Poisson structures on (non) associative noncommutative algebras and integrable Kontsevich type Hamiltonian systems

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Introduction
We present a classical Poisson manifold approach, closely related to construction of integrable Hamiltonian systems, generated by nonassociative and noncommutative algebras.

Poisson structures on non commutative functional manifolds
It is interesting to look at construction of the Hamiltonian operators and revisit it from the classical point of view, considering them as those defi ned on the naturally associated [4,[12][13][14][15][16][17], cotangent space  on the product and respectively, the gradient for any It is well known [18,19], that a linear

Abstract
We have revisited the classical Poisson manifold approach, closely related to construction of Hamiltonian operators, generated by nonassociative and noncommutative algebras.In particular, we presented its natural and simple generalization allowing effectively to describe a wide class of Lax type integrable nonlinear Kontsevich type Hamiltonian systems on associative noncommutative algebras.M Namely, this condition (2.3) was used in the investigations [18,20], to formulate criteria for the operator Then the derivation condition (2.4) can be equivalently rewritten [4,12,[15][16][17], as the strong Lie derivative  which can be rewritten as

Citation
where, by defi nition, with those, analyzed in some detail in [24] .
As it was already mentioned [18,24], based on the matrix Hamiltonian operators prove to be very useful [25,26,27], for describing a wide class of multicomponent hierarchies of integrable Riemann type hydrodynamic systems and their various physically reasonable reductions.

Poisson structures on manifolds generated by associative non commutativ e algebras
Proceed now to a slightly generalized construction of Hamiltonian operators on a phase space, generated by associative noncommutative algebra A -valued matrices, which was fi rst studied in [1][2][3][4], in case of the noncommutative operator algebras and continued later in [5][6][7][8][9][10][11], in case of general associative noncommutative algebras.This natural and simple generalization appeared to be very useful Proof.Symmetricity: We have:  a := = ...
kn Really, the sum of (3.5) under the  -mapping can be now rewritten, respectively, as : a = a : a , = 1, , with some D -coeffi cients from  for all , jn S   depending quadratically on coeffi cients of expansions, staying at uniform and symmetric basis elements of the algebra .
A As the  -mapping sends all of them, by defi nition, to zero, the resulting system (3.5)reduces to the set of algebraic equations   As the loop Lie algebra   allows natural direct sum splitting : a = a : a , , the related two-dimensional matrix loop Lie algebra ) .gl   Moreover, we also will assume that A-algebra valued coeffi cients of the phase space

Conclusion
In this work we succeeded in revisting the classical Poisson manifolds approach to Hamiltonian operators on functional noncommutative manifolds, as well as presented it simple and natural realization, generated by associative noncommutative group algebra.The latter appeared to be very useful for describing a wide class of new Lax type integrable nonlinear Hamiltonian systems on associative noncommutative algebras, interesting for diverse applications in modern quantum physics.
to some linear functional noncommutative manifold M   ,    where  is, in general, a (non)associative noncommutative algebra over a fi eld  ,   is its naturally adjoint space.Then, a Hamiltonian operator on M is defi ned[12,15], bracket (2.1) is determined on M by means of the natural convolution(, )

:
Hentosh OE, Balinsky AA, Prykarpatski AK (2020) Poisson structures on (non)associative noncommutative algebras and integrable Kontsevich type Hamiltonian systems.Ann Math Phys 3(1): 001-006.DOI: https://dx.doi.org/10.17352/amp.000010operator Yet these criteria appear to be very complicated and involve a large amount of cumbersome calculations even in the case of fairly simple differential expressions.So, we have reanalyzed this problem from a slightly different point of view.First, recall that the Jacobi identity for the bracket (2.1) is completely equivalent to the fact that the bracket operator taken arbitrary.This can be easily reformulated as follows: take any element

 11 (
As an example, one can check that a skew-symmetric matrix-differential operator on M of the form for all , = 1, .ijn Similarly, one can verify that the skew-symmetric inverse-differential operator


The condition (2.6) for the operator (2.10) to be Hamiltonian reduces to the constraints on the related nonassociative algebra


representations of the right Leibniz algebra and the new nonassociative Riemann algebra, one can construct many nontrivial Hamiltonian operators related with diverse types of nonassociative noncommutative algebras . These

[ 28 ,Lemma 3 . 1
29,30,31,[8][9][10], for describing a wide class of new Lax type integrable nonlinear Hamiltonian systems on associative noncommutative algebras, interesting for diverse applications in modern quantum physics.We start here with a free associative noncommutative algebra 12 = , ,..., >, m Au uu  generated by a fi nite set of elements {: = 1 , } , j uA j m  and defi ne its "abelianization" := / [ , ] AA A A  and the projection :, AA    where the space [, ] : = AA {uv vu A  :, } .uv A  Consider now a Citation: Hentosh OE, Balinsky AA, Prykarpatski AK (2020) Poisson structures on (non)associative noncommutative algebras and integrable Kontsevich type Hamiltonian systems.Ann Math Phys 3(1): 001-006.DOI: https://dx.doi.org/10.17352/amp.000010naturally related with A n -dimensional matrix Lie algebra := ( ; ) gl n A  over the fi eld  with entries in A subject to the usual matrix commutator [a,b] := ab ba  for all a,b . Being fi rst interested in the Lie-algebraic studying [14,15,32,33], of co-adjont orbits on the adjoint space ,   let us construct a bi- linear form <|> : A      on the Lie algebra  by means of the trace-type expression <a|b>:= tr(a b) The bilinear form (3.1) on  is symmetric, nondegenerate and ad-invariant.

3 )
Assume that <a|b>=0 A   for a fi xed a  and all b. state that a=0, let us put then b=a and obtain Taking into account that the associative algebra is generated by the fi nite set of elements {: = 1 , } ,

Proposition 3 . 2
The construted Lie algebra  is ad-invariant and  -metrized.Proof.Really, from the symmetry property (3.2) one easily obtains that < a|[b,c] >=< [a,b]|c > 0 mod  (3.9) modulo  -mapping for any elements a,b and c.  As the bilinear form (3.1 is non-degenerate, one has ,    that jointly with the ad-invariance property (3.9) means that the Lie algebra  is metrized.Being interested in constructing integrable noncommutative dynamical systems on the algebra , A we need to introduce into our analysis a "spectral" parameter ,    responsible for the existence of infi nite hierarchies of the corresponding dynamical systems invariants, guaranteeing their integrability.This wil be done in next Section, devoted to the Lie-algebraic analysis on loop-Lie-algebras, related with the Lie algebra ,  introduced above.Consider now the Lie algebra {, [ ,] } ,   constructed above, and the related loop Lie algebra the corresponding  -valued Laurent series with respect to the parameter ,

. 13 )
their adjoint spaces with respect to the bilinear form (3.11) split the adjoint loop space = where, by defi nitions, P     are the projections on the corresponding subspaces .    It is a well known property [14,15,32,33] that the deformed Lie product [a,b] := [ a,b] [a, b] any a,b     satisfi es the Jacobi condition and generates on the loop Lie algebra   a new Lie algebra structure.Within the classical Adler-Kostant-Symes Lie-algebraic approach, or its  -matrix structure generalization [14,15,32,33], the adjoint loop space    is then endowed with the modifi ed Lie-Poisson structure {l(a), l(b)} := (l|[a, b] ),      (3.15) for any basic functionals l(a), l(b) it possible to construct integrable Hamiltonian fl ows on the associative algebra A as Poissonian fl ows on the co-adjoint orbits on the adjoint space ,    generated by suitable loop Lie algebra   Casimir gradient elements.Namely, if an element l      is fi xed, the corresponding Hamiltonian fl ow on    subject to the deformed Poisson bracket (3.15) and a Casimir funcrtional () I      possesses the well known Lax type [33-40]t   is a related evolution parameter.The example of this construction and its Lie algebraic properties are discussed in the next Subsection.
is metrized subject to the bi-li near product (3.11) and generated by affi ne elements


fl ow on points l      with respect to the Poisson bracket (3.15) in the Lax type form (3.17).To analyze this fl ow in detail, let us put, by defi nition, that the seed orbit point l    is given by the following  -squared expression


.2) are representable subject to the basis of A as to the Lax type representation (3.15), the Kontsevich dynamical system (4.7)proves to be equivalent to the following matrix commutator equation l/ =[l, (l)any    in the Lie algebra ,   where the A-valued matrix a complete set of  -commuting to each other conservation laws of the Kontsevich dynamical system (4.7),thus proving its generalized integrability.Moreover, choosing both another group algebra and orbit elements l   ,  one can construct the same way many other integrable Hamiltonian systems on the associative noncommutative phase space ,A that is planned to be a topic of a next investigation.

Kontsevich type integrable systems on unital fi nitely ge- nerated free associative noncommutative algebras
The algebra A is infi nite dimensional with the countable basis