Deformations of nilHecke algebras
A nilHecke algebra \(NH_n\) is the graded ring generated by operators \(x_i\) in degree 2 and operators \(\partial_j\) in degree \(-2\), \(1\leq i, j\leq n\) subject to the relations \(\partial_i^2=0\), \(\partial_i \partial_{i+1} \partial_i =\partial_{i+1} \partial_i \partial_{i+1}\), \(x_i\partial_{i}-\partial_{i+1}x_{i+1}=1\), \(\partial_{i}x_i-x_{i+1}\partial_{i}=1\). The algebra \(NH_n\) acts on the polynomial ring \(H^*(BT^n)=\mathbb Z[x_1, \ldots , x_n]\) by the divided difference operators \[ \partial_{i}f=\frac{f-\sigma_i(f)}{x_i-x_{i+1}}, \quad f\in H^*(BT^n) \] where \(\sigma_i\) is the transposition \((j, j+1)\). Now, given a formal group law \(F(x,y)\in Rx,y\), the author defines the generalized nilHecke algebra \(GNH_n=R\otimes NH_n\) by setting \[ \partial_{i}f=\frac{f-\sigma_i(f)}{x_i-_F x_{i+1}}, \quad f\in H^*(BT^n) \] The main result (Theorem 3.11) gives an explicit presentation of the algebra \(GNH_n\). Furthermore, the author constructs a family of deformations of \(GNH's\).
scientific article; zbMATH DE number 6388892
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