Canonical maps between semidirect products with

Comrnun.Math.Phys.90, 235-250 (1983)

Communications in

Mathematical

Physk

© Springer-Verlag 1983

Canonical Maps Between Semidirect Products with Applications to Elasticity and Superfluids Boris A. Kupershmidt ~ and Tudor Ratiu 2 Department of Mathematics,Universityof Michigan,Ann Arbor, MI 48109, USA

Abstract. It is shown that two canonical maps arising in the Poisson bracket formulations of elasticity and superfluids are particular instances of general canonical maps between duals of semidirect product Lie algebras.

1. Introduction

In the last years many models in classical physics have been shown to possess Poisson structures. In many examples this Poisson structure is the canonical one on the dual of a Lie algebra, sometime s supplemented by a two-cocycle. It turns out that in almost all such cases, the Lie algebra in question is a semidirect product. Quite often the same physical system allows descriptions in different sets of variables, thus obtaining two Poisson structures for the same model. These structures are not equivalent but connected. Such double descriptions commonly occur, e.g., in systems coupled to the magnetic field by introducing magnetic potentials; magnetohydrodynamics is such a system. The relation between the two descriptions, when linear, is produced by a Lie algebra homomorphism (like in the above-mentioned magnetic case of magnetohydrodynamics, see [5, 8]). In this case the dual map of the Lie algebra homomorphism is naturally canonical between the two Poisson structures. However, it was recently I-5,6] observed experimentally that in two physical models, elastodynamics and superhydrodynamics, the transformation between the two Poisson structures, even though non-linear, is still canonical. We hasten to emphasize that this is not the standard case in the theory of finite dimensional Lie algebras: given two general Lie algebras S5and 15, there are no natural non-linear canonical maps from ~* to ~*. The problem of interpretation of the above-mentioned non-linear canonical maps is the topic of this paper. Our explanation turns out to be very natural and simple (see, e.g., Theorem 3.5 below): canonical maps with range the dual of a Lie algebra are realized as momentum maps. This underlying philosophy is closely related to [8]. 1 Partially supported by NSF Grant MCS-8003104 2 Partially supported by NSF Grant MCS-8106142

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Let us describe briefly our set-up. In the final analysis, a simplified version of the problem boils down to the following question. Let 15, -~1, -~z be Lie algebras and ~b:15~Der(.~ 0 O:15~Der(.~2) be Lie algebra homomorphisms, where Der denotes the Lie algebra of derivations. Let f :.~* ~ . ~ be a map. Under what conditions is the induced map id x f:(15 x ~1)* ~(15 x ~z)* between the duals of semidirect products canonical? Our answer, given in different versions needed for the treatment of the examples, is that f :.~* ~z must be a canonical map of Poisson manifolds compatible with the actions of 15 (see Theorem 3.3). The plan of the paper is the following. Section 2 recalls the definitions and formulas relevant for semidirect products of Lie groups and algebras. Section 3 describes the general set-up for getting non-linear canonical maps between semidirect products and gives the general theorems to be used in the next section. Section 4 starts with four theoretical examples which together with the main theorems of the previous section enable us to show that the non-linear canonical maps coming up in elastodynamics and superhydrodynamics are particular instances of our general theorems. Throughout the paper we employ the following conventions and notations. For a manifold P, ~(P), W(P) denote the ring of functions and the Lie algebra of vector fields respectively. A Poisson bracket on P is a multiplication {,} on ~ ( P ) making (~(P), {, }) into a Lie algebra and such that the map f ~ X i e W ( P ), XI(g): = {f,g}, is a Lie algebra homomorphism of ~ ( P ) into 5e(P), i.e. X~s,g ~= [X s, Xg]. A manifold P endowed with a Poisson bracket is called a Poisson manifold. a map a:(P~, {, }a)~(P2, {, }2) between Poisson manifolds is called canonical, if ct*{/, g}2 = {~*f, ~*g}l tbr any f , g ~ ( P 2 ) , where the upper star denotes the pull-back operation. A Lie group action on a manifold P is a group homomorphism 4):G ~ Diff(P), where Diff(P) denotes the group of diffeomorphisms of P, such that the map (g, p)~--~b0(p) is smooth. If P is a Poisson manifold, ~b is called canonical if all the diffeomorphisms ~bo, geG, are canonical maps of P. A Lie algebra action on a manifold P is a Lie algebra homomorphism qS:15o~r(P) such that the map (4, P)~--~q~(~)(P) is smooth. If ~i happens to be the (left) Lie algebra of a Lie group G acting on P, then q~= - 4~', where the upper prime denotes the Lie algebra homomorphism induced by ~.1 IfP is a Poisson manifold, the Lie algebra action q5is said to be canonical if for any ~E~i and f l , f 2 ~ ( P ) , 6({){fl, f2} = {~b({)L, fz} + {fl, q~({)f2}. If the Lie group G with Lie algebra @ acts canonically on the Poisson manifold P, a momentum mapping J :P-+ 15" is a map satisfying

¢(~) =

x~

for all ~05, where 3(~)e~(g) is defined by J(~)(p) = (J(p), ~), where ( , ) denotes the pairing between t5" and 15. J is said to be equivariant, if 1 The reasonfor the minusin front of¢' is due to the factthat Y'(P) is the right Lie algebraof Diff(P). (See [1], ex. 4.1G, page 274.)

Canonical Maps BetweenSemidirectProducts

237

J o q~ = Ad* , oJ for all g e G; here Adg :(5 ---}(5 denotes the adjoint action of G on (5 and Ad* :(5* ~ (5* is its dual map. If we deal with a canonical Lie algebra action gb of (5 on P, the definition of the momentum mapping is unchanged, but equivariance is replaced by

TpJ(~(~)(p)) = (ad ~)*(J(p)) for all ~e(5, peP; here TpJ: TpP-~ (5* denotes the tangent map (differential) of J at peP. (The formula has no minus signs since ~b= - rb'.) Lie group (algebra) actions on a Poisson manifold admitting equivariant momentum maps are called Hamiltonian actions. The dual (5* of a Lie algebra (5 is a Poisson manifold with respect to the LiePoisson bracket given by

for # e (5" and f , g functions on (5"; here ( , ) denotes the pairing between (5 and (5". The "functional derivative" 6 f / 6 # ~ (5 is the derivative D f(#) regarded as an element of (5 rather than (5"*, i.e.

Df(#)'v=

v,

for #, re(5*. For infinite dimensional (5, the pairing is with respect to a weakly nondegenerate form and the existence of 6 f / 6 # is a bona fide hypothesis on f . The same formula defines the Lie-Poisson bracket on polynomial functions on the dual of a Lie algebra over any ring. The Hamiltonian vector field defined by the function h on (5* is given by / ~h '~* Xa(#) = ad~ ~ ) ( # ) , where ad (0'~/= [~, r/] is the adjoint action of (5 on (5 and (ad (0)* (5* --* (5* its dual map. The following standard fact of great use in the paper can be found in the symplectic context in e.g. [-1, 3, 9].

Proposition 1.1. Let P be a Poisson manifold and ¢p:(5 --*3F(P) a canonical Lie algebra action. The following are equivalent: (i) the action is Hamiltonian; (ii) there exists a Lie algebra homomorphism t~ :(5--* ~ ( P ) such that q~(() = X,I,(¢) for all ( e(5 ; (iii) there exists a rin 9 and Lie algebra homomorphism Z : Y ((5 *) ~ ~ (P) such that X(xoj)(o=rp(~) for all ~e(5, where j:(5--*~((5") is the Lie algebra homomorphism given by J ( O ~ ) = ( # , ( ) In fact, if J is the momentum map of the action (p, then q = 3 and ~( = J*. A second standard fact to be used later on is the following.

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Proposition 1.2. Let (5,9 be Lie algebras and a:(5-~ ~ a linear map. The dual map a* :.~*~ (5* is canonical f and only if a is a Lie algebra homomorphism. For a study of the local structure of Poisson manifolds the reader is referred to

[10]. 2. Semidirect Products of Lie Groups and Lie Algebras Let G, H be Lie groups with Lie algebras (5 and .~ respectively. Denote by Aut (H) the Lie group ofautomorphisms of H and by Der (.~) the Lie algebra of derivations of

5. Let ~ : G - , A u t ( H ) be a Lie group homomorphism. The semidirect product G x H of G with H by • is a Lie group with underlying manifold G x H, composition law (gl, h0(g2,/12) = (g,92, hi ~(gl)(h2)),

(2.1)

identity element (e, e), and inverse (g, h)" 1 = (g- 1, ~(g- 1)(h - 1)). The homomorphism ~ induces a Lie algebra homomorphism ~b:(5 -~ Der(~) in the following way. For every gEG, cb(g):H-oH is an automorphism which thus induces a Lie algebra automorphism ~(g):=cb(g)':.~.~. In this way ~ : G ~ Aut(.~) becomes a Lie group homomorphism whose induced Lie algebra homomorphism q~:= $ ' :(5 ~ Der(.~) allows one to define the semidirect product (5 x 5 as the Lie algebra with underlying vector space 15 x .~ and bracket 0

[(~1, rh), (~2, q2)] = ([~1, ~2],¢(~0r/z - ~b(~2)r/1 + [ql,q2])-

(2.2)

where ~1, ~2 e(5, q~, q2 ~-~. It is well known that the Lie algebra of G x H is (5 x~b (see [t4]). Let ~(Aut(H)) denote the Lie algebra of Aut(H). To identify elements of A°(Aut (H)), let c:( - e, e) ~ Aut(H) be a smooth curve with c(0) the identity map of H. Then for any heH, d/dtlt=oC(t)(h)~ThH, i.e. c'(0) defines a vector field on H by h~--~c'(O)(h). Thus, if ~Y(H) denotes the vector space of all vector fields on H, A°(Aut(H))~ Y'(H). It can be shown from the fact that c(t)EAut(H), that c'(O)(e) = 0, but this fact will not be used in the sequel. If