Medical, Pharma, Engineering, Science, Technology and Business

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:

: Lie algebra product,

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

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

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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 denes a new Lie structure on the space of vector elds.

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 , which satisfies the Jacobi condition 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.

Customarily, one denes a Lie algebra as a linear space L with a product denoted (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":

(2.1)

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

(2.2)

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

. Similarly, the adjoint action of v ∈ L is dened simply as a (1,1)-tensor. Of course, from the structural point of view both denitions are equivalent, .

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 on the manifold L. In particular, the manifold L is equipped with a constant (1, 2)-type tensor field

(2.3)

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

(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) dened by

(2.5)

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

(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

(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 .

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 . Then, Theorem 2.1 states that every Lie algebra L possesses, as a manifold, a
unique natural tensor field , which at point x ∈ L is dened as an endomorphism
taking a tangent vector v ∈ T_{x}L to

(2.8)

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

(2.9)

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

**Remark 2.2.** The canonical endomorphism field A is dened 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 (in our language). Instead of the Nijenhuis bracket, we have the Schouten bracket. Thus, Ω denes a Poisson structure, which, moreover, restricts to the coadjoint orbits, on which its inverse denes a symplectic structure, =0. Section 8 summarizes these parallels.

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, and 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 at any direction. Turning and weaving in space do not have any meaning (unlike in some other conventions). For instance,

this representation is as good as this

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:

**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) =

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

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

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

**Lie algebra in pictures.** An algebra is dened 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 =

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:

(3.1)

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

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

(This one-form vanishes for semisimple algebras.)

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

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

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

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

Denition: Theorem:

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

Every element (vector) v ∈ L denes a "constant" vector field on manifold L
obtained by parallel transport; in coordinates, if v = v^{i}e_{i} then . The canonical
endomorphism field A on manifold L applied to such fields denes 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:

(4.1)

**Proposition 4.1.** The map denes a homomorphism (the
innitesimal representation of L in terms of XL):

(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:

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

(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 dene "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

(4.4)

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

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

First, rewrite (2.8) for X ∈ T_{x}L:

Vector of the vectorfield X_{v} at point x ∈ L can be expressed as follows:

.

Thus,

which was to be proven.

**Example 4.3.** Consider the 2-dimensional solvable algebra dened by . Then,

The adjoint orbits are lines parallel to e_{2}, and the canonical endomorphism-when restricted
to any of them-becomes a dilation.

**Example 4.4.** The Lie algebra of 3-dimensional rotations, so_{3}, is dened by relations . Thus,

The orbits are spheres dened 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 innitesimal representation, the biform (2.6) takes
at any point x a vector-value:

. (4.5)

Thus, A restricted to an orbit is (locally) integrable if 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.

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

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

**Corollary 5.1.** The endomorphismfield on a Lie algebra satisfies

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:

(5.1)

**Proof.** Use Leibniz rule to show that for every

**Proposition 5.3.** The endomorphismfield on a Lie algebra satisfies

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

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

(5.2)

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

etc. Clearly, the second invariant is a quadratic function related to Killing form and will be
denoted k = I_{2} = K(J; J) = k, but the third is obviously not related to the Lie 3-form.

**Corollary 5.4.** Dierentials of the trace functions are among the annihilators of A:

(5.3)

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:

Duality between tangent bundle TQ over a manifold M, which possesses enough structure so
that any ("regular") function L on TQ denes 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 dened 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 dene 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.

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 dened by afield of endomorphisms-(1, 1)-variant
tensorfield on M. Exploring further the analogy, we may study dynamical systems described
by vectorfields that are dened by their "potentials"-other vectorfields. Thus, instead of
Hamilton equations, we have a map:

(7.1)

This contrasts with symplectic geometry, where the potentials of dynamical systems are Differential forms, namely, dierentials of Hamiltonians. It would be natural to require that the set of all such dynamical systems, be closed under the Lie bracket of vectorfields. This way it would form a subalgebra of . The nal demand would be to have a well-dened 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 dened in the previous sections forms such a system. In particular, it is equipped with a bracket for potentials that we dene below.

Consider the underlying linear space L of a Lie algebra as a manifold. Any smooth
vectorfield B can be viewed as a generator (or "potential") of a dynamical system dened
by vectorfield X_{B} dened

(7.2)

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

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,

**Denition 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

(7.3)

The space of Lax vectorfields will be denoted by

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

**Corollary 7.2.** The dynamical system dened by a Lax vectorfield (7.2) leaves Casimir
polynomials I_{k} invariant, X_{B}I_{k} = 0, for any .

Proof (graphical). We show the reasoning for I_{2} = K(J, J) (quadratic polynomials dened
by Killing form):

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 is closed under the commutator of vectorfieldsA new bracket of vectorfields is implied.

**Theorem 7.3.** The space of Lax vectorfield forms a subalgebra of the algebra of smooth
vectorfields, In particular, if X_{B} and X_{C} are two (global) Lax vectorfields,
then their commutator is a Lax vectorfield with potential:

(7.4)

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

(7.5)

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

(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:

(7.7)

where

Notice that although the two right-most terms are dened in coordinates, their dierence has a coordinate-free meaning, as it can be dened by .

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 . Due to its involved nature, it may be rather surprising that it denes 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 and that extend vectors v,w ∈ L, it is Thus the bracket formula (7.5) reduces in this case to the innitesimal
representation .

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

(7.8)

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

where

Now, using the formula we get

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

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 between Lie
algebra of vectorfields on L with bracket dened 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:

hence the claim: .

**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 may be useful for such dynamical systems:

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 is used as a "potential", the resulting Lax vectorfield X = describes the dynamical system of "rotating body". In the case of the Lie algebra of 3-dimensional orthogonal group with the standard coordinates (x, y, z) and for a diagonal matrix R = diag(a, b, c), we get the standard Euler's equations:

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

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 dierent calculus on both spaces treated as manifolds.

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:

While the cotangent bundle T*Q over a manifold Q possesses a canonical Differential biform dening symplectic structure, the tangent bundle TQ possesses a canonical (1,1)-
variant tensorfield dening an endomorphismfield (endomorphisms of T(TQ)
and T*(TQ)). In the natural coordinates (x^{i}, v^{i}) on TQ, this tensor can be expressed as (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 denes a biformwhich
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:

.

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 that satisfies

(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.

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.

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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