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On 2-Plectic Lie Groups On 2-plectic Lie groups Mohammad Shafiee and Masoud Aminizadeh Abstract. A 2-plectic Lie group is a Lie group endowed with a 2-plectic structure which is left invariant. In this paper we provide some inter- esting examples of 2-plectic Lie groups. Also we study the structure of the set of Hamiltonian covectors and vectors of a 2-plectic Lie algebra. Moreover, the existence of i-isotropic and i-Lagrangian subgroups are in- vestigated. At last we obtain some results about the reduction of some 2-plectic structures. M.S.C. 2010: 53D05, 17B60. Key words: 2-plectic structure; quadratic Lie algebra; Hamiltonian vector. 1 Introduction A k-plectic manifold (M; !) is a smooth manifold M endowed with a (k + 1)-form ! which is closed and nondegenerate in the sense that ιX ! = 0, X 2 TM, implies that X = 0. In this case ! is called a k-plectic structure. These structures are general version of symplectic structures and they naturally appear in the Hamiltonian formulation of classical fields (see [4]and references therein ). Mathematically, in spite of general case, symplectic structures are very interesting and so they have been studied extensively. However, in recent years, 2-plectic structures also have been considered and these structures also are studied extensively ([1], [8]). An important class of 2-plectic manifolds are 2-plectic Lie groups. Similar to symplectic case, a 2-plectic Lie group is defined as follows. Definition 1.1. A 2-plectic Lie group (G; !) is a Lie group G endowed with a 2- plectic structure ! which is left invariant. In contrast to the symplectic case, there are a lot of canonical 2-plectic structures which are induced by quadratic forms and bialgebra structures (see below). However, symplectic Lie groups have been studied extensively ([2] and references therein). In this article we study 2-plectic Lie groups and since for simply connected Lie groups, the study of 2-plectic Lie groups reduces to the study of 2-plectic Lie algebras, we will study 2-plectic Lie algebras. Balkan* Journal of Geometry and Its Applications, Vol.26, No.2, 2021, pp. 112-125. © Balkan Society of Geometers, Geometry Balkan Press 2021. On 2-plectic Lie groups 113 Definition 1.2. A 2-plectic Lie algebra (g;!) is a Lie algebra g equipped with a 2-plectic structure ! which is closed, in the sense that ! : g × g × g ! R satisfying δ!(x; y; u; v) = !([x; y]; u; v) − !([x; u]; y; v) + !([x; v]; y; u) −!([y; u]; x; v) + !([y; v]; x; u) − !(x; y; [u; v]) = 0; for all x; y; u; v 2 g. The structure of the paper is as follows: In section 2 we provide some important examples. In section 3 we introduce Hamiltonian covectors and vectors and study their structures. In section 4 we study the existence of isotropic and Lagrangian ideals and subalgebras. In the last section we consider the reduction of some 2-plectic structures. 2 Examples In this section we provide some important examples of 2-plectic Lie groups. Definition 2.1. Let (g; [ ; ]) be a Lie algebra. A bilinear form B : g × g ! R is called i) symmetric if B(x; y) = B(y; x), for all x; y 2 g, ii) nondegenerate if B(x; y) = 0; for all y 2 g implies x = 0, iii) invariant if B([x; y]; z) = −B(y; [x; z]), for all x; y; z 2 g. If B is a bilinear form on g, which is symmetric, nondegenerate and invariant, then it is called a quadratic form and the pair (g;B) is called a quadratic Lie algebra. Quadratic form is a generalization of the Killing form. In general, a Lie algebra may not have a quadratic form. However, the existence and classifying quadratic forms on low dimensional Lie algebras has been studied extensively ([7], [6], [3]). Let (g;B) be a quadratic non Abelian Lie algebra and define the 3-form ! on g by (2.1) !(x; y; z) = B([x; y]; z); x; y; z 2 g: Since B is symmetric and invariant, then ! is totally skew symmetric and closed. But, in general, ! is not nondegenerate. Indeed, if !(x; y; z) = 0, for all y; z 2 g, then [x; y] = 0, for all y 2 g. So Ker ! = fx 2 g : ιx! = 0g = z(g); g where z(g) is the centre of g. However, ! induces a 2-plectic structure on g = z(g) . Theorem 2.1. The 3-form !¯ on g, defined by (2.2) !¯(x; y; z) = !(x; y; z); x; y; z 2 g is a 2-plectic structure, where x = x + z(g). 114 Mohammad Shafiee and Masoud Aminizadeh Proof. Let x2 = x1 + x0; y2 = y1 + y0; z2 = z1 + z0, where x0; y0; z0 2 z(g). Then !(x2; y2; z2) = !(x1 + x0; y1 + y0; z1 + z0) = !(x1; y1; z1) + !(x1; y1; z0) + !(x1; y0; z1) +!(x1; y0; z0) + !(x0; y1; z1) + !(x0; y1; z0) +!(x0; y0; z1) + !(x0; y0; z0): Since x0; y0; z0 2 z(g) and !(x; y; z) = B([x; y]; z), then all terms of the last equation, except the first one, are zero. Hence, !(x2; y2; z2) = !(x1; y1; z1). This shows that !¯ is well-defined. Moreover, if! ¯(x; y; z) = 0, for all y; z 2 g, then B([x; y]; z) = !(x; y; z) = 0, for all y; z 2 g. Thus x 2 z(g) and hence x = 0. At last, since ! is closed then! ¯ is closed. In the following, we will denote this 2-plectic Lie algebra by (g; !;¯ B). Using this theorem, we provide some important examples. Remark 2.2. In this paper all Lie algebras are real. However, if a Lie algebra g is a complex Lie algebra and B is a complex valued quadratic form on g, then Im B ( imaginary part of B) is a quadratic form on g, when it is considered as a real Lie algebra. So, in this case we can define ! and! ¯ by Im B on g (as a real Lie algebra). Example 2.3. Suppose g is a semisimple Lie algebra and B = K is its Killing form. Since z(g) = 0, then K induces a 2-plectic structure on g itself, defined by (2:1). We will show this 2-plectic Lie algebra by (g; !; K). Example 2.4. Let (g; [ ; ]) be a Lie bialgebra, i.e, a Lie algebra g whose dual g∗ also has a Lie algebra structure f ; g which satisfies in a compatibility condition (see [5]). ∗ Then g ⊕ g has a Lie algebra structure [ ; ]d defined by ∗ ∗ ∗ ∗ [x + α; y + β]d = [x; y] + fα; βg + adβx − adxβ − adαy + adyα; ∗ ∗ for all x + α; y + β, where adα : g ! g is the adjoint operator. The Lie algebra ∗ (g⊕g ; [ ; ]d) is called the double of g and it is denoted by (d; [ ; ]d). There is a natural symmetric and nondegenerate bilinear form B on d defined by B(x + α; y + β) = α(y) + β(x); 8x + α; y + β: It is well known that B is also invariant (see [5]). Therefor, (d;B) is a quadratic Lie algebra and hence (d; !;¯ B) is a 2-plectic Lie algebra. When g is semisimple, d itself is a 2-plectic Lie algebra ([9]). Example 2.5. Let (g; [ ; ]) be a simple Lie algebra and denote its Killing form by K. n For n > 1, define [ ; ]n on g = g ⊕ ::: ⊕ g by k−1 X [x; y]n = (z1; :::; zn); zk = [xj; yk−j]; j=1 n n where x = (x1; ::::xn); y = (y1; :::; yn) are in g . Then (g ; [ ; ]n) is a nilpotent Lie algebra ([7]). Define the bilinear form B on gn by n X n B(x; y) = K(xj; yn−j+1); x; y 2 g : j=1 On 2-plectic Lie groups 115 The bilinear form B is a quadratic form on gn ([7]). Thus (gn; !;¯ B) is a 2-plectic Lie algebra. Example 2.6. Assume that (gi;Bi), i = 1; :::; n, is a quadratic Lie algebra. On g = g1 ⊕ ::: ⊕ gn consider the bilinear form B = B1 ⊕ ::: ⊕ Bn defined by B(x1 + ::: + xn; y1 + ::: + yn) = B1(x1; y1) + ::: + Bn(xn; yn): Obviously, (g;B) is a quadratic Lie algebra. Suppose that gi contains a central isotropic element zi, i.e, Bi(zi; zi) = 0. Let j be an ideal in g spanned by the elements: ? z1 − zn; :::; zn−1 − zn, and j be the orthogonal complement of j with respect to B. 0 j? 0 0 Put g = j and define the bilinear form B on g by B0(x; y) = B(x; y); x; y 2 j?: Then (g0;B0) is a quadratic Lie algebra ([6]) which is called amalgamated product of quadratic Lie algebras. So, (g0; !;¯ B0) is a 2-plectic Lie algebra. 1 Example 2.7. Consider the set E = fej = − 2 iλj : j = 1; :::; 8g as a basis for the Lie algebra su(3) where, 0 0 1 0 1 0 0 −i 0 1 0 1 0 0 1 λ1 = @ 1 0 0 A ; λ2 = @ i 0 0 A ; λ3 = @ 0 −1 0 A ; 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 −i 1 0 0 0 0 1 λ4 = @ 0 0 0 A ; λ5 = @ 0 0 0 A ; λ6 = @ 0 0 1 A ; 1 0 0 i 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 λ7 = @ 0 0 −i A ; λ8 = p @ 0 1 0 A : 0 i 0 3 0 0 −2 If fe1; :::; e8g is the corresponding dual basis, then its easy to see that ! = δ(e1 ^ e8) is a 2-plectic structure.
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