The Erwin Schrodinger International Pasteurgasse 6/7. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Pasteurgasse 6/7 Institute for Mathematical Physics A-1090 Wien, Austria Poisson and Symplectic Structures on Lie Algebras. I. D.V. Alekseevsky A.M. Perelomov Vienna, Preprint ESI 328 (1996) May 2, 1996 Supported by Federal Ministry of Science and Research, Austria Available via

2 POISSON AND SYMPLECTIC STRUCTURES ON LIE ALGEBRAS. I. D.V. Alekseevsky * Max-Planck-Institute fur Mathematik, Gottfried-Claren-Strasse 26, D Bonn, Germany and A.M. Perelomov ** Departamento de Fisica Teorica, Univ. de Valencia Burjassot ( Valencia ), Spain * daleksee@mpim-bonn.mpg.de ** On leave from Institute of Theoretical and Experimental Physics, Moscow, Russia; perelomo@evalvx.ic.uv.es

3 Contents Introduction 1 Poisson brackets on a Lie algebra 1.1 The Schouten bracket in the space of polyvectors and Poisson bivectors 1.2 Poisson brackets induced by Poisson bivector on a G-manifold 1.3 Support of a Poisson bracket and symplectic structures on Lie algebras 2 Decompositions of a Lie algebra with a Poisson bivector or a closed 2-form 3 Classication of a Poisson bivectors on some Lie algebras 4 Decomposition of the Borel subalgebra B into a sum of elementary algebras and closed 2-forms and symplectic structures on B 2

4 Introduction The purpose of this paper is to describe a new class of Poisson brackets on simple Lie algebras and symplectic Lie algebras. This gives a new class of solutions of the classical Yang{Baxter equation. Let us rst recall some basic facts on the classical Yang{Baxter equation, referring the reader for more details to the well-known paper by Belavin and Drinfel'd [BD 1982]. The classical Yang{Baxter equation (CYBE) is the functional equation [X 12 ( 1 ; 2 ); X 13 ( 1 ; 3 )] + [X 12 ( 1 ; 2 ); X 23 ( 2 ; 3 )] + [X 13 ( 1 ; 3 ); X 23 ( 2 ; 3 )] = 0 (0:1) for the function X(; ) taking the values in G G, where G is the Lie algebra. In order to dene the quantity X 12 ( 1 ; 2 ), following [BD 1982], we x an associative algebra A with unit, which contains G and the linear maps ' 12 ; ' 23 and ' 13, so that ' 12 : G G! A A A; ' 12 (a b) = a b 1 (0:2) and analogously for maps ' 23 and ' 13. Note that if X(; ) is a solution of eq. (0.1) and '(u) is a function with values in G, then ~ X(; ) = ('() '()) X(; ) is also a solution of (0.1) and we will consider the solutions X and ~ X as equivalent. Let us introduce the following denition. Denition 0.1. The function X(; ) is invariant relative to g 2 Aut G, if (g g) X(; ) = X(; ): The set of all such g forms the group that is called the invariance group of X(; ). The function X(; ) is said to be invariant with respect to h 2 G if [h h; X(; )] = 0; i.e. if it is invariant relative to expfad hg for any t. Note that if X(; ) is a solution of (0.1), which is invariant relative to the subalgebra H G, and if a tensor r from H H satises the following Yang-Baxter equation [r 12 ; r 13 ] + [r 12 ; r 23 ] + [r 13 ; r 23 ] = 0; (0:3) r 21 =?r 12 ; (0:4) then the function ~ X(; ) = X(; ) + r is also a solution of (0.1). Note also that if the algebra H is Abelian, then (0.3) is satised automatically. It is usually supposed that the Lie algebra G is a nite-dimensional simple Lie algebra over C. In [BD 1982] the solutions of (0.1) have been studied in details, such that: (i) X(; ) is a meromorphic function; ; D; D is a domain in C ; (ii) the determinant of the matrix formed by the coordinates of the tensor X(; ) is not identically zero; (iii) X(; ) depends only on the dierence (? ). 3

5 In fact, as was shown in [BD 1983], the condition (iii) indeed follows from (i) and (ii). In the paper [BD 1982] it was shown that solutions of (0.1) are of three types: a) Elliptic solutions, b) Trigonometric solutions, c) Rational solutions, and all elliptic and trigonometric solutions were found. As for rational solutions, only few of them were found, and the main purpose of this paper is to extend this class. The paper is organized as follows. In section 1 we recall some standard denitions and facts about Poisson structures. In the section 2 a decomposition of a Lie algebra G into sum of two subalgebras is considered and relations between Poisson and symplectic structures on G and its subalgebras are studied. Their results are used in the section 3 to describe explicitely closed 2-forms and Poisson structures on the elementary Lie algebra E n+1 which is the Iwasawa subalgebra of su(2; n). The main result of this section is Theorem 3.7. It reduces the description of Poisson and symplectic structures of a Lie algebra G, which is semi-direct sum of a subalgebra F and the ideal E n+1 to the description of such structures on F. This gives an algorithm for description of all closed 2-forms and of symplectic structures on any Lie algebra which is decomposed into semi-direct sum of elementary subalgebras. In the section 4 we construct canonical decomposition of the Borel subalgebra B(G) of a semisimple Lie algebra G into semi-derect sum of elementary subalgebras (plus, may be, a commutative subalgebra of the Cartan subalgebra). Applying the results of the section 3, we obtain a description of closed 2-forms and symplectic forms (if they exist) on the Borel subalgebra B(G) of a semisimple Lie algebra G. As a biproduct, we get description of the second cohomology group H 2 (B (G)). Acknowledgments. Both authors thank the Erwin Schrodinger Institute for Mathematical Physics where the work has been began, D.A. thanks Max-Planck-Institute fur Mathematik, Bonn and A.P. thanks Depart. de Fisica Teorica of University of Valencia, Spain for hospitality and support. 4

6 1. POISSON BRACKETS ON A LIE ALGEBRA 1.1. The Schouten bracket in the space of polyvectors and Poisson bivectors Let G be a Lie algebra and ^G = P ^ig be the exterior algebra over G. The Lie bracket on G determines naturally the bracket on ^G : [x 1 ^ ::: ^ x p ; y 1 ^ : : : ^ y q ] = = (?1) p?i+j?1 x 1 ^ :::^x i ::: ^ x p ^ [x i ; y j ] ^ y 1 ^ :::^y j ::: ^ y q ; x 1 ; :::; x p ; y 1 ; :::; y q 2 G: This bracket is called the Schouten bracket and it turns the space ^G into a graded Lie superalgebra. Denition A bivector 2 ^2G is called a Poisson bivector (or a Poisson bracket in the Lie algebra G) if it commutes with itself [; ] = 0 (1:1) (this is equivalent to the classical Yang{Baxter equation 0.3.). 2. Two Poisson bivectors 1 ; 2 are called compatible if they commute: [ 1 ; 2 ] = 0: (1:2) Note that this is equivalent to the fact that any linear combination is a Poisson bivector. If fe i g is a basis in G, then may be written as = X ij e i ^ e j and [; ] = X ij kl [e i ^ e j ; e k ^ e l ]; where the bracket of two simple bivectors is given by [x ^ y; u ^ v] = [x; u] ^ y ^ v + x ^ [y; u] ^ v + y ^ [x; v] ^ u + x ^ u ^ [y; v]: 5

7 1.2. Poisson bracket induced by a Poisson bivector on a G-manifold Let M be a G-manifold, i.e. the manifold with xed homomorphism ' : G! X (M) of a Lie algebra G into the Lie algebra X (M) of vector elds on M. The homomorphism ' may be extended to a homomorphism ' : ^G! ^(M) of the Lie superalgebra of polyvectors on G into the Lie superalgebra of polyvector elds on M (relative to the Schouten bracket). In particular, the Poisson bivector = ij e i ^ e j 2 ^2G determines the bivector G() = ij '(e i ) ^ '(e j ) on the manifold, such that ['(); '()] = 0. This bivector denes the Poisson bracket in the space of functions on the manifold according to the formula ff; gg = '()(df; dg) = ij (X i f) (X j g); X i = '(e i ): In particular, because the Lie algebra G acts naturally in G and also in G, the Poisson bivector determines Poisson brackets in the spaces of functions on G and G. These Poisson brackets are dened by fe i ; e j g = ij ; (1:3) where fe j g is the basis of the space G, which is considered as the space of linear functions on G and respectively by f; g = X ij (ad e i )(ad e j ); ; 2 G : Note that bivector elds, corresponding to brackets, have the form G = ij e i ^ e j ; G = ij C a ik Cb jl xk x l e a ^ e b ; where e a is the basis of G dual to the basis e i of G, and C a ik are structure constants of G. 6

8 1.3. Support of a Poisson bracket and symplectic structures on Lie algebras Let be a bivector on a Lie algebra G. Then determines the linear mapping : G! G;! = i : Denition 1.2. The subspace G = (G ), which is the image of this linear mapping, is called the support of the bivector 2 ^2G. Lemma 1.3. The support G of the Poisson bivector of the Lie algebra G is the Lie subalgebra of G. Recall that a symplectic form (or a symplectic structure) on a Lie algebra G is a closed non-degenerate two-form! 2 ^2G. The closeness condition means that 0 = d!(x; y; z) = x;y;z!([x; y]; z); x; y; z 2 G; where x;y;z denotes the sum of cyclic permutations of x; y; z. If! is a symplectic form, then the tensor =!?1 is a Poisson bivector. More generally, we have Proposition 1.4. Let A be a subalgebra of the Lie algebra G and! be a symplectic form on A. Then the inverse tensor =!?1 2 ^2A ^2G is a Poisson bivector with support A, and any Poisson bivector may be obtained using this construction. Hence the classication problem for Poisson bivectors on the Lie algebra G reduces to the classication of Lie subalgebras A G with a symplectic form. 7

9 2. DECOMPOSITIONS OF A LIE ALGEBRA WITH A POISSON BIVECTOR OR A CLOSED 2-FORM Proposition 2.1. Let G be a Lie algebra with a non-degenerate Poisson bivector and! =?1 the associated symplectic form. Let = be a decomposition of into the sum of two bivectors i and A i = supp i. Assume that A 1 \ A 2 = 0. This means that G = A 1 + A 2 is a decomposition of G into the sum of!-non-degenerate subspaces and i = (!j Ai )?1. Then i) A 1 is a subalgebra $ 1 is a Poisson bivector, ii) A 1 ; A 2 are subalgebras $ 1 and 2 are commuting Poisson bivectors, iii) assertion (ii) holds if [ 1 ; 2 ] = 0 or A 1 is an ideal. Proof. It follows immediately from the remarks that i 2 ^2A i and [^2A i ; ^2A i ] ^2A i ^ [A i ; A i ]; [^2A 1 ; ^2A 2 ] A 1 ^ A 2 ^ [A 1 ; A 2 ]: The assertion (iii) implies the following Corollary 2.2. Let G be a Lie algebra with a symplectic form! and A is nondegenerate ideal of A. Then!-orthogonal complement A? to A in G is a subalgebra. Proposition 2.3. Let G = A 1 + A 2 ; A 1 \ A 2 = 0 be a decomposition of a Lie algebra G into a direct sum of two ideals, and = be the corresponding decomposition of a Poisson bivector ; i 2 ^2A i ; 0 2 A 1 ^ A 2. Then 1 ; 2 are commuting Poisson bivectors. Proof. This follows from relations [^2A i ; ^2A i ] ^3A i [^2A 1 ; A 1 ^ A 2 ] ^2A 1 ^ A 2 ; [A 1 ^ A 2 ; A 1 ^ A 2 ] A 1 ^ (^2A 2 ) + ^2A 1 ^ A 2 : Proposition 2.4. Let G = A + V be a semi-direct sum of a Lie subalgebra A and a commutative ideal V. Let be a Poisson bivector and = A + V + 0 be its decomposition. Then i) A ; V are Poisson bivectors, ii) [ A ; 0 ] = [ V ; 0 ] = 0; iii) [ 0 ; 0 ] + 2[ A ; V ] = 0: In particular, A ; V are commuting bivectors i 0 is a Poisson bivector. Corollary 2.5. Under the notation of Proposition 2.4, assume moreover that the sum is direct, i.e. V is a central subalgebra. Then a bivector with the decomposition = A + V + 0 is a Poisson bivector if and only if i) A is a Poisson bivector, ii) 0 2 C A ( A ) ^ V; where C A ( A ) = fa 2 A; ad a A = 0g 8

10 is the stability subalgebra of A, and iii) supp 0 \ A is a commutative Lie algebra. In this case A ; V ; 0 are mutually commuting Poisson bivectors. Proof. Calculating the bracket, we obtain [; ] = [ A ; A ; ] + [ A ; 0 ] + [ 0 ; 0 ]: Since the summands belong to the dierent homogeneous components, the left-hand side vanishes i all summands of the right-hand side are equal to zero. It is easy to check that the condition [ A ; 0 ] = 0 is equivalent to the condition (ii) and the condition [ 0 ; 0 ] = 0 gives (iii). Proposition 2.6. Let A = A 1 +A 2 ; A 1 \A 2 = 0 be a decomposition of a Lie algebra G into a sum of two subalgebras and! i be a symplectic form on A i ; i = 1; 2. Then! =! 1 +! 2 is a symplectic form on G i the natural representation ada i of A i (i = 1; 2) into the space G=A i A i 0; fi; i 0 g = f1; 2g is symplectic, i.e. it preserves the symplectic form! i 0. Corollary 2.7. Let (A i ;! i ); i = 1; 2 be two Lie algebras with symplectic forms and ' : A 1! Der(A 2 ) be a representation of A 1 by means of derivations of the Lie algebra A 2. If the linear Lie algebra '(A 1 ) is symplectic, i.e. if it preserves! 2, then the semi-direct sum G = G 1 + G 2 has the symplectic form! =! 1 +! 2. Proof of Proposition 2.6. Let a i ; b i ; c i 2 A i ; i = 1; 2. Then we have d!(a 1 ; b 1 ; c 1 ) = d! 1 (a 1 ; b 1 ; c 1 ) = 0; d!(a 1 ; b 1 ; c 2 ) =!([a 1 ; b 1 ]; c 2 ) +!([b 1 ; c 2 ]; a 1 ) +!([c 2 ; a 1 ]; b 1 ) =!(ad c2 a 1 ; b 1 ) +!(a 1 ; ad c2 b 1 ) = (adc 2! 1 )(a 1 ; b 1 ); d!(a 2 ; b 2 ; c 1 ) = (ad c 1! 2 )(a 2 ; b 2 ): Hence, d! = 0, ad A 1! 2 = ad A 2! 1 = 0: The following Lemma gives a description of closed two-forms on a semi-direct sum of two Lie algebras. Lemma 2.8. Let G = A + B be a semi-direct decomposition of a Lie algebra into a sum of a subalgebra A and an ideal B. Then i) the ^2A-component A of any Poisson bivector on G is a Poisson bivector, ii) any closed 2-form! on G has the canonical decomposition! =! A +! B +! 0 ; (2:1) where! A =! j A;! B =! j B are closed forms on A and B (considered in the natural way as forms on G) and! 0 2 A ^ B ^2G is a 2-form that satises conditions: for a; a 0 2 A; b; b 0 2 B:! 0 (a; [b; b 0 ]) = (ad a! B)(b; b 0 ) =! B ([a; b]; b 0 ) +! B (b; [a; b 0 ]); (2:2)! 0 ([a; a 0 ]; b) +! 0 ([a 0 ; b]; a) +! 0 ([b; a]; a 0 ) = 0 (2:3) 9

11 Conversely, for any closed 2-forms! A ;! B on A and B and a 2-form! 0 2 A ^ B that satises (2.2) and (2.3), formula (2.1) denes a closed 2-form on G. The proof is straightforward. We shall denote by z i (A) the space of closed i-forms on a Lie algebra A. Corollary 2.9. Let G = A + B be the direct sum of two ideals. Then In particular, if [A; A] = A then z 2 (G) = z 2 (A) + z 2 (B) + z 1 (A) ^ z 1 (B): z 2 (G) = z 2 (A) + z 2 (B): As another corollary of Lemma, we have Proposition Let G = A + B be a semi-direct decomposition of a Lie algebra G as in Lemma 2.8. i) Assume that [B; B] = 0. Then the space z 2 (G) of the closed 2-forms on G is given by z 2 (G) = z 2 (A) + z 2 (B) A + z 2 AB; where z 2 (B) A is the space of ada-invariant 2-forms on B and zab 2 is the space of 2-forms from A ^ B that satisfy (2.3). ii) Assume that [A; A] = A; [A; B] [B; B] and let! be a closed 2-form with the canonical decomposition (2.1). If! B is ad A-invariant form, then! 0 = 0 and the decomposition G = A + B is!-orthogonal:!(a; B) = 0. Applying this proposition to the Levi{Malcev decomposition G = S + R of a Lie algebra G, we obtain Theorem Let G = S + R be the Levi{Malcev decomposition of a Lie algebra G, where S is a semi-simple subalgebra and R is the radical. Let! be a closed 2-form on G. Assume that its restriction of! R to R is ad S-invariant. Then!(S; R) = 0 and! =! S +! R, where! S in the restriction of! to S. In particular, the form! is degenerate. Note that if the semi-simple part S is compact, then any closed 2-form on G is cohomologic to a closed ad S-invariant 2-form. Proposition Let A gl(v ) be a linear Lie algebra and G = A + V the associated inhomogeneous Lie algebra, which is the semi-direct sum of the subalgebra A and the vector ideal V. Then the space z 2 (G) of closed 2-forms on G is the direct sum of three subspaces: z 2 (G) = z 2 (A) ^2(V ) A z 1 (A; V ); where ^2(V ) A is the space of ad A-invariant 2-forms on V and z 1 (A; V ) = f! 2 A ^ V j!([a; B]; x) =!(A; Bx)?!(B; Ax); A; B 2 A; x 2 V g: We note that we may consider z 1 (A; V ) as the space of closed V -valued 1-forms on A, where the dierential d! of a 1-form! : G! V is given by d!(a; B) =!([A; B])?!(B)A +!(A)B: 10

12 (Here we denote by 7! A the action of A 2 A on the 1-form 2 V ; (A)(x) = (Ax) for x 2 V.) Remark that any 1-form 2 V may be considered as a 0-form on A with values in V and, hence, it denes the exact 1-form! = d 2 dz 0 (A; V ) z 1 (A; V ) :! (A; x) = (Ax): Corollary Assume that H 1 (A; V ) = 0: Then z 2 (G) = z 2 (A) ^2(V ) A dz 0 (A; V ): Corollary Assume that the action of A onto V preserves no non-zero 2-form on V, i.e. 2 (V ) A = 0 and dima < dimv. Then any closed 2-form! on G is degenerate. In particular A does not admit a symplectic form. Proof. Since ^2(V ) A = 0, any closed 2-form may be written as! =! A +! 0, where! A 2 z 2 (A);! 0 2 z 1 (A; V ). Since dim! 0 (A; V ) < dimv, there exists v 2 V such that! 0 (A; v) = 0. It belongs to the kernel of!. 11

13 3. CLASSIFICATION OF POISSON BIVECTORS ON SOME LIE ALGEBRAS Let A be a commutative subalgebra of a Lie algebra G. Then any bivector 2 ^2A ^2G is a Poisson bivector. It has commutative subalgebra supp A as the support. The following simple proposition gives the complete description of all Poisson bivectors in a compact Lie algebra. Proposition 3.1. Any Poisson bivector on a compact Lie algebra G has a commutative support. Proof. Since any subalgebra of a compact Lie algebra is a compact Lie algebra, i.e. the Lie algebra of a compact Lie group, the support A = supp of a Poisson bivector is a compact Lie algebra with a non-degenerate Poisson bracket. A compact Lie algebra A is the direct sum of a compact semi-simple Lie algebra A 0 and a commutative subalgebra B. By Corollary 2.9 the symplectic form! =?1 on A is the sum of symplectic form! 0 of A 0 and a symplectic form! B of A 0. To nish the Proof, we must show that A 0 = 0. This follows from the well-known Lemma 3.2. Any closed 2-form! on a semi-simple Lie algebra G is exact, i.e. it has the form! = d for some 1-form 2 G. Its kernel Ker! 6= 0 and it coincides with the centralizer in G of the vector X = B?1 2 G associated with by means of the Killing{Cartan form B of G. In particular, there is no symplectic form on G. This shows that A 0 = 0 and proves Proposition 3.1. Now we associate with a symplectic vector space (V; ) over eld k = R; C some (2n + 2)-dimensional Lie algebra E n+1 with the canonical symplectic form! can and the canonical Poisson bivector can =! can?1. It is dened as follows: E n+1 = ke 0 + ke 1 + V 2n = kh + kr + kfp j ; q k g; [e 1 ; V n ] = 0; [x; y] = (x; y)e 1 ; x; y 2 V n ; [e 0 ; e 1 ] = 2e 1 ; ad e 0 jv n = 1: can = 1 2 e 0 ^ e 1 + X (p i ^ q i );! can = de 1 = 2e 0 ^ e 1 + : Here fp j ; q k g denotes a standard symplectic base of V n :!(p i ; p j ) =!(q i ; q j ) = 0;!(p i ; q k ) = jk : The base fh = e 0 ; r = e 1 ; p i ; q j g will be called a standard base of E n+1. The dual base of the dual space is denoted by fe 0 = h ; e 1 = r ; p i ; q j g. 12

14 Following [PS 1962] we will call E n+1 an elementary Kahler algebra. It is the Lie algebra of the Iwasawa subgroup AN of the Lie group G = SU(n; 2) = KAN. Lemma 3.3. Any closed 2-form on E n+1 is exact and is a linear combination of the form! can and a form of the type dv = e 0 ^ v ; v 2 (V n ) : (3:1) The form is degenerate if and only if it is given by (3.1). Corollary 3.4. Any non-degenerate Poisson bivector on E n+2 may be written as Lemma 3.5. The stabilizer = can + e 1 ^ v; v 2 V n ; 0 6= 2 k: C En+1 () = fx 2 E n+1 ; (ad x) = 0g of any non-degenerate Poisson bivector = can + e 1 ^ v is equal to C En+1 () = ke 1 : We say that a bivector is homogeneous of weight k if (ad e 0 ) = k: Note that any symplectic subspace U 2m of the dimension 2m of the symplectic space V 2n denes a subalgebra E m+1 = ke 0 + U 2m + ke 1 of E n+1. It will be called a standard subalgebra of E n+1. Lemma 3.6. Let be a homogeneous Poisson bivector on E n+1 of weight 2. Then either A = supp () is a standard subalgebra of E n+1 and = can is the canonical Poisson bivector of the elementary algebra A, or A is a commutative subalgebra. Proof. We may write as = e 0 ^ e 1 + V ; where V 2 ^2V. If = 0, then C = supp is a commutative subalgebra of V. Assume now that 6= 0. Then C = supp = k fe 0 ; e 1 g + W; where W is a subspace of V. We can write C as a semi-direct sum C = A + B, where B is the kernel of the canonical symplectic form on W and A = C fe 0 ; e 1 g + U is a standard subalgebra. Since B is a commutative ideal, we can apply Proposition Note that ad e0 j ^2 B =?2 id: 13

15 Hence z 2 (B) A = 0. We claim that zab 2 = e 0 ^ B. Indeed, for! 0 2 zab 2 ; b 2 B we have Hence, moreover, the equations 0 = d! 0 (e 0 ; e 1 ; b) =! 0 ([e 0 ; e 1 ]; b) +! 0 ([e 1 ; b]; e 0 ) +! 0 ([b; e 0 ]; e 1 ) = 2! 0 (e 1 ; b) +! 0 (e 1 ; b) = 0:! 0 (e 1 ; B) = 0; d! 0 (e 1 ; a; b) = d! 0 (a; a 0 ; b) = 0 for a; a 0 2 U; b 2 B are satised automatically. The equation 0 = d! 0 (e 0 ; a; b) =! 0 ([e 0 ; a]; b) +! 0 ([a; b]; e 0 ) +! 0 ([b; e 0 ]; a) = 2! 0 (a; b) means that! 0 (U; B) = 0. Hence! 0 2 e 0 ^ B and z 2 AB = e 0 ^ B. Applying Proposition 2.10, we have z 2 (C) = z 2 (A + B) = z 2 (A) + e 0 ^ B. Lemma 3.3. shows that any closed form on C has the form! can + e 0 ^ v ; v 2 U + B ; where! can is the canonical symplectic form on A. It is degenerate if B 6= 0. On the other hand, the bivector denes a non-degenerate closed 2-form?1 on C = supp. Hence, B = 0 and Lemma is proved. The following theorem describes all closed 2-forms on Lie algebra which admits an ideal isomorphic to the elementary algebra. Theorem 3.7. Let G be a Lie algebra with semi-direct decomposition G = F + E; where the ideal E = ke 0 + ke 1 + V is isomorphic to the elementary Lie algebra and the sublagebra F commutes with e 0 and has semi-direct decomposition F = A + F 0 ; F 0 = [F; F]; [A; A] = 0: Then any closed 2-form! on G can be written as! =! F +! can + du + e 0 ^ a ; where 2 k;! F ;! can are the trivial extension to G of the restriction! j F and the canonical form of E; u 2 V ; a 2 A. The form! depends on 1 + dim A parameteres. The form! is non degenerate i 6= 0 and the system of equations! F (f; f i ) = 1= ([f i ; u]; [f; u]); where f i is a basis of F and u =?1 u has only trivial solution. 14

16 Proof. Changing the commutative subalgebra A if necessary, we may assume that it commutes also with e 1. By Lemmas 2.8 and 3.3, a closed form! on G can be written as! =! F +! E +! 0 ; where! F ;! E =! can + e 0 ^ u are closed forms on F; E respectively, considered as forms on G and! 0 2 F ^ E satises the equations (2.2), (2.3). A direct calculations show that these equations are equivalent to the following relations! 0 (F; e 1 ) = 0;! 0 (f; v) = u ([f; v]);! 0 (F 0 ; e 0 ) = 0 for all f 2 F; v 2 V. We can rewrite! in the following form: where! 00 2 E ^ F satises the relations! =! F +! can + du +! 00 ;! 00 (F; ke 1 + V ) = 0;! 00 (F 0 ; e 0 ) = 0: Hence,! 00 = e 0 ^ a for some a 2 A. It remains to study when! is non degenerate. We may assume that 6= 0, because in the opposite case e 1 belongs to the kernel of!. Assume that a vector z = f + e 0 + e 1 + v belongs to the kernel of!. Then 0 =!(z) =! F (f) + (e 0? e 1 + v)+ +ad f u? u + ad v u + a (f)e 0? a : Projecting this vector equation onto F ; e 0; e 1; V we obtain the following system:! F f + ad v u? a = 0; + a (f) = 0; = 0; v + ad f u? u = 0: Hence, = 0; =?1= a (f)? v =?1 ad f u = ad f and the kernel is determined by solutions f of the equation! F f = 1= u ad [f;u] : This proves Theorem. Corollary 3.8. For a closed 2-form! the following conditions are equivalent : 1)!j E =! can ; 2) u = 0; 3)! is the sum of eigenvectors of the operator ad e0 with the eigenvalues 0 and

17 If [F; E] = V, these conditions are equivalent to 4)!(F; E 0 ) =!(F; ke 1 + V ) = 0: Corollary 3.9. Assume that [F; E] = V. Then any closed form! on G with!(f; E) = 0 is given by! =! F +! can ; where! F is a closed form on F. It is non-degenerate i 6= 0 and! F is a non-degenerate closed form on F trivially extended to G. Proof. Assume that!(f; E) = 0. Then a = 0. Suppose that u = 0. Then there exist f 2 F and v 2 V such that u ([f; v]) 6= 0. Hence,!(f; v) = du (f; v) = u ([f; v]) 6= 0: We come to a contradiction. The Lie algebra G is called Frobenius one if it admits an exact symplectic form! = d. In other words, this means that the coadjoint action of the corresponding group G has an open orbit Ad G. Corollary Under the assumption of Theorem 3.7, the Lie algebra G is Frobenius one i the Lie algebra F is Frobenius. Moreover, any exact form on G can be written as! =! F +! E =! F + d(e 1 + v ); where! F is an exact form on F and v 2 V. In particular, a closed form! is exact i!(f; E) = 0 and! j F is exact. Proof. It follows from Theorem 3.7 and Lemma 3.3. Remark. This corollary reduces the problem of description of open coadjoint orbits of the group G with the Lie algebra G to the same problem for the subgroup F, corresponding to the subalgebra F. Denote by z 2 (G) (resp., dg ) the space of closed, (resp., exact) 2-forms on the Lie algebra G and by H 2 (G) = z 2 (G)=dG the corresponding cohomology group. Remark that the space A F is the space of closed 1-forms on F and such forms are never exact. Using this we derive from Theorem 3.7 and Corollary 3.10 the following Corollary Under the notation of Theorem 3.7, assume that the elementary algebra E = E n+1 has dimension 2n + 2. Then 1) dim z 2 (G) = dim z 2 (F) + dim A + 2n + 1; 2) dim dg = dim df + 2n + 1; 3) dim H 2 (G) = dim H 2 (F) + dim A = dim H 2 (F) + dim H 1 (F): 4) If F admits a symplectic structure then symplectic structures on G depend on dim z 2 (G) parameters. We say that a Poisson bivector is consistent with a semi-direct decomposition G = F + E if it is a sum of two bivectors F ; E with support in F and E, respectively. Then by Proposition 2.1 F ; E are commuting Poisson bivectors. We have Corollary Under the notation of Theorem 3.7, any Poisson bivector on G which is consistent with the decomposition G = F + E is given by = F + E = F + can + e 1 ^ v; 2 k; 16

18 where F is a Poisson bivector on F, can = 1=2 e 0 ^1 + p i ^ q i and v 2 V is a vector commuting with the subalgebra supp F. Proof. By Corollary 3.4, any Poisson bivector on E has the form E = F + can + e 1 ^ v for some v 2 V. Since (ad F) can = 0, we have [ F ; can ] = 0. Hence, the bivectors F ; E commute i [supp F ; v] = 0. This proves Corollary. 4. CLOSED 2-FORMS AND SYMPLECTIC STRUCTURES ON THE BOREL SUBALGEBRA OF THE SEMISIMPLE LIE ALGEBRA. Using the induction, we can apply the results of Section 3 to any Lie algebra G which is decomposed into semidirect sum G = E 1 + : : : + E k of elementary subalgebras such that for any i > 1, E 1 + : : : + E i is a subalgebra with the ideal E i and the complementary subalgebra E 1 + : : : + E i?1. Now we prove that the Borel subalgebra of the semisimple (complex or normal real) Lie algebra admits such semidirect decomposition (where, sometimes, also a subalgebra of the Cartan subalgebra appears). Let G be a semisimple (complex) Lie algebra and R corresponding root system with respect to the Cartan subalgebra H. Recall that a subset Q R is called to be closed if (Q + Q) \ R Q: Such subset denes a regular subalgebra G(Q) of G, generated by the root vectors E ; 2 Q. More generally, two closed subsets P; Q of R dene the regular subalgebra B = G(P )+ G(Q) with the ideal G(P ) i (P + Q) \ R P: Denote by R + a system of positive roots of G and by the highest root of R +. We set and R = f 2 R + j? 2 R + [ f0gg = f 2 R + j (; ) > 0g Q = R +? R = f 2 R + j(; ) = 0g: Proposition R ; Q are closed subsets of roots and (Q + R ) \ R R. 2. The Borel subalgebra B(G) = H + G(R + ) of G and G(R + )admits semi-direct decomposition G(R + ) = E + F ; where E = kh + G(R ) is an ideal and F = H 0 + G(Q ) is a subalgebra. Here H is the highest root vector and H 0 is the orthogonal complement to H into H. 17

19 3. The ideal E is isomorphic to the elementary Lie algebra E n+1, where jr j = 2n+1 and n = h _? 2, h _ is the dual Coxeter number. Proof 1. Note that the highest root is always a long root and we normalize it as (; ) = 2. Then the set R has the form R = R 1 [ fg where R 1 = f j (; ) = 1g since 2(; )=(; ) < 2. Let = + 2 R + for 2 R + ; 2 R +. Then If ; 2 R 1, then (; ) = (; ) + (; ) = 2; and =. If; 2 Q then (; rho) = (; ) + (; ) = = 0 and 2 Q. If 2 R 1 ; 2 Q then (; ) = (; ) + (; ) = 1; and 2 R 1. This proves 1. The statement 2 follows from 1 and the remarks before Proposition From the proof of 1, it follows that (R 1 + R 1 ) \ R + = fg. Hence we can write R = f 1 ; :::; n ; 1 ; :::; n ; g where i + i = ; i = 1; : : : n are the only non trivial relations between the roots from R. This shows that G(R ) is the Heisenberg Lie algebra. Moreover, E is the elementary algebra, because [H ; E i ] = (; i )E i = E i : One can check easily that i =?S i where S is the reection in the hyperplane orthogonal to the root and that n = h _? 2 where h _ is the dual Coxeter number. This proves 3. 18

20 Now we describe a decomposition B(G) = E + F of the Borel subalgebra of a semisimple Lie algebra G explicitly. It is sucient to consider only simple Lie algebras. Recall that there are four series and ve exceptional Lie algebras A n ; B n ; C n ; D n ; E 6 ; E 7 ; E 8 ; F 4 ; G 2 : The basic characteristics of these algebras are given in Table 1. Table 1 Type of group Rank Coxeter number Number of positive roots A n ; n 1 n n + 1 n(n + 1)=2 B n ; n 2 n 2n n 2 C n ; n 3 n 2n n 2 D n ; n 4 n 2(n? 1) n(n? 1) E E E F G According to the book [OV 1990], the next Table 2 contains enumeration of the root system of each simple Lie algebra together with description of the subsystem R = fg [ R 1 ; R 1 = f i ; i j i + i = g; associated with the highest root. 19

21 Table 2 Type of G Roots Highest root Decomposition of, = j + j A n ; n 1 e i? e j e 1? e n+1 j = e 1? e j ; j = e j? e n+1 ; j = 2; : : : ; n B n ; n 2 e i e j ; e j e 1 + e 2 j = e 1 + e j ; j = e 2? e j ; ~ j = e 1? e j ; ~ j = e 2 + e j ; 2n?3 = e 1 ; 2n?3 = e 2 j = 3; : : : ; n C n ; n 3 e i e j ; 2e j 2e 1 j = e 1 + e j ; j = e 1? e j j = 2; : : : ; n D n ; n 4 e i e j e 1 + e 2 j = e 1 + e j ; j = e 2? e j ~ j = e 1? e j ; ~ j = e 2 + e j j = 3; : : : ; n E 6 e i? e j ; 2e 2e jkl = e + e j + e k + e l ; e i + e j + e k e jkl = e? e j? e k? e l ; j; k; l = 1; : : : ; 6 E 7 e i? e j?e 7 + e 8 j =?e 7 + e j ; j = e 8? e j e i + e j + e k + e l jkl = e 8 + e j + e k + e l ; jkl =?e 7? e j? e k? e l j; k; l = 1; : : : ; 6 E 8 e i? e j e 1? e 9 j = e 1? e j ; j = e j? e 9 (e i + e j + e k ) jk = e 1 + e j + e k ; jk =?e 9? e j? e k ; j; k = 2; : : : ; 8 F 4 e i e j ; e j e 1 + e 2 = e 1 ; = e e 2 e 3 e 4 ) j = e 1 + e j ; j = e 2? e j ; j = 3; 4 ~ j = e 1? e j ; ~ j = e 2 + e j ; j = 3; 4 6;7 = 1 (e e 2 + e 3 e 4 ) 6;7 = 1 (e e 2? e 3 e 4 ) G 2 e i? e j ; e j e 1? e 3 1 = e 1 ; 1 =?e 3 2 = e 1? e 2 ; 2 = e 2? e 3 Recall that subsystem of roots S = R +? R consists of positive roots orthogonal to 20

22 the root, and so S is generated by simple roots orthogonal to. Hence S may be easily constructed from the extended Dynkin diagram of the Lie algebra G which corresponds to the set of simple roots and the minimal root (?). The simple roots connected to the root (?) are not orthogonal to the root. The rest roots form the extended Dynkin diagram, which generate subsystem S and the corresponding Borel subalgebra. Note that the number of roots in R is equal to 2h _? 3, where h _ is the dual Coxeter number. Using these remarks, we obtain the decompositions of the Borel subalgebra indicated into Table 3. Here H 0 is the element of the Cartan subalgebra H which corresponds to (n? 1)(e 1 + e n+1 )? 2(e 2 + : : : + e n ) under the identication H = H. Recall that dim (E n ) = 2n. Table 3 B(A n ) = E n + (B(A n?2 ) + kh) 0 B(B n ) = E 2n?2 + (B(B n?2 ) + B(A 1 )) B(C n ) = E n + B(C n?1 ) B(D n ) = E 2n?3 + (B(D n?2 ) + B(A 1 )) B(E 6 ) = E 11 + B(A 5 ) B(E 7 ) = E 17 + B(D 6 ) B(E 8 ) = E 29 + B(E 7 ) B(F 4 ) = E 8 + B(C 3 ) B(G 2 ) = E 3 + B(A 1 ) Using this Table, it is easy to write the explicit formulae for the decomposition of the Borel subalgebra of any semi-simple Lie algebra into the elementary subalgebras. We present the results in Table 4. 21

23 Table 4 B(A n ) = E n + E n?2 + : : : + E 2 (ore 1 ) + H 0 m; m = [ n] 2 B(B n ) = E 2n?2 + E 2n?6 + : : : + E 4 (ore 2 ) + me 1 ; m = [ n+1] 2 B(C n ) = E n + E n?1 + : : : + E 2 + E 1 B(D n ) = E 2n?3 + E 2n?7 + : : : + E 5 + (m + 1)E 1 ; n = 2m B(D n ) = E 2n?3 + E 2n?7 + : : : + E 3 + me 1 + H 1 ; n = 2m + 1 B(E 6 ) = E 11 + E 5 + E 3 + E 1 + H 2 B(E 7 ) = E 17 + E 9 + E 5 + 4E 1 B(E 8 ) = E 29 + E 17 + E 9 + E 5 + 4E 1 B(F 4 ) = E 8 + E 3 + E 2 + E 1 B(G 2 ) = E 3 + E 1 Here H m is the subalgebra of the dimension m of the Cartan subalgebra. Using the results of section 3, we derive now some corollaries from these results. By Corollary 3.10, any subalgebra B which admits a decomposition into a semi-direct sum of the elementary subalgebra is a Frobenius Lie algebra. This means that the coadjoint action of the corresponding Lie group has an open orbit or, in other words, B has an exact symplectic form. Checking Table 4 and using Corollary 3.11, we get Proposition ) The Borel subalgebra of a simple Lie algebra G is Frobenius i G is dierent from A n ; D 2m+1 and E 6. 2) The minimal dimension of the kernel of an exact 2-form (which is equal to the codimension of a regular coadjoint orbit) is equal to m = [n=2] for B(A n ), 1 for B(D 2m+1 ) and 2 for B(E 6 ). 3) The Borel subalgebra admits a symplectic form i its dimension is even. In the opposite case it admits a closed 2-form with one-dimensional kernel. Recall that for the elementary Lie algebra E n the dimension of the space of closed 2-forms is equal to 2n? 1 and H 2 (E n ) = 0, since any closed 2-form is exact (Lemma 3.3). Now we calculate the cohomology H 2 (B(G)) for each simple Lie algebra G. Proposition 4.3. Let G be a simple Lie algebra of rank n. Then dim H 2 (B(G)) = n(n? 1)=2: Proof. Let B = E 1 +: : :+E p +H q be a decomposition of the Lie algebra B(G) of rank n into semi-direct sum of elementary Lie algebras and the commutative q-dimensional Lie algebra H q. Then Corollary 3.11 implies the following formula for the dimension of the second cohomology group: dim H 2 (B) = (n? 1) + : : : + (n? p) + q(q? 1)=2 = (2n? p? 1)p=2 + q(q? 1)=2: 22

24 For the Frobenius Borel algebra, q = 0; p = n and we get the Proposition. Using this formula we check Proposition also for the cases G = A 2m+1 ; A 2m ; D 2m+1 and E 6 : Now the calculation of the dimension of the space of closed 2-forms reduces to the calculation of the dimension of the space of exact 2-form. For the Lie algebra with a semidirect decomposition B = E n + F we have dim d B = 2n? 1 + dim d F by Theorem 3.7 and Corollary More generally, for the Lie algebra with a semidirect decomposition B = E n1 + : : : + E np + H q we get formula dim d B = px i=1 (2n i? 1); i.e dim d B is equal to the number of positive roots of algebra G: Using this formula we calculate the dimension of the space of exact 2-forms db(g) and the space z 2 (B(G)) of closed 2-forms for all simple Lie algebras G. The results are presented in Table 5. Table 5 Type of group G dim z 2 (B(G)) dim db(g) dim H 2 (G) A n ; n 1 n 2 n(n? 1)=2 n(n + 1)=2 B n ; n 2 n(3n? 1)=2 n(n? 1)=2 n 2 C n ; n 3 n(3n? 1)=2 n(n? 1)=2 n 2 D n ; n 4 n(3n + 1)=2 n(n? 1)=2 n(n + 1) E E E F G Let B(G) = E n1 + E n2 + : : : + E nk + H m be the decomposition of the Borel subalgebra of the simple Lie algebra G into the sum of elementary Lie algebras and, may be, the commutative Lie algebra, described in Table 3 denotes by i the canonical Poisson bivector on elementary subalgebra E i and by 0 any bivector on the commutative subalgebra H m. Then Corollary 3.12 implies the following result. Proposition 4.4. The Poisson bivectors i ; i 0 commute each with other and dene the Poisson bevector = 1 + : : : + k + 0 on the Borel subalgebra B(G). 23

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