Assume that I have a finitely presented algebra $A$ over the complex numbers (by which I mean that $A$ is generated over $\mathbb{C}$ by finitely many elements $x\_1,...,x\_n$ subject to finitely many relations). Let now $y\_1,\ldots, y\_m$ be a finite subset of elements of $A$. Is there an algorithm which gives a presentation of the subalgebra $B$ of $A$ generated by $y\_1,...y\_m$ ? 
In particular, I am interested in the following example: Let $A = \mathbb{C}B\_3 = \mathbb{C}\langle \sigma\_1,\sigma\_2 | \sigma\_1\sigma\_2\sigma\_1 = \sigma\_2\sigma\_1\sigma\_2\rangle$ be the group algebra of the Braid group on 3 braids. Is there any nice presentation of the subalgebra generated by $\sigma\_1$ and $\sigma\_2 + \sigma\_2^{-1}$ ?



presentations of subalgebras
If we modify your first question only slightly, then the answer to the question is no, there is no algorithm. By Theorem 1 of the paper G. Baumslag, W. W. Boone and B. H. Neumann, Some unsolvable problems about elements and subgroups of groups, Math. Scand. 7, 191-201 (1959), let $G$ be a finitely presented group for which is it undecidable whether elements have finite order. Let $A=\mathbb{C}G$. For each element $g\in G$, we can take $B\_g=\mathbb{C}[g]$ to be the subalgebra generated by $g$. Any non-trivial relation in $B\_g$ would tell us $g$ has finite order, and so there can be no algorithm (which is independent of $g$) to yield a presentation of $B\_g$.


