Higher-order Sugawara operators for affine Lie algebras
Higher-order Sugawara operators for affine Lie algebras
复制标题
仿射李代数的高阶菅原算子
DOI：
10.1090/s0002-9947-1989-0958893-5
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发表时间：
1989
影响因子：
1.3
通讯作者：
N. Wallach
中科院分区：
文献类型：
--
作者：
R. Goodman;N. Wallach
令 0 为与简单李代数 0 关联的仿射李代数。 0 的表示由圆 T（X e 0 和 £ £ T ）上的当前场 X(Q ) 来描述。本文构造了从对称代数 5(0) 到适当类别 0 模上的（形式）算子域的线性映射 a。与 S($) 的 0 不变元素对应的算子域称为菅原域。证明它们满足 (*) [<t(h)(í) 形式的交换关系， X(t})] = cx,DS{C,ln)a(Vxu)(Q + 当前场的高阶项，其中 c» 是 0 中中心元素的重整化，Dô 是狄拉克 delta 函数的导数。(*) 中的高阶项是使用 0 的不变量理论和有限维表示理论的结果进行研究的。对于 4 次或更小的适当归一化不变量 u，这些项显示为零。此消失也被证明适用于 0 = sl(n,C) 和 u 运行对称不变量的特定选择，只要 c<> 用零表示，由此类不变量定义的菅原域就与当前域交换。此属性用于获取 0 的一类最高权模的交换环、组合级数和特征公式。 g 的表示理论与两个时空维度的量子场论模型密切相关，在所谓的菅原模型中，g 的表示对应于当前场，并且能量动量场作为当前场的二次函数获得[Fre，G-O，P-S，Sug]。在本文中，我们在适当的 g 模类别上构造了从对称代数 S(g) 到（形式）算子场的线性映射。 S(g) 菅原域的 g 不变元素。我们的主要结果涉及当前域和菅原域之间的交换关系。编辑于 1988 年 3 月 2 日收到。1989 年 4 月 16 日提交给马萨诸塞州伍斯特市第 848 届 AMS 会议。1980 年数学科目分类（1985 年修订版）。 17B67、22E47；中学 15A72、20C30、20G45、81E99。研究由罗格斯大学教师学术研究计划、澳大利亚国立大学数学分析中心和 NSF Grant DMS 86-03169（至 R.G.）和 NSF Grant DMS 84-02684（至 N.R.W.）部分支持。 ©1989 美国数学会 0002-9947/89 $1.00+ $.25 每页
Let 0 be the affine Lie algebra associated to a simple Lie algebra 0 . Representations of 0 are described by current fields X(Q on the circle T (X e 0 and £ £ T ). In this paper a linear map a from the symmetric algebra 5(0) to (formal) operator fields on a suitable category of 0 modules is constructed. The operator fields corresponding to 0-invariant elements of S($) are called Sugawara fields. It is proved that they satisfy commutation relations of the form (*) [<t(h)(í), X(t})] = cx,DS{C,ln)a(Vxu)(Q + higher-order terms with the current fields, where c» is a renormalization of the central element in 0 and Dô is the derivative of the Dirac delta function. The higher-order terms in (*) are studied using results from invariant theory and finite-dimensional representation theory of 0 . For suitably normalized invariants u of degree 4 or less, these terms are shown to be zero. This vanishing is also proved for 0 = sl(n,C) and u running over a particular choice of generators for the symmetric invariants. The Sugawara fields defined by such invariants commute with the current fields whenever c«> is represented by zero. This property is used to obtain the commuting ring, composition series, and character formulas for a class of highest-weight modules for 0 . Introduction Let q be the affine algebra associated with a finite-dimensional simple Lie algebra g. The representation theory of g is closely related to models for quantum field theory in two space-time dimensions. In the so-called Sugawara models, the representation of g corresponds to a current field, and the energymomentum field is obtained as a quadratic function of the current fields [Fre, G-O, P-S, Sug]. In this paper we construct a linear map from the symmetric algebra S(g) to (formal) operator fields on a suitable category of g modules. We call the operator fields corresponding to g-invariant elements of S(g) Sugawara fields. Our main results concern the commutation relations between current fields and Sugawara fields. Received by the editors March 2, 1988. Presented to the 848th meeting of the AMS, Worcester, Massachusetts, April 16, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 17B35, 17B67, 22E47; Secondary 15A72, 20C30, 20G45, 81E99. Research partially supported by Rutgers University Faculty Academic Study Program, Australian National University Centre for Mathematical Analysis, and NSF Grant DMS 86-03169 (to R.G.) and by NSF Grant DMS 84-02684 (to N.R.W.). ©1989 American Mathematical Society 0002-9947/89 $1.00+ $.25 per page