Deformations over non-commutative base (Q6553456)

The paper considers deformation theory over non-commutative (NC) base algebras. Such a theory is interesting because there are more deformations than the usual deformations over commutative bases. The deformations over commutative base can possibly be regarded as the `first order' approximation of more general `higher order' deformations. The formal theories of deformations over commutative and non-commutative bases are parallel and the extension to the non-commutative case is simple, but some new phenomena and invariants appear.\N\NTwo important results of the paper are:\N\NProposition 1. Let \(V \cong k^n\) with coordinate linear functions \(x_1,\dots,x_n\), and let \(W \cong k^m\) be defined by \(x_{m+1} = \dots = x_n = 0\). Then the formal semi-universal NC deformation of \(W\) as a linear subspace of \(V\) has the parameter algebra \(\hat R\) and the ideal \(\hat I\) given as follows: \N\[\N\begin{split}\N&\hat R = k\langle \langle a_{ij} \mid 1 \le i \le m < j \le n \rangle \rangle/\hat J \\\N&\hat J = (a_{ij_1}a_{ij_2} - a_{ij_2}a_{ij_1}, \,\,\,\Na_{i_1j_1}a_{i_2j_2} - a_{i_2j_2}a_{i_1j_1} + a_{i_1j_2}a_{i_2j_1} - a_{i_2j_1}a_{i_1j_2} \\\N&\mid 0 \le i \le m, 1 \le i_1 < i_2 \le m < j_1 < j_2 \le n) \\\N&\hat I = (x_j + \sum_{i=1}^m a_{ij}x_i \mid m+1 \le j \le n).\N\end{split}\N\]\NThe next result describes the base algebra of the semi-universal NC deformation of a deformation functor \(\Phi\) which has the tangent space \(T^1\) and the obstruction space \(T^2\).\N\NTheorem 1. Let \(\Phi: (\text{Art}_r) \to (\text{Set})\) be an NC deformation functor. Assume that the obstruction space \(T^2\) is finite dimensional. Then there is a \(k^r\)-linear map \(m: (T^2)^* \to \hat T_{k^r}(T^1)^*\) such that \(\hat R \cong \hat T_{k^r}(T^1)^*/(m((T^2)^*))\), a quotient algebra of the completed tensor algebra by a two-sided ideal generated by the image of \(m\).