplicity of a mantra on a japa mala.
Lie algebra in pictures. An algebra is defined by a (1,2)-variant tensor c, as shown below on the left. Also a product and adjoint representation is shown: 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 iden-tity, which may be written this way The labels a and b are only to discern between different entries.
Perhaps the simplest derived object is a characteristic one-form χ ∈ L * the value of which on a vector v ∈ L is χ(v) = −Tr ad v . Its pictograph is χ = (This one-form vanishes for semisimple algebras.) The Killing form is defined as an inner product K(v, w) = Tr ad v ad w . In the diagrammatic script it is easy to define the corresponding 2-covariant tensor K K = Every Lie algebra possesses a skew-symmetric exterior Lie 3-form ω that for any triple v, w, z ∈ L takes value ω(v, w, z) = Tr ad [[v, w]] ad z . Using diagrammatic script we may "draw" the form ω directly -here it is, simplified with the use of Jacobi identity (11): where a and b are merely labels to distinguish the covariant entries. If we use symbol the ∧ 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

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 Since the contraction with J is introduces dependence on poisition (coordinates x), we shall use rather notation that will be easier perceptually. Thus, for instance: Every element (vector) v ∈ L defines a "constant" vector field v ∈ X L on manifold L obtained by parallel transport; in coordinates, The canonical endomorphism field A on manifold L applied to such fields defines 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 If the center of L is trivial, the map presents a monomorphism.
Proof. The proposition readily follows from the Jacobi identity.
The image of A spans at every point a subspace of the tangent space of L, defining in this way a distribution 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 define "adjoint orbits" without reference to the Lie group simply as the integral manifolds O of D, satisfying T O = D.
Now we prove the theorem.
Proof of Theorem 2.1: Recall that the Nijenhuis bracket [K, K] of a vector-valued one-form (endomorphism field) K with itself is a vector-valued bi-form that, evaluated on two fields X and Y , takes the value according to (see, e.g., [13]). Evaluating (half of) the Nijenhuis bracket [A, A] on two constant vector fields v and w and using formulae of Proposition 4.1 and 4.2, one gets In particular, substitution X = ∂ a and Y = ∂ b leads to the coordinate formula (8). Now, let us show that A can be restricted to orbits, i.e., Vector of the vector field X v at point x ∈ L can be expressed as which was to be proven.
The adjoint orbits are lines parallel to e 2 and the canonical endomorphismwhen restricted to any of them -becomes a dilation.
The orbits are spheres defined by the Killing form. On the unit sphere, tensor A forms an almost complex structure, A • A = −id.

Remark 4.5.
Although the Nijenhuis bracket (7) vanishes for two-step nilpotent algebras, (including Heisenberg-type algebras [8,7]), in general it does not, and therefore endomorphism field A is in general not integrable. Note that for vector fields of infinitesimal representation, the bi-form (7) takes at any point x a vector-value ]]] = 0 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.

Other basic properties of the endomorphism field
The fundamental property of the canonical endomorphism field (Theorem 2.1) is Other basic properties of the geometry of a Lie algebra are summarized below: Corollary 5.1. The endomorphism field on a Lie algebra satisfies: 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 representation, that is
Proof. Use Leibniz rule to show that (£ Xv A)(w) = 0 for every w: Proposition 5.3. The endomorphism field on a Lie algebra satisfies: where the objects are as follows: K is the Killing form defined for two vectors as K(v, w) = Tr ad v • ad w . When evaluated for (J, J), it becomes a quadratic scalar function K(J, J) = x a x b c k ai c i bk . Similarly, χ ∈ L * is a characteristic form on L defined χ(v) = Tr ad v . Property (iii) states that the endomorphism A is skew-symmetric with respect to the Killing (possibly degenerated) scalar product.
The endomorphism defines for every k = 1, 2, . . . , a scalar function of the power trace that will be called Casimir polynomials on L. In the diagrammatical language they are: x x x x etc. Clearly, the second invariant is a quadratic function related to Killing form and will be denoted κ = I 2 = K(J, J) = κ, but the third is obviously not related to the Lie 3-form.

Corollary 5.4. Differentials of the trace functions are among the annihilators of A,
i.e., A dI k = 0 (21)

The endomorphism field 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 endomorphism field A. The candidate coming to mind first is Lagrangian mechanics, as suggested by this chain of correspondences: KKS theorem → symplectic → Hamilton (Lie coalgebras) geometry equations Theorem 2.1 → endomorphic → ? (Lie algebras) geometry Duality between tangent bundle T Q over a manifold M , which possesses enough structure so that any ("regular") function L on T Q defines a dynamical system via Lagrange equations, and the cotangent bundle T * M with its own symplectic structure ω 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 endomorphism field on the tangent fiber bundle (see Appendix B). Yet it seems that the most direct formalism at the question mark seemsmuch generalized -Lax equations of motion.
Although Lax equations are typically defined as matrix equations, the endomorphism A allows one to geometrize it in a new way. In the next sections we shall discuss "Lax vector fields" on a Lie algebra and will push the analogy with symplectic geometry to see how far it goes.
We show that, quite pleasantly, "Lax vector fields" form a closed subalgebra under vector field commutator. We shall also define a new "Poisson bracket" in the space of vector fields on Lie algebra, and prove a homomorphism between Lie algebra of vector fields with this bracket with the standard Lie algebra of vector field.

The algebra of Lax vector fields
Let us start with a general construction. By analogy to symplectic geometry dealing with manifolds equipped with symplectic structure, {M, ω}, we may consider a pair {M, A} where manifold M is equipped with a structure defined by a field of endomorphisms -(1,1)-variant tensor field on M . Exploring further the analogy, we may study dynamical systems described by vector fields that are defined by their "potentials" -other vector-fields. Thus, instead of Hamilton equations, we have a map This contrast with symplectic geometry, where the potentials of dynamical systems are differential forms, namely differentials of Hamiltonians. It would be natural to require that the set of all such dynamical systems , One may ask why one would want to replace one vector field 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 endomorphism field defined in the previous sections forms such a system. In particular, it is equipped with a bracket for potentials that we define below.
Consider the underlying linear space L of a Lie algebra {L, [[ , ]]} as a manifold. Any smooth vector field B can be viewed as a generator (or "potential") of a dynamical system defined by vector field X B defined The integral curves of X B satisfy the Lax equations, which in a somewhat imprecise way are expressedẋ where the x on the left side is understood as a point in L, while the x inside the bracket on the right side as a vector in L. More accurately, Definition 7.1. Vector fields on a Lie algebra L of form (23) will be called Lax vector fields generated by B, or Lax dynamical systems. In the diagrammatic representation, the Lax vector field is: The space of Lax vector fields will be denoted by X A L = A(X L) ⊂ X L.
A simple and a well-known fact is the existence of Casimir invariants: where first we used Jacobi identity (11) and then skewsymmetry of the resulting ω. The right side vanishes as ω has two identical entries, x. The argument for the other Casimir invariants is similar.
The geometric meaning of the fundamental Nijenhuis property of the endomorphism field becomes clear in the current context. Namely, it implies that the space of Lax vector fields X A L is closed under the commutator of vector fields [X A L, X A L] ⊂ X A L. A new bracket of vector fields is implied.
where in the last part we see that the endomorphism field A may be "factored out" thanks to Theorem 2.1. Thus the commutator is of the form (23) the formulas in the theorem follow.

Proposition 7.4. The bracket (25) can be calculated by the following formula
Notice that although the two right-most terms are defined in coordinates, their difference has a coordinate-free meaning, as it can be defined by The bracket { , } turns the space of vector fields on L into a Lie algebra and can be viewed as a "differential deformation" of the Lie algebra bracket [[ , ]]. Due to its involved nature, it may be a rather surprising that defines 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 vector fields. Proof. If X, Y ∈ X L are two vector fields, then we denote X ⊲ Y = X i (∂ i Y j )∂ j a vector field calculated in linear coordinate system. Thus, formula (28)    Proof. Use the Jacobi identity for vector fields Basic examples. What can be used as a Lax potential? The simplest are constant vector fields, in which case the homomorphism reduces to Proposition 4.1 (see Remark 7.5). Also, a Lax vector field may be "reused" as a potential for a new Lax vector field. The following formulas for bracket {·, ·} may be useful for such dynamical systems where v and w are understood as constant vector fields (the tilde is suppressed for simplicity). Another class consists of Lax vector fields generated from linear vector fields on L. Euler's equations of the motion a rigid body belongs to this category. Here is their -somewhat naïve -generalization to arbitrary Lie algebra: Let R ∈ End L be a matrix describing the tensor of inertia. If vector field J R is used as a "potential", the resulting Lax vector field X = A(J R) describes the dynamical system of "rotating body". In the case of the Lie algebra of 3-dimensional orthogonal group L = so(3, R), with the standard coordinates (x, y, z), and for a diagonal matrix R = diag(a, b, c), we get the standard Euler's equations X = (b − a)xy∂ z + (c − b)yz∂ x + (a − c)zx∂ y (orẋ = (c − b)zy, etc.). A more accurate description will be given elsewhere.

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 different calculus on both spaces treated as manifolds.
Lie algebra L Lie coalgebra L * as manifold as a manifold Coordinates  Table 1: Legend: ∂ i ≡ ∂/∂x i , ∂ i ≡ ∂/∂x i . Tilde ∼ denotes extension of tensors to tensor fields on L and on L * , defined by the affine structure on linear spaces.
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:

Remark on Lagrange equations
While the cotangent bundle T * Q over a manifold Q possesses a canonical differential biform ω ∈ Λ 2 Q defining symplectic structure, the tangent bundle T Q possesses a canonical (1,1)-variant tensor field S ∈ T (1,1) T Q defining an endomorphism field (endomorphisms of T (T Q) and T * (T Q)). In the natural coordinates (x i , v i ) on T Q, this tensor can be expressed as S = ∂ ∂v i ⊗ dx i (sum over i). Its basic property is Ker S = Im S (implying nilpotence S • S = 0). If L is a function on T Q (a Lagrangian), then one defines a biform ω = d • S • d L, which for a "regular" Lagrangian is nondegenerate and therefore forms a symplectic structure. It is easy to see that Lagrange equations may be written as The existence of S and its role in Lagrangian mechanics was noticed rather late [11]; it replaces a rather awkward notion of "vertical derivative" used before in an attempt 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 S ∈ T 1 1 M that satisfies where the second condition (ii) is a generalization of the Schouten-Nijenhuis bracket to "vector-valued differential forms" (see e.g. [17] and [18]), which assures (local) integrability of the distribution Ker S. As a result, one obtains all of the structure of the tangent bundle (Ker S gives the fibering) 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 (30) are relaxed. 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 co-algebras.