仿射Chevalley代数的中心

设R是具有恒等元的可换环,在1]、2]中.J.F.Hurley对R上Chevalley代数gR=Rg.计算了它们的中心. 本文,对仿射李代数g(A)3].我们得到了R上仿射Chevllcy代数g_R=Rg_(A)4]的中心C_R:C_R=Rh_C_R (R-模直和)其中:h_是张成g(A)的一维中心之生成元C_R是R上loop代数Rt,t~(-1)]g_z的中心由此知道.在中心扩张:0→Rh_→g_R→Rt,t~(-1)]g_z→0中,若对环R作适当限制,则Rh_就是g_R的中心,这一结果.与5]中获得的关于g_R的全部“理想”结构是一致的;若去掉对环R的限制,则Rh_被真正包含在g_R的中心C_R里.

Centers of Chevalley Algebras

Let R be a commutative ring with identity. In1],2] J. F. Hurley calculated the center of Chevalley algebra g_R=R(?)g_z over R, where g_z is the Z-span of Chevalley basis of simple Lie algebra g.In the present paper, we have determined the center C_r of Affine Chevalley algebra g_R(A)=R(?)g_z(A)4]5] over R, where gz(A) is the Z-span of Chevalley basis of Affine Lie algebra g(A) 3], and obtain:C_r=Rh_1(?)C_R (direct sum of R-modules),whereh_r is the generator of 1-dimensional center of g(A),C_r is the center of loop algebra Rt,t~(-1)](?)gz.Subject to certain restriction on ring R, we have C_r=Rh_r which coincides with the general ideal structure in5]. Our result shows that R h_r is properly contained in C_r if the restriction on R is excluded.