Non-Commutative Ricci and Calabi Flows

Abstract

Starting from an improved version of the bicomplex structure associated the continual Lie algebra with non-commutative base algebra, we obtain dynamical systems resulting from the bicomplex conditions. General expressions for conserved currents associated to a continual Lie algebra bicomplex are found explicitly in two first orders. The Moyal-product counterparts for two-dimensional Ricci and Calabi flow equations depending on non-commutative variables are introduced.

Noncommutative Root Space Witt, Ricci Flow, and Poisson Bracket Continual Lie Algebras

“…In particular, these algebras can define generalizations of certain exactly solvable models. Integrable models defined in noncommutative spaces with ordinary Lie algebras as an algebraic origin were constructed in [8,9,10,24]. Using the bicomplex construction [22] for continual Lie algebras with noncommutative root spaces one derives associated dynamical systems in noncommutative spaces.…”

“…c) The identities ( 5) and ( 8) are trivially satisfied by K 0,0 (24). Then we check ( 6) for ( 22) and ( 24):…”

“…It is easy to see that both sets obey the conditions ( 5)-( 8). The bicomplex construction [4,24] based on generators of this Lie algebra leads to the simplest example of the Ricci flow equation.…”

“…In this paper we introduce generalizations for the Witt [6], Ricci flow [2,3,4,24], and Poisson bracket [18], as well as other examples of continual Lie algebras with noncommutative root spaces. The noncommutative root space E we use is the space of tensor product powers of a space E which is endowed with a noncommutative product.…”

“…Many interesting properties of continual Lie algebras have been found useful in applications within the group-theoretical approach [13] to the construction of exactly solvable dynamical systems [18,19,2,3,4,23,22]. The structure of commutations relations and bilinear mappings K turned out to be very helpful in the development of noncommutative generalizations for known integrable models [23,22,24]. Continual Lie algebras with noncommutative root space we introduce inspire various new applications in integrable models.…”