A note on $2$-plectic homogeneous manifolds

In this note we study the existence of $2$-plectic structures on homogenous spaces. In particular we show that $S^{5}=\frac{SU(3)}{SU(2)}$, $\frac{SU(3)}{S^{1}}$, $\frac{SU(3)}{T^{2}}$ and $\frac{SO(4)}{S^{1}}$ admit a $2$-plectic structure. Furthermore, If $G$ is a Lie group with Lie algebra $\mathfrak{g}$ and $R$ is a closed Lie subgroup of $G$ corresponding to the nilradical of $\mathfrak{g}$, then $\frac{G}{R}$ is a $2$-plectic manifold.

and SO(4) S 1 admit a 2-plectic structure. Furthermore, If G is a Lie group with Lie algebra g and R is a closed Lie subgroup of G corresponding to the nilradical of g, then G R is a 2-plectic manifold.
1. Introduction. Let V be a real vector space. A 3-form ω ∈ ∧ 3 V * is called a 2-plectic form if ω is nondegenerate in the sense that ι v ω = 0 if and only if v = 0.
If ω is a 2-plectic form on V , the pair (V, ω) is called a 2-plectic vector space. A smooth manifold M is called a 2-plectic manifold if there is a closed 3-form ω on M such that (T x M, ω x ) is a 2-plectic vector space, for all x ∈ M . 2-plectic structures (and in general multisymplectic structures) in the above sense, appeared in [3] for the first time. In the same paper, The authors introduced three important classes of 2-plectic manifolds as follows: 1. Compact semisimple Lie groups with the 2-plectic structure induced by the Killing form.
2. The bundle of exterior 2-forms E on a smooth manifold M with the 2-plectic structure ω = dΘ, where Θ is the canonical 2-form on E, characterized by α * (Θ) = α, for all 2-forms α on E.
3. Cosymplectic manifolds (M, θ, η) of dimension 2n + 1 with the 2-plectic structure ω = θ ∧ η, where θ is a closed 2-form and η is a closed 1-form on M such that θ n ∧ η = 0. Use of multisymplectic structures in the covariant Hamiltonian formulation of classical field theories have already been considered extensively ( [6,5,4,2]). In particular, 2-plectic manifolds are used to describe a classical string ( [1,9]). However, geometrically, 2-plectic geometry, in contrast to symplectic case, have not been considered. A possible reason is that the Darboux theorem does not hold in this case ( [10,8]). Another reason, can be the fact that 2-plectic manifolds are not, in comparison to symplectic manifolds, do not enjoy variety. In this note, using homogeneous spaces, we try to introduce new 2-plectic manifolds. 2. Existence of 2-plectic structures on homogenous manifolds. Let G be a connected Lie group with Lie algebra g. We recall that for a p-form α ∈ ∧ p g * , δ(α) is defined by For a p-form α on g, the kernel of α is defined by If α is closed, i.e δ(α) = 0, then k is a sub lie algebra of g. In this case α, as a left invariant form induces a foliation on G with leaves diffeomorphic to K, where K is the connected Lie group corresponding to k. Furthermore if K is closed then α induces a nondegenerate p-form α on G K , the space of left cosets of K, which is invariant under left action of G on G K and satisfying π * α = α, where π : G → G K is the canonical projection. Using this fact , in this note we are interested to construct 2-plectic structures on some homogenous manifolds. In this section, at first we prove some general results about such structures. These results are similar to the results which have been proved in [11] for symplectic case. Then some existence results will be proved.
Let H be a closed Lie subgroup of G.
Notice that if ω is a closed form on g with kernel k, then the 2-plectic form ω on G K induced by ω as above is G-invariant. In fact: Consider the decomposed Lie algebra g = g 1 ⊕ g 2 ⊕ ... ⊕ g n ( it means that g is the direct sum of Lie algebras g 1 , g 2 ...g n ) and let p i : g → g i , q i : g i → g be the corresponding projection and injection of Lie algebras respectively. The 3-form ω on g is called decomposable if ω = (q 1 • p 1 ) * ω + ... + (q n • p n ) * ω. It is easy to see that ω is decomposable if and only if ω(g i , g j , g k ) = 0, whenever i = j = k. Lemma 2.2. ( [11])Let the Lie algebra g of the Lie group G has a decomposition g = g 1 ⊕ g 2 ⊕ ... ⊕ g n and ω be a decomposable 3-form on g. Then the Lie algebra h of the Lie subgroup The following result is a direct consequence of Lemma 2.3. Corollary 1. If g = g 1 ⊕ g 2 ⊕ ... ⊕ g n is a semi simple Lie algebra, g i simple ideal for i = 1, ..., n, then any closed 3-form on g is decomposable.
Corollary 2. Let G be a connected semi simple Lie group with Lie algebra g = g 1 ⊕ g 2 ⊕...⊕g n , and G i be the connected Lie subgroup corresponding to g i , i = 1, ..., n.  Proof. The first statement is proved similar to Lemma 4.1 of [11]. To prove the second statement, let X ∈ n, Y ∈ b, Z ∈ c and W ∈ a. since ω is closed, then Now the first statement implies that the first three terms of the right hand side are zero. Since a ∩ b = 0 = a ∩ c, therefore the last two terms are also zero. Thus the result holds.
Theorem 2.5. Let G be a Lie group with Lie algebra g and r be the radical of g. If the connected Lie subgroup R corresponding to r is closed then G R is a 2-plectic manifold.
Proof. Consider a Levi decomposition g = r ⊕ s for g and let B denote the Killing form on s. Define the 3-form ω 0 on s by ω 0 is closed on s. Define the 3-form ω on g with ω = p * ω 0 , where p : g → s is the projection. ω is closed with kerω = r. Indeed, since then, if at least one of the vectors X, Y, Z and W is in r, the right hand side is zero. Otherwise, δ(ω)(X, Y, Z, W ) = δ 0 (ω)(X, Y, Z, W ) = 0. Thus ω induces a G-invariant 2-plectic form on G R .
Theorem 2.6. Let G be a Lie group with Lie algebra g = g 1 ⊕ g 2 ⊕ ... ⊕ g n , g i semi simple, and G i be the connected Lie subgroup corresponding to g i , i = 1, ..., n. Furthermore, let J ⊂ {1, ....n} and G j be closed for j ∈ J. Then G i∈J Gj is a 2-plectic manifold.
Proof. Let the 3-form ω j be defined on g j by its Killing form as Theorem 2.6 for j ∈ J c . Put ω = p * (⊕ j∈J c ω j ), where p : g → j∈J c g j is the projection. ω is a closed 3-form on g with Kerω = ⊕ i∈J g i . Thus ω induces a 2-plectic structure on G i∈J Gj .
3. Examples. In this section we consider the homogeneous spaces of SU (3) and SO(4). In particular we show that S 5 is a 2-plectic manifold.  Proof. Consider one of the following closed 3-forms on SU (3) The kernel of each of these forms at e is isomorphic to su (2). Thus each of them induces a foliation on SU (3) with leaves diffeomorphic to SU (2). Hence they induce 2-plectic structures on SU (3) SU (2) . In the same way the closed 3-forms induce 2-plectic structures on SU (3)  Thus α induces a foliation on SO(4) with leaves diffeomorphic to SO(3) and hence it induces a 2-plectic structure on SO(4) SO(3) S 3 . A similar argument works for SO(4) SO(2) , when we consider the 3-form β = d(Θ 2 ∧ Θ 4 − Θ 3 ∧ Θ 5 ).