On symmetric Poisson structure and Lie bracket in linear algebres

(1)

ON SYMMETRIC POISSON STRUCTURE AND LIE BRACKET IN LINEAR ALGEBRES

Jerzy Grochulski

Institute of Mathematics and Computer Science, Czestochowa University of Technology

Abstract. In the paper the symmetric Poisson structure on linear has been applied. A con- nection of this structure with Lie bracket has been detined.

Let V be a linear algebra over R and let

V V

A: × V→

be a skew - symmetric 2-linear mapping satisfying the conditions

(

α β,γ

)

αA

(

β,γ

)

βA

(

α,β

)

A ⋅ = + (i)

( )

(

Aα^,β ^,γ

)

+A

(

A

(

γ^,α

)

^,β

)

+A

(

A

(

β^,γ

)

^,α

)

=⁰

A (ii)

for any α,β,γ∈V.

The mapping A is said to be a Poisson structure on V and the pair (V, A) we will called a Poisson linear algebra.

From definition it follows that for any α∈V the mapping

( )

^V ^V

A

Dα := ⋅,α : → is a derivation of the algebra V.

It is easily to prove.

Proposition 1. The set D(V) of all derivations Dα of V is a linear space over R.

Moreover D(V) is a Lie algebra with the Lie bracket given by

[

Dα,Dβ

]

=Dα ⋅Dβ −Dβ⋅Dα (2) for any ^D_α^,^D_β_∈^D

( )

^V .

Proposition 2. For any α,β∈V

[

α β

]

₍β,α₎

,D D_A

D = (3)

(2)

An elementα∈V is said to be a Casimir element of V with respect to A, if

(

α^,β

)

=⁰

A for any β∈V.The set of all Casimir element of V with respect to A we denote by VC^A. Evidently the pair

(

^V^,^A

)

is a Lie algebra and VC^A is its ideal.

Now let T:V →V be a mapping satisfying the condition

( )

(

T α β

)

A

(

α T

( )

β

)

A , =− , (4)

for any α,β∈V.

Proposition 3. A mapping T:V →V satisfying the condition (4) has the following properties:

(

α+β

)

=T

( )

α +T

( )

β +γ

T (i)

(

^x_⋅^α

)

₌^xT

( )

^α ₊^δ

T (ii) for anyα,β∈V and ,

A

VC

x∈ whereγ andδ are some elements of .

A

VC

Proof. For anyα,β∈V by (4) we have.

( )

( ) ( ( ) ) ( ( ) ) ( ( ) )

( )

(

α γ

) ( ( )

β γ

)

γ β γ

α γ

β α γ

β α

, ,

T A T

A

T A T

A T

A

+

=

−

= +

−

= +

Hence

( ) ( ) ( )

(

T α+β −T α −T β ,γ

)

=0 A

which gives

(

^α +^β

)

=T

( )

^α +T

( )

^β +^γ T

for some .

A

VC

∈ γ

Similarly we have

( )

( ) ( ( ) ) ( ( ) )

( )

(

α β

) ( ( )

α β

)

β α β

α β

α

, ,

,

xT A T

xA

T xA T

x A x

T A

=

−

=

−

=

Hence

( ) ( )

(

T x^,α −xT α ^,β

)

=⁰ A

which gives T

(

xα

)

= xT

( )

α +δ for any α∈V, x∈VC^A where δ is some element of .

A

VC

(3)

One can easily top prove

Proposition 4. A mapping T:V →V satisfying the condition (4) satisfies also the conditions.

(

Tⁿ

( )

^α ^β

) ^{( )}

ⁿA

(

^α Tⁿ

^{( )}

^β

)

A , = −1 , (i)

(

α+β

)

= ⁿ

( )

α + ⁿ

( )

β +γ

n T T

T (ii)

(

^α

)

= ⁿ

( )

^α +^δ

n x xT

T (iii)

for any α,β∈V,x∈VC^A and n ∈N,whereγ andδ are some elements of .

A

VC

Proposition 5. If α∈VC^A then T

( )

α ∈VC^A^. In consequence VC^A is a T-invariant linear subspace of the linear space V.

Proof. Let ,

A

VC

∈

α then for any β∈VA

(

α^,β

)

=⁰^, for any β∈V. Therefore

( )

VC^A^.

T α ∈ Let us put

(

α,β

)

A

(

T

( )

α ,β

)

S = (5)

for anyα,β∈V.

Evidently the formula (5) defines a 2-linear mapping S:V×V →V. Lemma 6. The mapping S defined by (5) is symmetric one.

Prof. From (4) and (5) it follows

(

α,β

)

A

(

T

( )

α ,β

)

A

(

α,T

( )

β

)

A

(

T

( )

β ,α

)

S

(

β,α

)

S = =− = =

for any α,β∈V . Now we will prove

Proposition 7. The mapping S defined by (5) satisfies the identities

( )

(

T α β

)

s

(

α T

( )

β

)

S , =− , (i)

(

α β,γ

)

αS

(

β,γ

)

βS

(

α,γ

)

S ⋅ = + (ii)

( )

(

ST α ,β ,γ

)

+S

(

S

(

T

( )

γ ,α

)

,β

)

+S

(

S

(

T

( )

β ,γ

)

,α

)

=0

S (iii)

(4)

Proof. (i). Using (4) and (5) we get

( )

(

α,T β

)

A

(

TT

( ) ( )

α ,T β

)

A

(

T

( ) ( )

β ,T α

)

S

(

T

( )

α ,β

)

S = =− =−

for any α,β∈V.

(ii) From (4) and (5) as well as from definition of A we get

( ) ( ( ) )

( )

(

β γ

)

β

(

α

( )

γ

)

α

(

β γ

)

β

(

α γ

)

α

γ β α γ

β α

, ,

,

S S

T A T

A

T A S

+

=

−

=

⋅

−

=

⋅

(iii) Analogically we get

( ) ( )

( ) ( ( ( ) ( ) ) ( ) )

( ) ( )

( ) ( ( ( ) ( ) ) )

( ) ( )

( )

( ) ( ( ( ) ( ) ) ) ( ( ( ) ) )

( )

(

^, ^,

) ( ( ( )

^,

)

^,

)

⁰

, , ,

, ,

,

, , ,

,

, , ,

,

= +

+

=

−

+

−

= +

+ +

α γ β β

α γ

γ β α α

γ β β

α γ

γ β α α

γ β

β α γ γ

β α

T S S T

S S

T S S T

T A S T

T A S

T T A S T

T T A A

T T T A A T

T T A A

for any α,β,γ∈V. So, we may accept

Def. 1. A mapping S, defined by (5) is said to be a symmetric Poisson structure on a linear algebra V over R.

From proposition 5 (ii) it follows that for anyα∈V the mapping

( )

^V ^V

S ⋅ →

= ,α : δ

α (6)

is a derivation of the algebra V.

Proposition 8. The set _∆

( )

^V of all derivationsδα of α∈V,is a linear space over R. Moreover ∆

( )

^V is a Lie algebra with a Lie bracket given by

[

δα,δβ

]

=δα⋅δβ−δβ⋅δα

for anyδα^,δβ∈∆

( )

^V ^.

From (1), (5) and (6) it follows the relation

−D

= δα

for anyα∈Vand consequently

[

δα δβ

]

δ ₍ _{( )}α β₎ ,

, ⋅T = _S_T for any α,β∈V.

Def. 2. An element α∈V is said to be a Casimir element of V with respect to S, if

(

α,β

)

=0

S for any β∈V.

(5)

The set of all Casimir elements of V with respect to S we denote by .

S

VC

We shall prove.

Lemma 9. If α∈VC^A then T

( )

α ∈VC^S^.

Proof. Let .

A

VC

∈

α By Proposition 5 T

( )

α ∈VC^S^. Hence by (5)

(

α^,β

)

=A

(

T

( )

α ^,β

)

=⁰ S

for any β∈V.Therefore .

S

VC

∈ α

Lemma 10. α∈VC^S in and only if T

( )

α ∈VC^A^.

Proof. It follows from S

(

α,β

)

= A

(

T

( )

α ,β

)

forβ∈V. Lemma 11. If α∈VC^S then T

( )

α ∈VC^S^.

Proof. Let α∈VC^S then S

(

α,β

)

=0 for any β∈V. Hence S

(

α TT,

( )

β

)

₌

( )

(

,

)

=0

−

= S T α β for any β∈V.ThereforeT

( )

α ∈VC^S^.

Corollary 12. VC^S is T-invariant subspace of the linear space V.

Evidently, if T:V →V is onto then .

A C S

C V

V = In general case there is the inclusion .

A C S

C V

V ⊃

Let us observe also that (V, S) is an algebra, which we shall call a symmetric Lie algebra. Of courseVC^S is an ideal of this algebra.

Let T:V →V be a mapping satisfying the condition

( )

(

α,T β

)

A

( ( )

α ,β

)

A =−

for any α,β∈V . This mapping induces the mapping

( )

^V ^D

( )

^V

D

T : →

* (7)

given by

(

Dα

)

DT_{( )}α

T_* = (8)

for any _{D ∈}α ^D

( )

^V ^.

Lemma 13. The mapping T_* Defined by (8) satisfies the condition

[

TDα Dβ

] [

Dα TDβ

]

*

* , =− , (9)

for any^Dα^,^Dβ∈^D

( )

^V ^.

(6)

Proof. Using from (5) we get for any^Dα^,^Dβ∈^D

( )

^V ^.

[ ] [

_{( )}

]

₍ _{( )}₎ ₍ _{( )} ₎

[

α ( )β

] [

α β

]

α β α

β β

α β

α

D T D D

D

D D

D D D D T

T

T A T

A T

*

, ,

*

, ,

−

=

−

=

−

=

Now let us put

( )

[

Dα₀,Dβ

] [

= T_*Dα,Dβ

]

(10) for any ^Dα^,^Dβ∈^D

( )

^V ^.

It is easily to observe that the formula (10) defines a 2-linear mapping.

[ ] ( )

^⋅^,^⋅ ^:^D

( )

^V ^×^D

( )

^V ^→^D

( )

^V

Lemma 14. The mapping

[ ] ( )

⋅,⋅ defined by (10) is a symmetric one.

Proof. By (9) and (10) we have

( )

[

Dα,Bβ

] [

= T_*Dα,Dβ

] [

=−Dα,T_*Dβ

] [

= T_*Dβ,Dα

]

=

[ (

Dβ,Dα

) ]

for any ^Dα^,^Dβ∈^D

( )

^V ^.

Proposition 15. The mapping

[ ] ( )

⋅,⋅ defined by (10) the following properties

( )

[

TDα Dβ

] [ (

Dα TDβ

) ]

*

* , =− , (i)

( )

[ ]

( )

[

T_*Dα^,Dβ ^,Dγ

]

+

[ ( [ (

T_*Dγ^,Dα

) ]

^,Dβ

) ]

+

[ ( [ (

T_*Dβ^,Dγ

) ]

^,Dα

) ]

=⁰ (ii) for any^Dα^,^Dβ∈^D

( )

^V ^.

Proof. (i) From (9) and (10) we get for any^Dα,^Dβ∈^D

( )

^V

( )

[

^D_α^,^T_*^D_β

] [

= ^T_*^D_α^,^T_*^D_β

]

=−

[

^T_*^D^β^,^T_*^D_α

]

=−

[

^T_*^D_α^,^D_β

]

(ii) Now for any^Dα,^Dβ,^Dγ ∈^D

( )

^V we get

[ ]

[ ] [ [ ] ] [ [ ] ]

[ ]

( )

[ ] [ [ ( ] ) ] [ [ ( ] ) ]

( )

[ ]

( )

[

^, ^,

] [ ( [ (

^,

) ]

^,

) ] [ ( [ (

^,

) ]

^,

) ]

⁰

, , ,

, ,

,

, , ,

, ,

,

*

= +

+

=

−

=

= +

+

α γ β β

α γ γ

β α

α γ β β

α γ γ

β α

α γ β β

α γ γ

β α

D D D T D

D D T D

D D T

D D T D T D

D T D T D D T D T

D T B T D T D T B T D T D T B T D T

(7)

So, we shall accept

Def. 3. The mapping

[ ] ( )

_⋅,_⋅ defined by (10) is said to be a symmetric Lie bracket.

It is easily to prove.

Proposition 16. The mapping ^T_*^:^D

( )

^V ^×^D

( )

^V ^→^D

( )

^V defined by (8) is a linear one over .

S

VC

Let (V, A) be a Poisson linear algebra and let D(V) denotes the Lie algebra of all derivations of V defined by (1). Now, let

( )

^V ^D

( )

^V

D →

ψ: be a mapping satisfying the condition

( )

[

ψ Dα ,Dβ

]

=−

[

Dαψ

( )

Dβ

]

(11) for any^Dα^,^Dβ∈^D

( )

^V ^.

One can easily prove

Lemma 17. A mappingψ ^:^D

( )

^V ^→^D

( )

^V satisfying the condition (11) is a linear one over R.

References

[1] Abraham R., Marsden J.E., Foundations of mechanics, Benjamin, New York 1967.

[2] Sasin W., Żekanowski Z., Some relations between almost symplectic, pseudoriemannien and almost product structures on differential spaces, Demonstr. Math. 1988, 21, 1139-1152.

[3] Multarzyński P., Żekanowski Z., On general Hamiltonian dynamical systems in diferential spaces, Demonstr. Math. 1991, 24 539-555.