Paweł Matraś
supervisor: Michał Ziembowski
We will be interested in subalgebras of nxn matrices with entries in field F. It is structure closed on multiplication and substraction which is also linear subspace. Elements of it are some nxn matrices.
Classical theorem gives bound on dimension of commutative subalgebra of nxn matrices. Any elements x, y of it satisfies xy = yx. Defining [x,y] = xy-yx we say that commutative subalgebra satisfies identity [x,y] = 0.
We present some results generalizing above theorem by considering other identities of similar type. We also give some examples of subalgebras with maximal dimension. Specially we will explore identity
[x1,y1][x2,y2]...[xk,yk] = 0,
which is satisfied by upper triangular matrices kxk. Moreover it was shown that other identities of upper triangular matrices are consequence of such identity.
Other motivations are connected with Lie nilpotency and Lie solvablity. Such properties are important in theory of Lie algebras. They are defined in terms of similar identities and some results from this theory are useful for us.
Lastly for matrices over algebras satisfying similar to written identity we can get theorem generalizing Caley-Hamilton theorem for matrices over field. This subject was explored in quite recent paper.