In 1990 Kantor introduced the conservative algebra $\mathcal{ W}(n)$ of all algebras (i.e. bilinear maps) on the $n$-dimensional vector space. In case $n >1$ the algebra $\mathcal{ W}(n)$ does not belong to well known classes of algebras (such as associative, Lie, Jordan, Leibniz algebras). We describe $\frac{ 1}{ 2}$derivations, local (resp. $2$-local) $\frac{ 1}{ 2}$-derivations and biderivations of $\mathcal{ W}(2)$. We also study similar problems for the algebra $\mathcal{ W}_2$ of all commutative algebras on the two-dimensional vector space and the algebra $\mathcal{ S}_2$ of all commutative algebras with trace zero multiplication on the two-dimensional space.
Categories: math.RA