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SU(3) as a 2 -plectic manifold
Journal of Geometry and Physics 93 (2015) 33–39
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Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp
SU (3) as a 2-plectic manifold Mohammad Shafiee Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, P.O.Box 518, Iran
article
info
Article history: Received 12 August 2014 Received in revised form 24 February 2015 Accepted 6 March 2015 Available online 18 March 2015 MSC: 53Dxx 53D05 53D20 11F22
abstract The Killing form induces a 2-plectic structure on a compact semisimple Lie group. The associated Lie group of canonical transformations (2-plectomorphisms) is compact. This 2-plectic structure induces a Cartan connection on the Lie group. The curvature and torsion tensor of this connection have been calculated for the special unitary Lie group SU (3). It is SU (3) SU (3) SU (3) shown that the homogeneous spaces SU (2) , S 1 and T 2 also, admit 2-plectic structures, SU (3)
SU (3)
which are induced by closed left invariant 3-forms on SU (3), whereas U (2) and SO(3) do not admit such 2-plectic structures. © 2015 Elsevier B.V. All rights reserved.
Keywords: 2-plectic structure Compact semisimple Lie group Special unitary Lie group
1. Introduction A k-plectic structure on a smooth manifold M is a closed (k + 1)-form ω on M which is nondegenerate, i.e., if ιX ω = 0, X ∈ TM, then X = 0 (see [1]). If ω is a k-plectic structure on M, (M , ω) is called a k-plectic manifold or a multisymplectic manifold of order k + 1. A multisymplectic manifold of order 2 is called a symplectic manifold. Multisymplectic structures arise in the geometric formulation of classical field theory much in the same way that symplectic structures emerge in the Hamiltonian description of classical mechanics [2–4]. In this formulation, a (k + 1)-dimensional field theory is described by a finite dimensional k-plectic manifold (M , ω) as a ‘‘multi-phase space’’ instead of an infinite dimensional phase space. Using the k-plectic form ω, one can define a system of partial differential equations which are the analogue of Hamilton equations in classical mechanics. The solutions of these equations correspond to special submanifolds of the multi-phase space M. Compact semisimple Lie groups are important examples of 2-plectic manifolds. The 2-plectic structure on these Lie groups is induced by the Killing form. It is well known that the space of smooth functions on a symplectic manifold forms a Lie algebra under Poisson bracket induced by the symplectic form. In [5], Baez and Rogers showed that any 2-plectic manifold gives rise to a Lie 2-algebra. They constructed hemistrict and semistrict Lie 2-algebras from any compact semisimple Lie group G. Moreover, they showed that these Lie 2-algebras are isomorphic to the string Lie 2-algebra of G. The latter structure is important in the theory of gerbs on Lie groups [6]. Another important result about these 2-plectic structures is that the Darboux’s theorem is not true [7,8]. Despite the above mentioned results, geometrically, multisymplectic geometry is a field worth investigating in its own right.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.geomphys.2015.03.003 0393-0440/© 2015 Elsevier B.V. All rights reserved.
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M. Shafiee / Journal of Geometry and Physics 93 (2015) 33–39
In this paper, we consider the Lie group SU (3) as a 2-plectic manifold. The paper is organized as follows. In Section 2, some general results on a compact semisimple Lie group G, as a 2-plectic manifold, are presented. In Section 3, we show that this 2-plectic structure induces a Cartan connection on G. In Section 4, some results about the geometry of SU (3) as a 2-plectic manifold have proved. In particular we show that ∗ω is a 4-plectic structure on SU (3), where ω is the 2-plectic structure on SU (3). Also we show that T (X , Y ) = −[X , Y ],
2
[[X , Y ], Z ], 9 where T and R are the torsion and the curvature tensor of the Cartan connection, respectively. In Section 5 we are trying to know whether special subgroups of SU (3) are special submanifolds or not. In the last section, we show that the homogeneous SU (3) SU (3) SU (3) spaces SU (2) , S 1 and T 2 , admit 2-plectic structures, which are induced by closed left invariant 3-forms on SU (3), whereas SU (3) U (2)
and
SU (3) SO(3)
R(X , Y )Z =
do not admit such 2-plectic structures.
2. The 2-plectic structure on a compact semisimple Lie group Let G be a compact semisimple Lie group with Lie algebra g and let ⟨., .⟩ denotes the positive definite adjoint invariant inner product on g defined by
⟨A, B⟩ = −K (A, B), where K is the Killing form on g and A, B ∈ g. Assume that σ is the biinvariant Riemannian metric on G induced by ⟨., .⟩. Define the 3-form ω on G by
ω(X , Y , Z ) = σ (X , [Y , Z ]),
X , Y , Z ∈ TG.
Since σ is nondegenerate and biinvariant then so is ω. Thus ω is closed. Furthermore, since K (A, [B, C ]) = K (B, [C , A]) = K (C , [A, B]), for all A, B, C in g, then ω is totally antisymmetric. So ω is a 2-plectic structure on G [1]. In the following we present some general results about the geometry of this 2-plectic structure. For more details refer to [8]. Choose a basis E = {e1 , . . . , em } for g and let E1 , . . . , Em be left invariant vector fields induced by e1 , . . . , em , and Θ 1 , . . . , Θ m be left invariant 1-forms dual to E1 , . . . , Em , respectively. Denote by Cijk the structure constants of g and by σij the components of σ with respect to E . Then the 2-plectic form ω reads
ω = Σijkl σij Cklj Θ i ∧ Θ k ∧ Θ l . A diffeomorphism ϕ : G → G is called a 2-plectomorphism provided ϕ ∗ ω = ω. We denote by Symp2 (G, ω), the group of all 2-plectomorphisms on G and by Symp20 (G, ω), its identity component. Let I denote the closed interval [0, 1]. A time dependent 1-form H : G × I → T ∗ G is called Hamiltonian, if there exists a time dependent vector field XH on G, the so called Hamiltonian vector field [9], such that
ιXH ω = dH . The flow ϕt , t ∈ I, of the time dependent Hamiltonian vector field XH is called the Hamiltonian flow generated by H. If ϕt is a Hamiltonian flow, then every individual diffeomorphism ϕt is called a Hamiltonian diffeomorphism. Note that every Hamiltonian diffeomorphism preserves ω. Hence the set of all Hamiltonian diffeomorphisms Ham(G, ω) is a subset of Symp20 (G, ω). In fact, as the following theorem shows, they are the same. Theorem 2.1 (See [8]). Ham(G, ω) = Symp20 (G, ω). Proof. Let ϕ ∈ Symp20 (G, ω) and consider a path {ϕt } ⊂ Symp20 (G, ω) with ϕ0 = id and ϕ1 = ϕ . Assume that Xt is the vector field induced by {ϕt }. Then ιXt ω is closed. Since G is simply connected then its second cohomology group H 2 (G, R) is trivial. Thus ιXt ω is exact. Let ιXt ω = dHt , for some 1-form Ht . Now ϕ is the time 1-map of time dependent Hamiltonian 1-form H on G, defined by H (x, t ) = Ht (x). Let Iso(G) denote the isometry group of G with respect to σ . The following theorem shows that Symp2 (G, ω) and Iso(G) are the same. Theorem 2.2 (See [8]). Symp2 (G, ω) = Iso(G). Proof. By definition of the 2-plectic structure ω and its relation with σ it is trivial that any isometry preserves ω. Conversely, let ϕ be a 2-plectomorphism and X , Y be in TG. It is well known that the Lie algebra of compactly supported vector fields on a smooth manifold is prefect [10]. So there are vector fields W and Z on G such that Y = [W , Z ]. Thus
ϕ ∗ σ (X , Y ) = ϕ ∗ σ (X , [W , Z ]) = ϕ ∗ ω(X , W , Z ) = ω(X , W , Z ) = σ (X , Y ). Therefore ϕ is an isometry.
M. Shafiee / Journal of Geometry and Physics 93 (2015) 33–39
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Thus left and right translations are 2-plectomorphisms and hence left and right invariant vector fields are Hamiltonian. A consequence of Theorems 2.1 and 2.2 is the existence of Hamiltonian vector fields which are neither left nor right invariant. To find such vector fields, let g (t ), 0 ≤ t ≤ 1, be any differentiable curve in G with g (0) = id. Define ϕt : G → G by ϕt (x) = g −1 (t )xg (t ). According to Theorems 2.1 and 2.2, ϕt is a Hamiltonian diffeomorphism. The vector field X defined by {ϕt } is Hamiltonian but neither left nor right invariant. Another consequence of these theorems is that, in contrast to symplectic case, the 2-plectic and Hamiltonian groups have no new results about the topology of the Lie group. Furthermore Theorem 2.2 shows that Symp2 (G, ω) is compact since Iso(G) is compact [11,12]. 3. The Cartan connection induced by the 2-plectic structure We recall that a connection ∇ on G is called left invariant if for any two left invariant vector fields X , Y , the vector field ∇X Y is also left invariant. A left invariant connection is called a Cartan connection, if for any A ∈ g, the curve t → etA is a geodesic. It is well known that there is a one-to-one correspondence between the left invariant connections on G and bilinear forms on g with value in g [13]. In this section, using the 2-plectic structure ω, we construct a left invariant connection on a compact semisimple Lie group G. To do this, let ♭ : g∗ → g be the isomorphism induced by σ . The 2-plectic form ω induces a bilinear form α : g × g → g as follows
α(A, B) = (ιA ιB ωe )♭ , where e is the identity element of G. Now the left invariant connection ∇ induced by ω is defined by
∇Ei Ej = (α(ei , ej ))L , where AL denotes the left invariant vector field induced by A ∈ g. The connection ∇ is a Cartan connection, since α is skew-symmetric. In next section we calculate the torsion and curvature tensor of this connection, when G = SU (3). 4. The 2-plectic structure on SU (3) Consider the set E = {ej = − 12 iλj : j = 1, . . . , 8} as a basis for the Lie algebra su(3), where
λ1 = λ4 = λ7 =
0 1 0
1 0 0
0 0 , 0
0 0 1
0 0 0
1 0 , 0
0 0 0
0 0 i
λ2 =
0
λ5 =
0 i 0
−i
0 0 i
0 0 0
λ8 = √
3
0
0 0
1
−i ,
0 0 , 0
−i
1 0 0
,
0 1 0
1 0 0
0 −1 0
0 0 0
0 0 1
λ3 =
0 0
λ6 =
0 0 , 0
0 1 , 0
0 0 . −2
Note that the following relation holds between λi and λj tr (λi λj ) = 2δij .
(4.1)
The nonvanishing structure constants Cijk of su(3) with respect to the basis E are as follows 3 C12 = 1,
7 6 7 5 C14 = C24 = C25 = C34 =
√ 1 6 7 C15 = C36 =− , 2
8 8 C45 = C67 =
3 2
1 2
,
.
Since on SU (3) the Killing form K is K (X , Y ) = 6tr (XY ),
X , Y ∈ su(3),
by using (4.1) we have
σij = σ (ei , ej ) = 3δij .
(4.2)
Using (4.2), the 2-plectic structure ω on SU (3) reads 1
1
1
ω = Θ1 ∧ Θ2 ∧ Θ3 + Θ1 ∧ Θ4 ∧ Θ7 − Θ1 ∧ Θ5 ∧ Θ6 + Θ2 ∧ Θ4 ∧ Θ6 2 2 2 √ 1
1
1
+ Θ ∧Θ ∧Θ + Θ ∧Θ ∧Θ − Θ ∧Θ ∧Θ + 2
2
5
7
2
3
4
5
2
3
6
7
3 2
√
Θ ∧Θ ∧Θ + 4
5
8
3 2
Θ 6 ∧ Θ 7 ∧ Θ 8.
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M. Shafiee / Journal of Geometry and Physics 93 (2015) 33–39
Remark 4.1. Notice that there are 2-plectic structures (3) which are not 2-plectomorphic to ω. For example, d(Θ 1 ∧ on SU k i C Θ ∧ Θ j , then i
Θ 8 ) is an exact 2-plectic structure. Since dΘ k = −
d(Θ 1 ∧ Θ 8 ) = −2Θ 2 ∧ Θ 3 ∧ Θ 8 − Θ 4 ∧ Θ 7 ∧ Θ 8 + Θ 5 ∧ Θ 6 ∧ Θ 8
√
+
3Θ 1 ∧ Θ 4 ∧ Θ 5 +
√
3Θ 1 ∧ Θ 6 ∧ Θ 7 .
If V = V i Ei and ιV (d(Θ 1 ∧ Θ 8 )) = 0, the linear independency of {Θ i ∧ Θ j : i < j} imply that V = 0. Thus d(Θ 1 ∧ Θ 8 ) is nondegenerate and hence a 2-plectic structure. In the same way, d(Θ 2 ∧ Θ 8 ) and d(Θ 3 ∧ Θ 8 ) are 2-plectic structures. These 2-plectic structures are not 2-plectomorphic to ω since ω is not exact.
Theorem 4.2. The Hodge dual ∗ω of ω with respect to σ is a 4-plectic structure on SU (3). Proof. An easy computation shows that
√ ∗ω = −
√ 3
2
3
Θ 12345 −
2
Θ 12367 −
1 2
Θ 12458 +
where Θ ijklm = Θ i ∧ Θ j ∧ Θ k ∧ Θ l ∧ Θ m . And if X =
1 2
Θ 12678 +
1 2
Θ 13468 +
1 2
Θ 13578 −
1 2
Θ 23478 +
1 2
Θ 23568 + Θ 45678 ,
X i Ei , then
√ 3
ιX ω ∧ ∗ω =
2
3+
(X 1 Θ 2345678 − X 2 Θ 1345678 + X 3 Θ 1245678 ) +
× (−X Θ 4
1235678
+X Θ 5
1234678
−X Θ 6
1234578
3
4
√
+X Θ 7
12134568
)−
3 2
X 8 Θ 1234567 .
Furthermore ω ∧ ∗ω = c Θ 12345678 , where c is a nonzero constant, and
i . . . ∧ Θ 8 . (−1)i−1 X i Θ 1 ∧ . . . Θ
ιX (ω ∧ ∗ω) = c
Thus, if ιX (∗ω) = 0, then ιX (ω ∧ ∗ω) = ιX ω ∧ ∗ω and hence X = 0. On the other hand since dΘ i = − easy to see that d ∗ ω = 0.
Cjki Θ j ∧ Θ k , it is
Theorem 4.3. Let ∇ be the Cartan connection induced by the 2-plectic structure on SU (3) and X , Y , Z are left invariant vector fields. Then a. ∇X Y = − 31 [X , Y ],
b. T (X , Y ) = −[X , Y ], R(X , Y )Z =
2 9
[[X , Y ], Z ],
where T and R are torsion and curvature tensors respectively. Proof. Since by definition ∇Ei Ej = (α(ei , ej ))L an easy computation shows 1
1
1
∇E1 E2 = − E3 ,
∇E1 E3 =
∇E1 E6 = − E5 ,
∇E 1 E7 =
∇E2 E4 = − E6 ,
∇E2 E5 = − E7 ,
3 1
6 1 6
3 1 6
E2 ,
∇E1 E4 = − E7 ,
E4 ,
∇E1 E8 = 0,
1
6
1
∇E2 E8 = 0,
∇E3 E4 = − E5 , 1
∇E3 E7 = − E6 , 6
1
∇E4 E7 = − E1 , 6 √ ∇E5 E8 = −
3 6
E4 ,
∇E3 E8 = 0, √ ∇E4 E8 =
3 6
∇E6 E7 =
∇E2 E6 =
∇E3 E5 =
6
1 6
∇E1 E5 =
6
1 6
1
1 6
E4 ,
E3 −
6
E8 ,
Thus 1
∇Ei Ej = − [Ei , Ej ]. 3
Now the assertions follows from the last equation.
1 6
E5 ,
∇E3 E6 = E7 , 6 √
6
3
∇E2 E7 = 1
E4 ,
∇E5 E6 =
E6 ,
3
1
√
6
∇E2 E3 = − E1 ,
∇E4 E5 = − E3 − E5 ,
1
1 6
3 6
1
E8 ,
E1 ,
∇E6 E8 =
∇E4 E6 = − E2 , 6
1
∇E5 E7 = − E2 , 6 √ 3 6
E7 ,
∇E7 E8 = −
√ 3 6
E6 .
M. Shafiee / Journal of Geometry and Physics 93 (2015) 33–39
37
5. Important subgroups Let (M , ω) be a 2-plectic manifold and N is a submanifold of M. For x in N let Tx N ω,1 = {v ∈ Tx M : ω(v, w, .) = 0 ∀w ∈ Tx N }. The submanifold N is called 1-isotropic (res. 1-coisotropic) if Tx N ⊂ Tx N ω,1 (res. Tx N ω,1 ⊂ Tx N), for all x in N. In this section we determine that whether important subgroups of SU (3) are isotropic or coisotropic. 5.1. S 1 Every nontrivial circle in SU (3) is of the form e2π ikθ Tk,l = 0 0
0
e2π ilθ 0
0 , 0 e − 2π i(k + l)θ
where k, l are two integers with |k| + |l| ̸= 0. Tk,l is a 1-isotropic submanifold of SU (3). Indeed, Te (Tk,l ) = span{ae3 + be8 }, where a, b are constant real numbers. Thus Te (Tk,l )ω,1 spans by one of the following sets
{e1 , . . . , e4 }, {e3 , e8 }, {e3 , e4 , e8 }, {e3 , e7 , e8 }. 5.2. SU (2) The Lie group SU (2) is imbedded into SU (3) in tree ways g ∈ SU (2) → g ∈ SU (2) →
g11 g21
g12 g22
g 0
0 1
1 0
0 g
∈ SU (3), ∈ SU (3),
∈ SU (2) →
g11 0 g21
0 1 0
g12 0 g22
∈ SU (3).
Thus Te (SU (2)) spans respectively by
{e1 , e2 , e3 }, √ 3 −1 e3 + e8 , e6 , e7 , 2
e4 , e5 ,
1 2
2
3
e3 +
2
e8 .
So, in the first and third cases SU (2) is a 1-symplectic submanifold in the sense that Tx (SU (2)) and Tx (SU (2))ω,1 has empty intersection. In these two cases the restriction of ω to SU (2) is a 2-plectic form. In the second case, SU (2) is not a special submanifold. 5.3. U (2) As a subgroup of SU (3), U (2) has one of the following forms g ∈ U (2) → g ∈ U (2) →
g =
g11 g21
g 0
0 det g −1
g12 g22
det g −1 0
0 g
g11 0 → g21
∈ SU (3), ∈ SU (3), 0 det g −1 0
g12 0 ∈ SU (3). g22
In the first case, Te (U (2)) = span{e1 , e2 , e3 , e8 } and so Te (U (2))ω,1 = span{e8 }. Thus U (2) is a 1-coisotropic submanifold. In this case
∆g = Te (U (2))ω,1 ,
g ∈ U (2)
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M. Shafiee / Journal of Geometry and Physics 93 (2015) 33–39
is an integrable distribution on U (2), and hence induces a foliation on U (2). The leaves of this distribution are the integral curves of E8 in U (2). Indeed
−√it g
e2
0
3
0
e
√it 3
det g
−1
,
t ∈R U (2)
is the leaf throw g ∈ U (2). The space ∼ induced by this foliation is diffeomorphic to SU (2) and ω induces a 2-plectic structure on it which is proportional to standard 2-plectic structure of SU (2). In the second and third cases, Te (U (2)) is spanned respectively by
{e3 , e6 , e7 , e8 },
{e3 , e4 e5 , e8 }.
Thus U (2) is 1-symplectic submanifold in these two cases. 5.4. SO(3) The Lie algebra so(3) as a Lie subalgebra of su(3) is spanned by {e2 , e5 , e7 }. Therefore SO(3) is a 1-symplectic submanifold. 6. The SU (3)-homogeneous 2-plectic manifolds In this section, using the closed 3-forms on SU (3), we show that some homogeneous spaces induced by SU (3) are 2-plectic manifolds. Theorem 6.1. The homogeneous spaces
SU (3) SU (3) , S1 SU (2)
and
SU (3) T2
are 2-plectic manifolds.
Proof. Consider one of the following closed 3-forms on SU (3)
ω1 = d(Θ 1 ∧ Θ 4 − Θ 2 ∧ Θ 5 ),
ω2 = d(Θ 1 ∧ Θ 6 + Θ 2 ∧ Θ 7 ),
ω3 = d(Θ 1 ∧ Θ 7 − Θ 2 ∧ Θ 6 ).
Kernel of each of these forms at e is isomorphic to su(2). Thus each of them induce a foliation on SU (3) with leaves SU (3) diffeomorphic to SU (2). Hence they induces 2-plectic structures on SU (2) . In the same way the closed 3-forms
ν = d(Θ 5 ∧ Θ 6 + Θ 5 ∧ Θ 7 − Θ 4 ∧ Θ 6 + Θ 4 ∧ Θ 7 ), υ = d(Θ 1 ∧ Θ 2 − Θ 4 ∧ Θ 5 + Θ 6 ∧ Θ 7 ) induce 2-plectic structures on
SU (3) S1
and
SU (3) , T2
respectively.
The statement of Theorem 6.1 has been treated before by T.B. Madsen and A. Swann in Example 5.7 of [14] they showed SU (3) SU (3) that SU (2) and T 2 are 2-plectic manifolds. Theorem 6.2. There are no 2-plectic structures on
SU (3) U (2)
and
SU (3) SO(3)
which are induced by a closed left invariant 3-form on SU (3).
Proof. By 5.3, Te (U (2)) is spanned by one of the following sets
{e1 , e2 , e3 , e8 }, {e3, e6 , e7 , e8 }, {e3 , e4 , e5 , e8 }. i j k Now let Ω = i
Ω = ω456 Θ 4 ∧ Θ 5 ∧ Θ 6 + ω457 Θ 4 ∧ Θ 5 ∧ Θ 7 + ω467 Θ 4 ∧ Θ 6 ∧ Θ 7 + ω567 Θ 5 ∧ Θ 6 ∧ Θ 7 . Now, imposing the condition dΩ = 0 shows that Ω = 0. For two other cases, similarly, one can prove the statement. Again, if Ω is a closed 3-form as above with kernel so(3) = span{e2 , e5 , e7 }, then imposing the conditions ιei ω = 0, i = 2, 5, 7, reduce ω to
Ω = ω134 Θ 1 ∧ Θ 3 ∧ Θ 4 + ω136 Θ 1 ∧ Θ 3 ∧ Θ 6 + ω138 Θ 1 ∧ Θ 3 ∧ Θ 8 + ω346 Θ 3 ∧ Θ 4 ∧ Θ 6 + ω348 Θ 3 ∧ Θ 4 ∧ Θ 8 + ω468 Θ 4 ∧ Θ 6 ∧ Θ 8 . Finally, since dΩ = 0, then Ω = 0.
M. Shafiee / Journal of Geometry and Physics 93 (2015) 33–39
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References [1] F. Cantrijn, A. Ibort, M. DeLeon, On the geometry of multisymplectic manifolds, J. Aust. Math. Soc. A 66 (1999) 303–330. [2] J.F. Carinena, M. Crampin, L.A. Ibort, On the multisymplectic formalism for first order field theories, Differential Geom. Appl. 1 (1991) 345–374. [3] M. Gotay, J. Isenberg, J. Marsden, R. Montgomery, Momentum maps and classical relativistic fields, part I: covariant field theory. Available as arXiv: Physics/9801019. [4] J. Kijowski, A finite-dimensional canonical formalism in the classical field theory, Comm. Math. Phys. 30 (1973) 99–128. [5] J.C. Baez, A.E. Hoffnung, C.L. Rogers, Categorified symplectic geometry and the classical string, Comm. Math. Phys. 293 (2010) 701–725. [6] C.L. Rogers, Higher symplectic geometry (Reverside D. Phil. Thesis), University of California. Available as arXiv:1106.4068v1. [7] G. Martin, A Darboux theorem for multisymplectic manifolds, Lett. Math. Phys. 27 (1988) 571–585. [8] M. Shafiee, On compact semisimple Lie groups as 2-plectic manifolds, J. Geom. 105 (3) (2014) 615–623. [9] F. Cantrijn, A. Ibort, M. DeLeon, Hamiltonian structures on multisymplectic manifolds, Rend. Semin. Mat. Univ. Politec. Torino 54 (3) (1996) 225–236. [10] A. Banyaga, The Structure of Classical Diffeomorphism Groups, in: Mathematics and its Applications, vol. 400, Kluwer Academic Publishers Group, 1997. [11] S. Kobayashi, Transformation Groups in Differential Geometry, in: Erg Math. Grenzgeb, vol. 70, Springer-Verlag, 1977. [12] S.B. Meyer, N.E. Steenrod, The group of isometries of a Riemannian manifold, Ann. of Math. 40 (1939) 400–416. [13] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, first ed., in: GSM, vol. 34, AMS, 2001. [14] T.B. Madsen, A. Swann, Multi-moment maps. Available as arXiv: 1012.2048v2.
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Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp
SU (3) as a 2-plectic manifold Mohammad Shafiee Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, P.O.Box 518, Iran
article
info
Article history: Received 12 August 2014 Received in revised form 24 February 2015 Accepted 6 March 2015 Available online 18 March 2015 MSC: 53Dxx 53D05 53D20 11F22
abstract The Killing form induces a 2-plectic structure on a compact semisimple Lie group. The associated Lie group of canonical transformations (2-plectomorphisms) is compact. This 2-plectic structure induces a Cartan connection on the Lie group. The curvature and torsion tensor of this connection have been calculated for the special unitary Lie group SU (3). It is SU (3) SU (3) SU (3) shown that the homogeneous spaces SU (2) , S 1 and T 2 also, admit 2-plectic structures, SU (3)
SU (3)
which are induced by closed left invariant 3-forms on SU (3), whereas U (2) and SO(3) do not admit such 2-plectic structures. © 2015 Elsevier B.V. All rights reserved.
Keywords: 2-plectic structure Compact semisimple Lie group Special unitary Lie group
1. Introduction A k-plectic structure on a smooth manifold M is a closed (k + 1)-form ω on M which is nondegenerate, i.e., if ιX ω = 0, X ∈ TM, then X = 0 (see [1]). If ω is a k-plectic structure on M, (M , ω) is called a k-plectic manifold or a multisymplectic manifold of order k + 1. A multisymplectic manifold of order 2 is called a symplectic manifold. Multisymplectic structures arise in the geometric formulation of classical field theory much in the same way that symplectic structures emerge in the Hamiltonian description of classical mechanics [2–4]. In this formulation, a (k + 1)-dimensional field theory is described by a finite dimensional k-plectic manifold (M , ω) as a ‘‘multi-phase space’’ instead of an infinite dimensional phase space. Using the k-plectic form ω, one can define a system of partial differential equations which are the analogue of Hamilton equations in classical mechanics. The solutions of these equations correspond to special submanifolds of the multi-phase space M. Compact semisimple Lie groups are important examples of 2-plectic manifolds. The 2-plectic structure on these Lie groups is induced by the Killing form. It is well known that the space of smooth functions on a symplectic manifold forms a Lie algebra under Poisson bracket induced by the symplectic form. In [5], Baez and Rogers showed that any 2-plectic manifold gives rise to a Lie 2-algebra. They constructed hemistrict and semistrict Lie 2-algebras from any compact semisimple Lie group G. Moreover, they showed that these Lie 2-algebras are isomorphic to the string Lie 2-algebra of G. The latter structure is important in the theory of gerbs on Lie groups [6]. Another important result about these 2-plectic structures is that the Darboux’s theorem is not true [7,8]. Despite the above mentioned results, geometrically, multisymplectic geometry is a field worth investigating in its own right.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.geomphys.2015.03.003 0393-0440/© 2015 Elsevier B.V. All rights reserved.
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M. Shafiee / Journal of Geometry and Physics 93 (2015) 33–39
In this paper, we consider the Lie group SU (3) as a 2-plectic manifold. The paper is organized as follows. In Section 2, some general results on a compact semisimple Lie group G, as a 2-plectic manifold, are presented. In Section 3, we show that this 2-plectic structure induces a Cartan connection on G. In Section 4, some results about the geometry of SU (3) as a 2-plectic manifold have proved. In particular we show that ∗ω is a 4-plectic structure on SU (3), where ω is the 2-plectic structure on SU (3). Also we show that T (X , Y ) = −[X , Y ],
2
[[X , Y ], Z ], 9 where T and R are the torsion and the curvature tensor of the Cartan connection, respectively. In Section 5 we are trying to know whether special subgroups of SU (3) are special submanifolds or not. In the last section, we show that the homogeneous SU (3) SU (3) SU (3) spaces SU (2) , S 1 and T 2 , admit 2-plectic structures, which are induced by closed left invariant 3-forms on SU (3), whereas SU (3) U (2)
and
SU (3) SO(3)
R(X , Y )Z =
do not admit such 2-plectic structures.
2. The 2-plectic structure on a compact semisimple Lie group Let G be a compact semisimple Lie group with Lie algebra g and let ⟨., .⟩ denotes the positive definite adjoint invariant inner product on g defined by
⟨A, B⟩ = −K (A, B), where K is the Killing form on g and A, B ∈ g. Assume that σ is the biinvariant Riemannian metric on G induced by ⟨., .⟩. Define the 3-form ω on G by
ω(X , Y , Z ) = σ (X , [Y , Z ]),
X , Y , Z ∈ TG.
Since σ is nondegenerate and biinvariant then so is ω. Thus ω is closed. Furthermore, since K (A, [B, C ]) = K (B, [C , A]) = K (C , [A, B]), for all A, B, C in g, then ω is totally antisymmetric. So ω is a 2-plectic structure on G [1]. In the following we present some general results about the geometry of this 2-plectic structure. For more details refer to [8]. Choose a basis E = {e1 , . . . , em } for g and let E1 , . . . , Em be left invariant vector fields induced by e1 , . . . , em , and Θ 1 , . . . , Θ m be left invariant 1-forms dual to E1 , . . . , Em , respectively. Denote by Cijk the structure constants of g and by σij the components of σ with respect to E . Then the 2-plectic form ω reads
ω = Σijkl σij Cklj Θ i ∧ Θ k ∧ Θ l . A diffeomorphism ϕ : G → G is called a 2-plectomorphism provided ϕ ∗ ω = ω. We denote by Symp2 (G, ω), the group of all 2-plectomorphisms on G and by Symp20 (G, ω), its identity component. Let I denote the closed interval [0, 1]. A time dependent 1-form H : G × I → T ∗ G is called Hamiltonian, if there exists a time dependent vector field XH on G, the so called Hamiltonian vector field [9], such that
ιXH ω = dH . The flow ϕt , t ∈ I, of the time dependent Hamiltonian vector field XH is called the Hamiltonian flow generated by H. If ϕt is a Hamiltonian flow, then every individual diffeomorphism ϕt is called a Hamiltonian diffeomorphism. Note that every Hamiltonian diffeomorphism preserves ω. Hence the set of all Hamiltonian diffeomorphisms Ham(G, ω) is a subset of Symp20 (G, ω). In fact, as the following theorem shows, they are the same. Theorem 2.1 (See [8]). Ham(G, ω) = Symp20 (G, ω). Proof. Let ϕ ∈ Symp20 (G, ω) and consider a path {ϕt } ⊂ Symp20 (G, ω) with ϕ0 = id and ϕ1 = ϕ . Assume that Xt is the vector field induced by {ϕt }. Then ιXt ω is closed. Since G is simply connected then its second cohomology group H 2 (G, R) is trivial. Thus ιXt ω is exact. Let ιXt ω = dHt , for some 1-form Ht . Now ϕ is the time 1-map of time dependent Hamiltonian 1-form H on G, defined by H (x, t ) = Ht (x). Let Iso(G) denote the isometry group of G with respect to σ . The following theorem shows that Symp2 (G, ω) and Iso(G) are the same. Theorem 2.2 (See [8]). Symp2 (G, ω) = Iso(G). Proof. By definition of the 2-plectic structure ω and its relation with σ it is trivial that any isometry preserves ω. Conversely, let ϕ be a 2-plectomorphism and X , Y be in TG. It is well known that the Lie algebra of compactly supported vector fields on a smooth manifold is prefect [10]. So there are vector fields W and Z on G such that Y = [W , Z ]. Thus
ϕ ∗ σ (X , Y ) = ϕ ∗ σ (X , [W , Z ]) = ϕ ∗ ω(X , W , Z ) = ω(X , W , Z ) = σ (X , Y ). Therefore ϕ is an isometry.
M. Shafiee / Journal of Geometry and Physics 93 (2015) 33–39
35
Thus left and right translations are 2-plectomorphisms and hence left and right invariant vector fields are Hamiltonian. A consequence of Theorems 2.1 and 2.2 is the existence of Hamiltonian vector fields which are neither left nor right invariant. To find such vector fields, let g (t ), 0 ≤ t ≤ 1, be any differentiable curve in G with g (0) = id. Define ϕt : G → G by ϕt (x) = g −1 (t )xg (t ). According to Theorems 2.1 and 2.2, ϕt is a Hamiltonian diffeomorphism. The vector field X defined by {ϕt } is Hamiltonian but neither left nor right invariant. Another consequence of these theorems is that, in contrast to symplectic case, the 2-plectic and Hamiltonian groups have no new results about the topology of the Lie group. Furthermore Theorem 2.2 shows that Symp2 (G, ω) is compact since Iso(G) is compact [11,12]. 3. The Cartan connection induced by the 2-plectic structure We recall that a connection ∇ on G is called left invariant if for any two left invariant vector fields X , Y , the vector field ∇X Y is also left invariant. A left invariant connection is called a Cartan connection, if for any A ∈ g, the curve t → etA is a geodesic. It is well known that there is a one-to-one correspondence between the left invariant connections on G and bilinear forms on g with value in g [13]. In this section, using the 2-plectic structure ω, we construct a left invariant connection on a compact semisimple Lie group G. To do this, let ♭ : g∗ → g be the isomorphism induced by σ . The 2-plectic form ω induces a bilinear form α : g × g → g as follows
α(A, B) = (ιA ιB ωe )♭ , where e is the identity element of G. Now the left invariant connection ∇ induced by ω is defined by
∇Ei Ej = (α(ei , ej ))L , where AL denotes the left invariant vector field induced by A ∈ g. The connection ∇ is a Cartan connection, since α is skew-symmetric. In next section we calculate the torsion and curvature tensor of this connection, when G = SU (3). 4. The 2-plectic structure on SU (3) Consider the set E = {ej = − 12 iλj : j = 1, . . . , 8} as a basis for the Lie algebra su(3), where
λ1 = λ4 = λ7 =
0 1 0
1 0 0
0 0 , 0
0 0 1
0 0 0
1 0 , 0
0 0 0
0 0 i
λ2 =
0
λ5 =
0 i 0
−i
0 0 i
0 0 0
λ8 = √
3
0
0 0
1
−i ,
0 0 , 0
−i
1 0 0
,
0 1 0
1 0 0
0 −1 0
0 0 0
0 0 1
λ3 =
0 0
λ6 =
0 0 , 0
0 1 , 0
0 0 . −2
Note that the following relation holds between λi and λj tr (λi λj ) = 2δij .
(4.1)
The nonvanishing structure constants Cijk of su(3) with respect to the basis E are as follows 3 C12 = 1,
7 6 7 5 C14 = C24 = C25 = C34 =
√ 1 6 7 C15 = C36 =− , 2
8 8 C45 = C67 =
3 2
1 2
,
.
Since on SU (3) the Killing form K is K (X , Y ) = 6tr (XY ),
X , Y ∈ su(3),
by using (4.1) we have
σij = σ (ei , ej ) = 3δij .
(4.2)
Using (4.2), the 2-plectic structure ω on SU (3) reads 1
1
1
ω = Θ1 ∧ Θ2 ∧ Θ3 + Θ1 ∧ Θ4 ∧ Θ7 − Θ1 ∧ Θ5 ∧ Θ6 + Θ2 ∧ Θ4 ∧ Θ6 2 2 2 √ 1
1
1
+ Θ ∧Θ ∧Θ + Θ ∧Θ ∧Θ − Θ ∧Θ ∧Θ + 2
2
5
7
2
3
4
5
2
3
6
7
3 2
√
Θ ∧Θ ∧Θ + 4
5
8
3 2
Θ 6 ∧ Θ 7 ∧ Θ 8.
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M. Shafiee / Journal of Geometry and Physics 93 (2015) 33–39
Remark 4.1. Notice that there are 2-plectic structures (3) which are not 2-plectomorphic to ω. For example, d(Θ 1 ∧ on SU k i C Θ ∧ Θ j , then i
Θ 8 ) is an exact 2-plectic structure. Since dΘ k = −
d(Θ 1 ∧ Θ 8 ) = −2Θ 2 ∧ Θ 3 ∧ Θ 8 − Θ 4 ∧ Θ 7 ∧ Θ 8 + Θ 5 ∧ Θ 6 ∧ Θ 8
√
+
3Θ 1 ∧ Θ 4 ∧ Θ 5 +
√
3Θ 1 ∧ Θ 6 ∧ Θ 7 .
If V = V i Ei and ιV (d(Θ 1 ∧ Θ 8 )) = 0, the linear independency of {Θ i ∧ Θ j : i < j} imply that V = 0. Thus d(Θ 1 ∧ Θ 8 ) is nondegenerate and hence a 2-plectic structure. In the same way, d(Θ 2 ∧ Θ 8 ) and d(Θ 3 ∧ Θ 8 ) are 2-plectic structures. These 2-plectic structures are not 2-plectomorphic to ω since ω is not exact.
Theorem 4.2. The Hodge dual ∗ω of ω with respect to σ is a 4-plectic structure on SU (3). Proof. An easy computation shows that
√ ∗ω = −
√ 3
2
3
Θ 12345 −
2
Θ 12367 −
1 2
Θ 12458 +
where Θ ijklm = Θ i ∧ Θ j ∧ Θ k ∧ Θ l ∧ Θ m . And if X =
1 2
Θ 12678 +
1 2
Θ 13468 +
1 2
Θ 13578 −
1 2
Θ 23478 +
1 2
Θ 23568 + Θ 45678 ,
X i Ei , then
√ 3
ιX ω ∧ ∗ω =
2
3+
(X 1 Θ 2345678 − X 2 Θ 1345678 + X 3 Θ 1245678 ) +
× (−X Θ 4
1235678
+X Θ 5
1234678
−X Θ 6
1234578
3
4
√
+X Θ 7
12134568
)−
3 2
X 8 Θ 1234567 .
Furthermore ω ∧ ∗ω = c Θ 12345678 , where c is a nonzero constant, and
i . . . ∧ Θ 8 . (−1)i−1 X i Θ 1 ∧ . . . Θ
ιX (ω ∧ ∗ω) = c
Thus, if ιX (∗ω) = 0, then ιX (ω ∧ ∗ω) = ιX ω ∧ ∗ω and hence X = 0. On the other hand since dΘ i = − easy to see that d ∗ ω = 0.
Cjki Θ j ∧ Θ k , it is
Theorem 4.3. Let ∇ be the Cartan connection induced by the 2-plectic structure on SU (3) and X , Y , Z are left invariant vector fields. Then a. ∇X Y = − 31 [X , Y ],
b. T (X , Y ) = −[X , Y ], R(X , Y )Z =
2 9
[[X , Y ], Z ],
where T and R are torsion and curvature tensors respectively. Proof. Since by definition ∇Ei Ej = (α(ei , ej ))L an easy computation shows 1
1
1
∇E1 E2 = − E3 ,
∇E1 E3 =
∇E1 E6 = − E5 ,
∇E 1 E7 =
∇E2 E4 = − E6 ,
∇E2 E5 = − E7 ,
3 1
6 1 6
3 1 6
E2 ,
∇E1 E4 = − E7 ,
E4 ,
∇E1 E8 = 0,
1
6
1
∇E2 E8 = 0,
∇E3 E4 = − E5 , 1
∇E3 E7 = − E6 , 6
1
∇E4 E7 = − E1 , 6 √ ∇E5 E8 = −
3 6
E4 ,
∇E3 E8 = 0, √ ∇E4 E8 =
3 6
∇E6 E7 =
∇E2 E6 =
∇E3 E5 =
6
1 6
∇E1 E5 =
6
1 6
1
1 6
E4 ,
E3 −
6
E8 ,
Thus 1
∇Ei Ej = − [Ei , Ej ]. 3
Now the assertions follows from the last equation.
1 6
E5 ,
∇E3 E6 = E7 , 6 √
6
3
∇E2 E7 = 1
E4 ,
∇E5 E6 =
E6 ,
3
1
√
6
∇E2 E3 = − E1 ,
∇E4 E5 = − E3 − E5 ,
1
1 6
3 6
1
E8 ,
E1 ,
∇E6 E8 =
∇E4 E6 = − E2 , 6
1
∇E5 E7 = − E2 , 6 √ 3 6
E7 ,
∇E7 E8 = −
√ 3 6
E6 .
M. Shafiee / Journal of Geometry and Physics 93 (2015) 33–39
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5. Important subgroups Let (M , ω) be a 2-plectic manifold and N is a submanifold of M. For x in N let Tx N ω,1 = {v ∈ Tx M : ω(v, w, .) = 0 ∀w ∈ Tx N }. The submanifold N is called 1-isotropic (res. 1-coisotropic) if Tx N ⊂ Tx N ω,1 (res. Tx N ω,1 ⊂ Tx N), for all x in N. In this section we determine that whether important subgroups of SU (3) are isotropic or coisotropic. 5.1. S 1 Every nontrivial circle in SU (3) is of the form e2π ikθ Tk,l = 0 0
0
e2π ilθ 0
0 , 0 e − 2π i(k + l)θ
where k, l are two integers with |k| + |l| ̸= 0. Tk,l is a 1-isotropic submanifold of SU (3). Indeed, Te (Tk,l ) = span{ae3 + be8 }, where a, b are constant real numbers. Thus Te (Tk,l )ω,1 spans by one of the following sets
{e1 , . . . , e4 }, {e3 , e8 }, {e3 , e4 , e8 }, {e3 , e7 , e8 }. 5.2. SU (2) The Lie group SU (2) is imbedded into SU (3) in tree ways g ∈ SU (2) → g ∈ SU (2) →
g11 g21
g12 g22
g 0
0 1
1 0
0 g
∈ SU (3), ∈ SU (3),
∈ SU (2) →
g11 0 g21
0 1 0
g12 0 g22
∈ SU (3).
Thus Te (SU (2)) spans respectively by
{e1 , e2 , e3 }, √ 3 −1 e3 + e8 , e6 , e7 , 2
e4 , e5 ,
1 2
2
3
e3 +
2
e8 .
So, in the first and third cases SU (2) is a 1-symplectic submanifold in the sense that Tx (SU (2)) and Tx (SU (2))ω,1 has empty intersection. In these two cases the restriction of ω to SU (2) is a 2-plectic form. In the second case, SU (2) is not a special submanifold. 5.3. U (2) As a subgroup of SU (3), U (2) has one of the following forms g ∈ U (2) → g ∈ U (2) →
g =
g11 g21
g 0
0 det g −1
g12 g22
det g −1 0
0 g
g11 0 → g21
∈ SU (3), ∈ SU (3), 0 det g −1 0
g12 0 ∈ SU (3). g22
In the first case, Te (U (2)) = span{e1 , e2 , e3 , e8 } and so Te (U (2))ω,1 = span{e8 }. Thus U (2) is a 1-coisotropic submanifold. In this case
∆g = Te (U (2))ω,1 ,
g ∈ U (2)
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M. Shafiee / Journal of Geometry and Physics 93 (2015) 33–39
is an integrable distribution on U (2), and hence induces a foliation on U (2). The leaves of this distribution are the integral curves of E8 in U (2). Indeed
−√it g
e2
0
3
0
e
√it 3
det g
−1
,
t ∈R U (2)
is the leaf throw g ∈ U (2). The space ∼ induced by this foliation is diffeomorphic to SU (2) and ω induces a 2-plectic structure on it which is proportional to standard 2-plectic structure of SU (2). In the second and third cases, Te (U (2)) is spanned respectively by
{e3 , e6 , e7 , e8 },
{e3 , e4 e5 , e8 }.
Thus U (2) is 1-symplectic submanifold in these two cases. 5.4. SO(3) The Lie algebra so(3) as a Lie subalgebra of su(3) is spanned by {e2 , e5 , e7 }. Therefore SO(3) is a 1-symplectic submanifold. 6. The SU (3)-homogeneous 2-plectic manifolds In this section, using the closed 3-forms on SU (3), we show that some homogeneous spaces induced by SU (3) are 2-plectic manifolds. Theorem 6.1. The homogeneous spaces
SU (3) SU (3) , S1 SU (2)
and
SU (3) T2
are 2-plectic manifolds.
Proof. Consider one of the following closed 3-forms on SU (3)
ω1 = d(Θ 1 ∧ Θ 4 − Θ 2 ∧ Θ 5 ),
ω2 = d(Θ 1 ∧ Θ 6 + Θ 2 ∧ Θ 7 ),
ω3 = d(Θ 1 ∧ Θ 7 − Θ 2 ∧ Θ 6 ).
Kernel of each of these forms at e is isomorphic to su(2). Thus each of them induce a foliation on SU (3) with leaves SU (3) diffeomorphic to SU (2). Hence they induces 2-plectic structures on SU (2) . In the same way the closed 3-forms
ν = d(Θ 5 ∧ Θ 6 + Θ 5 ∧ Θ 7 − Θ 4 ∧ Θ 6 + Θ 4 ∧ Θ 7 ), υ = d(Θ 1 ∧ Θ 2 − Θ 4 ∧ Θ 5 + Θ 6 ∧ Θ 7 ) induce 2-plectic structures on
SU (3) S1
and
SU (3) , T2
respectively.
The statement of Theorem 6.1 has been treated before by T.B. Madsen and A. Swann in Example 5.7 of [14] they showed SU (3) SU (3) that SU (2) and T 2 are 2-plectic manifolds. Theorem 6.2. There are no 2-plectic structures on
SU (3) U (2)
and
SU (3) SO(3)
which are induced by a closed left invariant 3-form on SU (3).
Proof. By 5.3, Te (U (2)) is spanned by one of the following sets
{e1 , e2 , e3 , e8 }, {e3, e6 , e7 , e8 }, {e3 , e4 , e5 , e8 }. i j k Now let Ω = i
Ω = ω456 Θ 4 ∧ Θ 5 ∧ Θ 6 + ω457 Θ 4 ∧ Θ 5 ∧ Θ 7 + ω467 Θ 4 ∧ Θ 6 ∧ Θ 7 + ω567 Θ 5 ∧ Θ 6 ∧ Θ 7 . Now, imposing the condition dΩ = 0 shows that Ω = 0. For two other cases, similarly, one can prove the statement. Again, if Ω is a closed 3-form as above with kernel so(3) = span{e2 , e5 , e7 }, then imposing the conditions ιei ω = 0, i = 2, 5, 7, reduce ω to
Ω = ω134 Θ 1 ∧ Θ 3 ∧ Θ 4 + ω136 Θ 1 ∧ Θ 3 ∧ Θ 6 + ω138 Θ 1 ∧ Θ 3 ∧ Θ 8 + ω346 Θ 3 ∧ Θ 4 ∧ Θ 6 + ω348 Θ 3 ∧ Θ 4 ∧ Θ 8 + ω468 Θ 4 ∧ Θ 6 ∧ Θ 8 . Finally, since dΩ = 0, then Ω = 0.
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