Complex and bi-Hermitian structures on four-dimensional real Lie algebras

Abstract

We give a new method for calculation of complex and biHermitian structures on low dimensional real Lie algebras. In this method, using non-coordinate basis, we first transform the Nijenhuis tensor field and biHermitian structure relations on Lie groups to the tensor relations on their Lie algebras. Then we use adjoint representation for writing these relations in the matrix form; in this manner by solving these matrix relations and using automorphism groups of four dimensional real Lie algebras we obtain and classify all complex and biHermitian structures on four dimensional real Lie algebras. * AbstractWe give a new method for calculation of complex and biHermitian structures on low dimensional real Lie algebras. In this method by use of non-coordinate basis, we first transform the Nijenhuis tensor field and biHermitian structure relations on Lie groups to the tensor relations on their Lie algebras. Then we use adjoint representation for writing these relations in the matrix form and by solving these matrix relations and use of automorphism groups of four dimensional real Lie algebras we obtain and classify all complex and biHermitian structures on four dimensional real Lie algebras. *

Towards Kaluza-Klein Dark Matter on nilmanifolds

“…by any M ∈ Aut that leaves (2.2) invariant. The M 's such that m a b e b satisfy (2.2) are constrained by 14) condition that, for the Heisenberg algebra, is solved by a matrix of the form (see also [44]): We want to mod out by such elements, i.e. we want to consider in (2.13) only those L's that are not related to each other by some M .…”

Classification of four-dimensional real Lie bialgebras of symplectic type and their Poisson–Lie groups

“…The list of four dimensional real Lie algebras with symplectic structure is given in [13] (see also [14] ); and we brought it in table 1 for self containing of the paper 2 Non-zero commutation relations g Non-zero commutation relations…”

“…In section two, we briefly review the definitions and notations. In section three, after giving the list of four dimensional real Lie algebra of symplectic type [13], [14] based in to [15] (classification of real four dimensional Lie algebras); we classify four dimensional real Lie bialgebras of symplectic type according to the method given in [12]. In section four, we determine the coboundary Lie bialgebras from the list obtained in section three.…”

Perturbed Wess-Zumino-Witten models and N=(2,2) supersymmetric sigma models on Lie groups with complex structure

“…7 For N=(2,2) supersymmetric WZW models on Lie group G, we have H ABC = f ABC . In this case, relations (8) and (9) show that we have the Lie bialgebra structures on g; 4-6 and relation (10) reduces to (8), and (11) is automatically satisfied, i.e., Lie bialgebra structure is a special case of algebraic bi-Hermitian structure (J, G, H ) with H ABC = f ABC .…”

“…[4][5][6] Furthermore, recently the algebraic structure associated with the bi-Hermitian geometry of the N=(2,2) supersymmetric sigma models on Lie groups has been found in Ref. 7. In Ref.…”