A word of warning: I have no background in algebraic geometry, so please excuse my ignorance. References welcome (but please refrain from saying things like "read Hartshorne's Algebraic Geometry, it is explained in there somewhere" — this would not be particularly helpful.)
Let $k$ be an algebraically closed field and $n\in \mathbb{N}$. Consider the affine algebraic $S$ set given by points $(c_{i,j}^p)\in k^{n^3}$ (for $1\leq i,j,p\leq n$) that satisfy $$\sum_{p}c_{ij}^pc_{p,m}^t=\sum _{t}c_{i,p}^t c_{j,m}^p$$ for all $1\leq i,j,m\leq n$.
In these notes on page 17, Kontsevich and Soibelman claim that this affine algebraic set is an algebraic variety. There seem to be different conventions around: According to wikipedia some people mean by an algebraic variety an irreducible one, some do not require irreducibility.
- What do the authors mean by the term algebraic variety? Or rather how would one verify irreducibility in this context?
Let $(A,\mu)$ be an $n$-dimensional $k$-vector space, for $n\in \mathbb{N}$. Choose a basis $(e_i)_{1\leq i\leq n}$ of $A$. The above algebraic variety can be seen as the variety of structure constants of associative products on the vector space $A$. In particular, we can view the associative algebra $A$ with its structure constants $(c_{i,j}^p)$ (with respect to the basis $(e_i)_{1\leq i\leq n}$ of $A$) as a point in $S$. We have an action of the group $G=GL_k(A)$ on the set $S$ given by conjugation: By identifying the point $(c_{i,j}^p)\in S$ with an element in $\{f\in\operatorname{Hom}_k(A\otimes_k A,A)\ \vert \ f \text{ associative}\}$ by viewing it as the structure constants of an associative multiplication, we can define the action as $$\phi.f(a,b):=\phi(f(\phi^{-1}(a),\phi^{-1}(b))$$ for $a,b\in A$ and $\phi\in GL_k(A), f \in \{f\in\operatorname{Hom}_k(A\otimes_k A,A)\ \vert \ f \text{ associative}\}.$
- Now, $GL_n(A)$ is an algebraic group. Is the action just defined an algebraic $G$-action (that is, is the $G$-action map a morphism of varieties)?
Consider the orbit space $M=S/G$.
The authors write:
Let us consider the tangent space $T_{[A]}M$ at a given point $[A]=(A,(c_{i,j}^m))\in M$. For a one-parameter first-order deformation of an associative product we can write $c_{i,j}^m(h)=c_{i,j}^m+ \tilde{c}_{i,j}^mh+O(h^2).$
- What notion of tangent space is meant here? I have read on the Zariski tangent space of an algebraic variety, but here $M$ is a quotient space; and I don't know why I should expect it to be an algebraic variety again.