Anna Cichocka
supervisor: Michał Ziembowski
The history of Leavitt path algebras dates back to the 1960s, when William G. Leavitt asked the question about the existence of R rings satisfying the equality R i ≅ R j as right-hand modules over R.
The first time Leavitt path algebra was defined the first time in years 2005-2007. Their construction was introduced to algebraize combinatorial objects related to Cuntz-Krieger algebras and C* - algebras. Their subject matter is of interest to both mathematicians dealing with algebra and those interested in functional analysis, in particular C* - algebras. Due to the flexibility of Leavitt path algebras construction, they are the source of various examples of algebras with fixed properties. The matrix algebras Mn(F) dimension n times n over a field F for n ∈ ℕ ∪ { ∞ } (where M∞(F) are infinite matrices of countable size with finite number of non-zero elements in each column) are an special example of Leavitt paths algebras. During the presentation, I will present Leavitt algebras and their properties will be presented. Next I will demonstrate the construction of some of maximal commutative subalgebras Leavitt paths algebras. Finally, I will display the construction of its maximal commutative subalgebra, inspired by some known maximal commutative subalgebra of the matrix algebra with the greatest dimension.