arXiv 1206.18352012-06-08math.KTmath.AT

SU的基环群的等变K理论的积结构（2）

The product structure of the equivariant K-theory of the based loop group of SU(2)

Megumi Harada, Lisa C. Jeffrey, Paul Selick

备注：: 34 pages

摘要：

设G=SU（2），ΩG表示G中的连续基环空间，G具有点态共轭作用.普通上同调H^*（ΩG）是可分多项式代数Γ[x]是拓扑学中的经典事实。代数Γ[x]可以被描述为外代数Λ（x_1，.，x_k）在变量x_1，.，x_k。我们计算了ΩG的G-等变K理论的R（G）-代数结构，从而将ΩG的普通上同调环的经典计算自然地推广为一个可除多项式代数Γ[x].具体地说，我们证明了K^* G（ΩG）是K^* G（（P^1）^{2 r}）的对称（S_{2r} -不变量）子代数的逆极限，其中对称群S_{2r}以自然方式作用于2 r重积（P^1）^{2 r}的因子，G通过标准作用对角地作用于每一条复射影线P^1.

英文摘要：

Let G=SU(2) and let ΩG denote the space of continuous based loops in G, equipped with the pointwise conjugation action of G. It is a classical fact in topology that the ordinary cohomology H^*(ΩG) is a divided polynomial algebra Γ[x]. The algebra Γ[x] can be described as an inverse limit as k goes to infinity of the symmetric subalgebra in the exterior algebra Λ(x_1, ...,x_k) in the variables x_1, ..., x_k. We compute the R(G)-algebra structure of the G-equivariant K-theory of ΩG in a way which naturally generalizes the classical computation of the ordinary cohomology ring of ΩG as a divided polynomial algebra Γ[x]. Specifically, we prove that K^*_G(ΩG) is an inverse limit of the symmetric (S_{2r}-invariant) subalgebra of K^*_G((P^1)^{2r}), where the symmetric group S_{2r} acts in the natural way on the factors of the 2r-fold product (P^1)^{2r} and G acts diagonally via the standard action on each complex projective line P^1.

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