Algebraic Inverses on Lie Algebra Comultiplications

In this note, we investigate algebraic loop structures and inverses of elements of a set of all homomorphisms of Lie algebras with a binary operation derived from a Lie algebra comultiplication. As a symmetry phenomenon, we show that if l ( 1 ) c and r ( 1 ) c are the left and right inverses of the identity 1 : L → L on a free graded Lie algebra L , respectively, based on the Lie algebra comultiplication ψ c : L → L ⊔ L , then we have l ( 1 ) = l ( 1 ) c and r ( 1 ) = r ( 1 ) c , where c : L → L ⊔ L is a commutator.

Algebraic loop structures on algebra comultiplications

In this paper, we study the algebraic loop structures on the set of Lie algebra comultiplications. More specifically, we investigate the fundamental concepts of algebraic loop structures and the set of Lie algebra comultiplications which have inversive, power-associative and Moufang properties depending on the Lie algebra comultiplications up to all the possible quadratic and cubic Lie algebra comultiplications. We also apply those notions to the rational cohomology of Hopf spaces.

A Battery Dynamic Model for the Power System Simulation

This paper analyzes the battery dynamic characteristics as well as some existing battery models, then presents a universal battery model applicable to micro-grid simulation. The model is composed of a controlled voltage source in series with a constant resistance .The voltage of the controlled voltage source is a one-to-one correspondence with the state of charge (SOC) of the battery, which can effectively avoid the algebraic loop problem. The parameters of the model can be easily extracted from the battery discharge curve. The simulation results shows that the biggest advantage of this model is that the initial SOC of the battery can be set accordingly, which allows the battery to be charged or discharged from any SOC conveniently.