Connections on Smooth Lie Algebra Bundles

Home , Adjoint bundle, Ehresmann connection

Indian J. Pure Appl. Math., 50(4): 891-901, December 2019 °c Indian National Science Academy DOI: 10.1007/s13226-019-0362-3

CONNECTIONS ON SMOOTH LIE ALGEBRA BUNDLES

K. Ajaykumar and B. S. Kiranagi

Department of Studies in Mathematics, University of Mysore, Mysore 570 006, India e-mails: [email protected]; [email protected]

(Received 25 April 2018; accepted 4 September 2018)

We deﬁne the notion of Lie Ehresmann connection on Lie algebra bundles and show that a Lie connection on a Lie algebra bundle induces a Lie Ehresmann connection. The converse is proved for normed Lie algebra bundles. We then show that the connection on adjoint bundle corresponding to the connection on principal G−bundle to which it is associated is a Lie Ehresmann connection. Further it is shown that the Lie Ehresmann connection on adjoint bundle induced by a universal G-connection is universal over the family of adjoint bundles associated to G-bundles.

Key words : Adjoint bundle; Lie algebra bundle; Lie connection; Lie Ehresmann connection; parallel transport, universal connection.

1. INTRODUCTION

Transporting a certain motion in the base manifold of a Lie algebra bundle to the motion in its fibres, needs an additional structure. This is provided by the notion of connection on Lie algebra bundle. Mackenzie [9], defines Lie connection on a Lie algebra bundle in terms of covariant derivative by defining Lie connection on Lie algebroids and proves the existence. A proof of the existence of Lie connection on Lie algebra bundles is also in Hasan Gu¨ndog˘an’s diploma thesis [5].

We define the notion of connection on smooth Lie algebra bundles in a more geometric way in terms of horizontal fields, which we call Lie Ehresmann connection. We prove that our notion is equivalent to the notion of Lie connection defined in [9] for normed Lie algebra bundles.

Further we deﬁne connection on adjoint bundle adP associated to a smooth principal bundle P in terms of a connection on P and show that it is Lie Ehresmann connection in a natural way. 892 K. AJAYKUMAR AND B. S. KIRANAGI

Parallel transport on adjoint bundles is deﬁned in terms of parallel transport on principal bundles. We then show that the connection recovered by such a parallel transport is a Lie Ehresmann connection.

A universal G−connection has been constructed by Narasimhan and Ramanan [10, 11] and Schlaﬂy [12]. Later in 2012, Biswas et al. [1] have given a simple and transparent construction of universal principal G−connection. Here we study the Lie Ehresmann connection on adjoint bundle induced by universal G−connection and show that such an induced Lie Ehresmann connection is G−universal over the family of adjoint bundles.

Notations and Terminologies

Unless speciﬁed otherwise, all bundles are smooth bundles over smooth manifolds that are haus- dorff, second countable and paracompact. All principal bundles have a structure group that is a ﬁnite dimensional Lie group. All Lie algebras are real.

2. PRELIMINARY DEFINITIONS

Deﬁnition 2.1 — A real normed vector space (L, | ∗ |) is said to be a normed Lie algebra [14] w.r.t a Lie product [, ] on L if it has the property

|[x, y]| ≤ 2|x||y|

Deﬁnition 2.2 — A smooth weak (normed) Lie algebra bundle (ξ, π, M) is a smooth vector bundle together with a smooth morphism Θ: ξ ⊕ ξ → ξ inducing a (normed) Lie algebra structure on each

ﬁbre ξx [2].

Definition 2.3 — A locally trivial smooth (normed) Lie algebra bundle [4, 7], for short a smooth Lie algebra bundle, is a smooth vector bundle ξ = (ξ, π, M) whose standard fibre is a (normed) Lie algebra say L, in which each fibre is a (normed) Lie algebra such that for each x in M there is an open set U in M containing x and a diffeomorphism ϕ : U × L → π−1(U) such that for each x in −1 U, ϕx : {x} × L → π (x) is a (normed) Lie algebra isomorphism.

Remark 2.1 : Every locally trivial smooth Lie algebra bundle is a weak Lie algebra bundle. But the converse is not true in general [8].

Deﬁnition 2.4 — A connection on a principal bundle π : P → M is a collection H = {Hp | p ∈ P } of subspaces Hp ⊂ TpP called horizontal subspaces, such that

1. the assignment p 7→ Hp depends smoothly on p ∈ P CONNECTIONS ON SMOOTH LIE ALGEBRA BUNDLES 893

2. TpP = Vp ⊕ Hp, where Vp = ker((π∗)p : TpP → Tπ(p)M) called vertical subspace

3. HRg(p) = ((Rg)∗)pHp for all g ∈ G and p ∈ P , where (Rg)∗ is the differential of Rg, the right action on P

Deﬁnition 2.5 — For a vector bundle π : E → M, let a : E → E be the diffeomorphism deﬁned by a(e) = a · e for every non-zero a ∈ R.A linear Ehresmann connection on a vector bundle

π : E → M is a collection H = {He | e ∈ E} of subspaces He ⊂ TeE called horizontal subspaces, such that

1. the assignment e 7→ He depends smoothly on e ∈ E

2. TeE = Ve ⊕ He, where Ve = ker((π∗)e : TeE → Tπ(e)M) called vertical subspace

3. Ha·e = ((a)∗)eHe, for all e ∈ E and non-zero a ∈ R, where (a)∗ is the differential of a

3. CONNECTIONON LIE ALGEBRA BUNDLES

Deﬁnition 3.1 — (Lie Ehresmann connection). Let ξ = (ξ, π, M, Θ) be a smooth Lie algebra bundle.

A Lie Ehresmann connection on ξ is a collection H = {Hl | l ∈ ξ}, such that

1. H is a linear Ehresmann connection on ξ

2. Θ∗(H £ H) = H, where H = ∪l∈ξHl and H £ H is the corresponding horizontal subbundle on ξ ⊕ ξ.

That is, Θ∗(Hl1 × Hl2 ) = (Θ∗)(l1,l2)(H(l1,l2)) = HΘ(l1,l2) = H[l1,l2], where Θ∗ is the differential of the smooth morphism Θ: ξ ⊕ ξ → ξ.

In otherwords, a linear Ehresmann connection on ξ for which the differential of the morphism Θ carries horizontal tangent vectors of ξ ⊕ξ to the corresponding horizontal tangent vector of ξ is called a Lie Ehresmann connection.

Deﬁnition 3.2 — (Lie Connection). Let ξ = (ξ, π, M, Θ) be a smooth Lie algebra bundle. A Lie connection [9] is a linear map

∇ : Γ(TM) → End(Γ(ξ)),X 7→ ∇X satisfying the following conditions:

∞ 1. ∇X (aA) = (X.a)A + a∇X A for all X ∈ Γ(TM), a ∈ C (M, R),A ∈ Γ(ξ) 894 K. AJAYKUMAR AND B. S. KIRANAGI

∞ 2. ∇(aX+bY )A = a∇X A + b∇Y A for all X,Y ∈ Γ(TM), a, b ∈ C (M, R),A ∈ Γ(ξ)

3. ∇X [A, B] = [∇X A, B] + [A, ∇X B] for all X ∈ Γ(TM), A, B ∈ Γ(ξ)

In short a Lie connection on a Lie algebra bundle ξ is a covariant derivative ∇X on vector bundle ξ such that it is a derivation on Γ(ξ), the Lie algebra of smooth sections of ξ.

We now establish the inter-relation between Lie connection and Lie Ehresmann connection on Lie algebra bundles.

Theorem 3.1 — Let ξ = (ξ, π, M, Θ) be a smooth Lie algebra bundle. Then a Lie connection ∇ on ξ induces a Lie Ehresmann connection on ξ and the converse holds if ξ is a normed Lie algebra bundle.

PROOF : Let ∇ be a Lie connection on ξ. Deﬁne, Hl = {A∗(Xp): p = π(l) and ∇Xp (A) = 0} for each l ∈ ξ. That is Hl = A∗(TpM) where π(l) = p and A ∈ Γ(ξ) is such that ∇A(p) = 0. Then

Hl does not depend on the choice of A ∈ Γ(ξ) and H = {Hl : l ∈ ξ} forms a linear Ehresmann connection on the vector bundle ξ [13, Addendum 3]. Now to show H is a Lie Ehresmann connection, it sufﬁces to show Θ∗H(l1,l2) = HΘ(l1,l2) for any (l1, l2) ∈ ξp × ξp ⊂ ξ ⊕ ξ. The horizontal subspace 0 H(l1,l2) = S∗(TpM) where S = (A1,A2) ∈ Γ(ξ ⊕ ξ) such that ∇ S(p) = (∇A1(p), ∇A2(p)) = 0. Consider

Θ∗(H(l1,l2)) = Θ∗(S∗(TpM))

= (Θ ◦ S)∗(TpM)

= [A1,A2]∗(TpM)

= H[l1,l2] = HΘ(l1,l2) (3.1)

since, ∇[A1,A2](p) = [∇A1(p),A2(p)] + [A1(p), ∇A2(p)] = 0 and π([l1, l2]) = p. Thus, H =

{Hl : l ∈ ξ} is a Lie Ehresmann connection on ξ.

Now suppose ξ is a normed Lie algebra bundle. We show that every Lie Ehresmann connection H on ξ is induced by some Lie connection ∇ on ξ. Suppose H is a Lie Ehresmann connection on ξ. Then clearly H is a vector bundle connection on ξ. Let hX and vX be the horizontal and vertical component of any tangent vector X ∈ ξl induced by the direct sum decomposition ξl = Vl ⊕ Hl.

Then ∇Xp A = vA∗(Xp), for Xp ∈ TpM, p ∈ M and A ∈ Γ(ξ) deﬁnes a connection ∇ on the vector bundle ξ [13, Addendum 3]. Then ∇ is a Lie connection on ξ since ∇X [A, B] = [∇X A, B] + CONNECTIONS ON SMOOTH LIE ALGEBRA BUNDLES 895

[A, ∇X B] for any A, B ∈ Γ(ξ) and X ∈ Γ(TM). For,

∇Xp [A, B] = v[A, B]∗(Xp)

= v([A∗(Xp),B] + [A, B∗(Xp)]) (as [ , ] is bilinear and ξ is normed)

= v[A∗(Xp),B] + v[A, B∗(Xp)]

= [vA∗(Xp),B] + [A, vB∗(Xp)] (by 2 in the Deﬁnition 3.1)

= [∇Xp A, B] + [A, ∇Xp B]2 (3.2)

4. CONNECTIONON ADJOINT BUNDLE

We deﬁne connection on adjoint bundle adP associated to a principal bundle P in terms of connection on P .

Let π : P → M be a principal G−bundle and πAd : adP → M be the adjoint bundle [6, 8] corresponding to P . Explicitly, the elements of the adjoint bundle are equivalence classes of pairs {(p, x)}, p ∈ P and x ∈ g, the Lie algebra of G, such that

{(p.g, x)} = {(p, Adg(x))}, for all g ∈ G

For w ∈ adP , choose (p, x) ∈ P × g such that w = {(p, x)}. Then βx : P → adP by

βx(p) = {(p, x)} = w is a smooth map for each x ∈ g.

Deﬁnition 4.1 — (Connection on adjoint bundle). Let π : P → M be a principal G−bundle with a connection, H = {Hp | p ∈ P }. The collection HadP = {Hw | w ∈ adP } is called the connection on adP corresponding to H, where Hw := ((βx)∗)p(Hp), w = {(p, x)}.

Clearly Hw is independent of the choice of (p, x). For all g ∈ G, x ∈ g and p ∈ P , we have

βx(p) = {(p, x)} = {(Rg(p), Ad −1 (x))} = β (Rg(p)) g Adg−1 (x)

So β = βx ◦ R −1 . If we had chosen (Rg(p), Ad −1 (x)) instead of (p, x), then still we Adg−1 (x) g g have

((β )∗) (H ) = ((βx ◦ R −1 )∗) ((Rg)∗)p(Hp) Adg−1 (x) Rg(p) Rg(p) g Rg(p)

−1 = ((βx)∗)p ◦ ((Rg )∗)Rg(p) ◦ ((Rg)∗)p(Hp)

= ((βx)∗)p(Hp).

Theorem 4.1 — Let πAd : adP → M be the adjoint bundle of a principal bundle π : P → M with a connection, H = {Hp | p ∈ P }. Then the connection HadP = {Hw | w ∈ adP } on adP corresponding to H is a Lie Ehresmann connection. 896 K. AJAYKUMAR AND B. S. KIRANAGI

PROOF : We first show that TwadP = Vw ⊕Hw, where Vw = ker((πAd)∗)w. Since πAd ◦βx = π, it is clear that (π∗)p = ((πAd)∗)w ◦ ((βx)∗)p. Then by definition of Hw, TwadP = Vw ⊕ Hw. Let a : adP → adP be the diffeomorphism defined by a(w) = a · w = {(p, a · x)} for a non-zero a ∈ R. Then

Ha(w) = Ha·w

= (dβa·x)p(Hp)

= ((a ◦ βx)∗)p(Hp)

= ((a∗)w ◦ ((βx)∗)p)(Hp) = (a∗)w(Hw)

Thus HadP is a linear Ehresmann connection on adP .

Let Θ: adP ⊕ adP → adP be the smooth morphism inducing the Lie bracket, Θm : adPm × adPm → adPm, Θm({(p, x1)}, {(p, x2)}) = {(p, [x1, x2])} on each ﬁbre. Let (w1, w2) ∈ adPm × adPm and π(p) = m. Then by deﬁnition of Hw it follows that

Θ∗(H(w1,w2)) = Θ∗(Hw1 × Hw2 )

= Θ∗(((βx1 )∗)pHp × ((βx2 )∗)pHp)

= Θ∗(((βx1 )∗, (βx2 )∗)p(Hp))

= (Θm ◦ (βx1 , βx1 ))∗(Hp)

= [βx1 , βx2 ]∗(Hp) = H [βx1 (p),βx2 (p)]

= H[w1,w2]

= HΘ(w1,w2) (4.1)

Hence, HadP is a Lie Ehresmann connection on adP . 2

Every principal bundle admits a connection [1, Lemma 2.2.]. Thus we have

Corollary 4.1 — There exists a Lie Ehresmann connection on the adjoint bundle adP of any principal bundle π : P → M.

Now, suppose H = {Hw : w ∈ adP } is a Lie Ehresmann connection on the adjoint bundle, adP of the principal bundle π : P → M. Now deﬁne for each p ∈ P ,

Hp = {Xp ∈ TpP :(βx∗ )p(Xp) ∈ H{[p,x]} = Hw, ∀ x ∈ g}

0 We claim that H = {Hp : p ∈ P } is a connection on the principal bundle P . CONNECTIONS ON SMOOTH LIE ALGEBRA BUNDLES 897

1. Suppose Vp = Ker(π∗). Then, TpP = Vp + Hp as πAd({[p, x]}) = π(p) and TwadP =

Vw ⊕ Hw where Vw = Ker(πAd∗ )w. Then Vp ∩ Hp = {0} for 0 6= Xp ∈ Vp we have

(π∗)p(Xp) = 0

⇒ (πAd ◦ βx)∗p (Xp) = 0

⇒ (πAd∗ )w(βx∗p (Xp)) = 0

⇒ (βx∗ )p(Xp) ∈ Vw

⇒ (βx∗ )p(Xp) ∈/ Hw

⇒ Xp ∈/ Hp

Hence TpP = Vp ⊕ Hp for all p ∈ P .

2. To show that (Rg∗ )p(Hp) = HRg(p). Suppose Xp ∈ Hp. Then

(β )((Rg )p(Xp)) = (β ◦ Rg)∗ (Xp) Adg−1 (x)∗ ∗ Adg−1 (x) p

= (βx∗ )p(Xp) ∈ H{[p,x]} = Hw

Thus, (Rg∗ )p(Xp) ∈ HRg(p) and therefore

(Rg∗ )p(Hp) ⊆ HRg(p) (4.2)

Now suppose XRg(p) ∈ HRg(p). Then there exists Xp ∈ TpP such that (Rg∗ )p(Xp) = XRg(p).

Now we claim that Xp ∈ Hp:

Since (Rg )p(Xp) = X we have (R −1 ) (X ) = Xp. Thus, ∗ Rg(p) g∗ Rg(p) Rg(p)

(βx )p(Xp) = (βx )p((R −1 ) (X )) ∗ ∗ g∗ Rg(p) Rg(p)

= (β ◦ R −1 ) (X ) x g ∗Rg(p) Rg(p)

= (βAd −1 )Rg(p)(XRg(p)) ∈ H{[Rg(p),Ad −1 ]} = Hw g∗ (x) g

Therefore Xp ∈ Hp and thus

HRg(p) ⊆ (Rg∗ )pHp (4.3)

So from (4.2) and (4.3), HRg(p) = (Rg∗ )pHp.

0 Hence H = {Hp : p ∈ P } is a connection on the principal bundle P . Thus we have

Theorem 4.2 — Let ξ = adP be the adjoint bundle of a principal bundle π : P → M. Then every Lie Ehresmann connection on ξ is induced by a principal bundle connection on P . 898 K. AJAYKUMAR AND B. S. KIRANAGI

4.1 Parallel transport in adjoint bundles

Deﬁnition 4.2 — Let π : P → M be a smooth principal G-bundle with a connection, H = {Hp | p ∈ P }. Let γ : [0, 1] → M be a curve in the base manifold. A curve γP : [0, 1] → P is called the horizontal lift of γ if:

1. π(γP ) = γ

2. All tangent vectors XP to γP are horizontal i.e XP ∈ HγP

Deﬁnition 4.3 — Let π : P → M be a principal G-bundle and adP be the adjoint bundle corresponding to P . Suppose γP (t) is a horizontal lift of γ(t) ∈ M in P , then a smooth curve

γadP (t) is said to be the horizontal lift of γ(t) in adP if

γadP (t) = {(γP (t), l)} for a ﬁxed element l of g, the Lie algebra of G.

0 This is independent of the lift chosen in P . For another horizontal lift in P say γP we have 0 γP (t) = γP (t)a, for a constant element a ∈ G. So we have,

0 γadP (t) = {(γP (t), l)} = {(γP (t), Ada−1 (l))} where Ada−1 (l) is a ﬁxed element in g. So this would still be a horizontal lift of γ(t) in adP , but through different element of g.

Choosing a trivialization for γP , γP (t) = φP (γ(t), g(t)) we have the corresponding trivialization for γadP ,

γadP (t) = φadP (γ(t), Adg(t)(l))

Deﬁnition 4.4 — (Parallel transport on principal bundle). Let π : P → M be a principal G−bundle. A parallel transport map P [3] associates smoothly to each path γ in M a

G− map, P(γ): Pi(γ) → Pe(γ) such that:

1. P(γm) = 1m, where γm is a constant map at m ∈ M

2. P(γ ◦ α) = P(γ), where α is an orientation-preserving diffeomorphism of intervals

3. P(γ1 ? γ2) = P(γ1) ◦ P(γ2) where γ1 ? γ2 is the juxtaposition of the paths γ2 and γ1 CONNECTIONS ON SMOOTH LIE ALGEBRA BUNDLES 899

A connection H on principal bundle π : P → M yields unique horizontal lift of every path γ in M. This horizontal lift deﬁnes a unique parallel transport on P [3, Theorem 1].

Deﬁnition 4.5 — (Parallel transport on adP). Let π : P → M be a principal G-bundle and adP be its adjoint bundle. Let P be a parallel transport map on P . We deﬁne the corresponding parallel transport Pad on adP by

Pad(γ):(adP )i(γ) → (adP )e(γ)

Pad(γ)(w) = {(P(γ)(p), x)} for each path γ in M where w = {(p, x)}.

Remark 4.1 : For each path γ in M, Pad(γ):(adP )i(γ) → (adP )e(γ) is a Lie algebra isomorphism.

4.2 Recovering connection on adP from parallel transport Pad

Let S(M, 0, m) be the set of all smooth paths in M starting at m. That is,

S(M, 0, m) := {γ : [0, 1] → M | γ is smooth and γ(0) = m} and for p ∈ P with π(p) = m, S(P, 0, p) be deﬁned similarly.

Then we know that the parallel transport P on P deﬁnes a smooth map Φp : S(M, 0, m) → S(P, 0, p), α 7→ Φp(α) where

p Φ (α): I → P, t 7→ P(α|t)(p)

d Then for p ∈ P , Hp := {v˜ :v ˜ is the lift to p of v, v ∈ TmM} where v˜ = d² |²=0Φ(α)(²) = ˙ ^p Φ (α)(0), α ∈ S(M, 0, m) forms a horizontal subspace of TpP . Further H := {Hp : p ∈ P } is a connection on P [3, Page 4].

w w Now deﬁne Φad : S(M, 0, m) → S(adP, 0, w), α 7→ Φ (α) where

w Φad(α): I → adP, t 7→ Pad(α|t)(w)

But observe that for w = {(p, l)},

Pad(w) = Pad({(p, l)}) = {(P(γ)(p), l)}

= βl(P(γ)(p))

= (βl ◦ P(γ))(p) 900 K. AJAYKUMAR AND B. S. KIRANAGI

Thus w p Φad(α) = βl ◦ Φ (α) ˙ g˙w ^p Now, v˜ad = Φad(0) = βl ◦ Φ (α)(0) = (βl)∗(˜v). Therefore Hw = (βl)∗(Hp), recovering the connection on adP induced by the connection on P . Thus we have

Theorem 4.3 — The Lie Ehresmann connection induced on the adjoint bundle by a principal bundle connection is same as the connection recovered from the parallel transport corresponding to it.

4.3 Universal G-connection on adjoint bundles

We now recall the deﬁnition of universal G-connection from [1] keeping up their notations.

Deﬁnition 4.6 — (Universal principal G-connection). Let G be a ﬁnite dimensional Lie group. A universal G-connection [1] is a pair (EG → BG, D), where EG → BG is a smooth principal G-bundle and D is a smooth connection on EG, such that for any smooth principal G-bundle FG → X, where

X is a ﬁnite dimensional manifold, and for any connection D on FG, there is a smooth map

ϕ : X → BG

∗ ∗ such that the pulled back principal G−bundle with connection (ϕ EG, ϕ D) is isomorphic to (FG,D).

∞ Here, D is a C splitting of the Atiyah exact sequence for EG [1, Page 369]. Such a splitting D determines a unique horizontal bundle H(D) and vice-versa [1, Page 370]. So in the succeeding deﬁnition and the result, we use H in place of D.

Deﬁnition 4.7 — (Universal G-connection on adjoint bundles). A Lie connection H0 on the adjoint bundle adP of a principal G−bundle π : P → M is said to be a universal G-connection if for the adjoint bundle adP 0 of any principal G-bundle π0 : P 0 → N and a Lie Ehresmann connection H on adP 0, there exists a smooth map f : N → M such that f ∗(adP ) is isomorphic to adP 0 as Lie ∗ algebra bundles and further an isomorphism between them may be so chosen that f (H0) is carried to H.

Theorem 4.4 — There exists an adjoint bundle admitting a universal connection.

0 PROOF : A universal G−connection, say (P → M, H0) exists [1, Theorem 4.2]. Let H0 be the 0 Lie Ehresmann connection induced by H0 on the adjoint bundle adP and let P → N be any principal bundle. Suppose H0 is any Lie Ehresmann connection on the adjoint bundle adP 0. Then by Theorem 4.2 there exists a principal bundle connection H on P 0 that induces H0 on adP 0. There exists a smooth ∗ 0 ∗ map f : N → M such that there is an isomorphism from f (P ) to P that carries f (H0) to H, since CONNECTIONS ON SMOOTH LIE ALGEBRA BUNDLES 901

H0 is a universal G-connection. This isomorphism induces a Lie algebra bundle isomorphism from ∗ 0 ∗ 0 0 ∗ ∗ ad(f (P )) to adP that carries f (H0) to H . But f (adP ) is isomorphic to ad(f (P )). Therefore ∗ 0 ∗ 0 0 0 we have an isomorphism from f (adP ) to adP that carries f (H0) to H . Hence H0 is a universal G-connection on adP over the family of adjoint bundles associated to G-bundles. 2

ACKNOWLEDGEMENT

The second author is thankful to the SERB/DST, New Delhi, India for the ﬁnancial assistance SR/S4/MS:856/13. We thank R. Rangarajan for his support.

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