Radical And Finiteness Of (Braided) Lie Algebras

In this paper, we research and discuss the radical theory and the finiteness of(braided) Lie algebra. The theory of Nichols algebras is relatively young, but it is affected by various research areas of mathematics and theoretical physics.In paper [1], it is classified that finite dimensional Nichols algebras over finite cyclic groups when B(V) is an arithmetic root. We describe YD- module on the elementary group when B(V) is an arithmetic root in this paper. It is established radical theory of infinite dimensional Lie algebras in paper [2]. We calculate the radical of untwisted affine Lie algebras in this paper. Calculation of puiujpuj uiis helpful to figure up braided Lie algebra.In chapter 2, we give the necessary and sufficient condition of YD- module over elementary group is limited,namely, if ? ? t, G is a elementary group and p = ord(a1), then(qij)?×?is a finite YD- module if and only if ord(q) = p and(qij)?×?is finite Cartan type. we also show the necessary and sufficient condition of YD- module over weak elementary group,namely, if ? ? t, G is a weak elementary group and N = ord(a1), then(qij)?×?is G-YD module if and only if ord(qij) | N.In chapter 3, we study the radical of untwisted affine Lie algebras. The theory of the radicals of Lie algebras is established, Baer radical is introduced. We obtain the Baer radicals of Witt algebras, Virasoro algebra, loop algebras and untwisted affine algebras. In section 1, we give the characterization of Baer radical for Lie algebras. That is, Baer radical is the low radical of all nilpotent Lie algebras; Baer radical is also Baer ideal LE. In section 2, we show that if L is a finite dimensional Lie algebra, then rb(L) is the maximal solvable ideal of L and L = S ? rb(L), where S is a semi-simple Lie algebra. In section 3, we find Baer radicals of untwisted affine Lie algebras.In chapter 4, we give the Lie operation formula of connected braided algebra of braided vector space V whose rank is two with arithmetic root systems and give the expression of the root system of E8.