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    Hausdorff series in a semigroup ring
    (World Scientific Publ Co Pte Ltd, 2020) Findik, Sehmus; Kelekci, Osman
    Let A = R < a(1), ..., a(n)> and B = R < b(1), ..., b(n)> be the semigroup rings spanned on the right zero semigroup RZ(n) = {a(1), ..., a(n)}, and on the left zero semigroup LZ(n) = {b1, ..., b(n)}, respectively, together with the identity element 1. We suggest a closed formula solving the equation w = log(e(u)e(v)) which is the evolution of the CampbellBaker-Hausdorff formula given by the Hausdorff series w = H(u, v) = u + v + 1/2[u, v] + 1/12 [u, [u, v]] + 1/12 [v, [u, v]] + ..., where [u, v] = uv - vu, in the algebras A and B.
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    Symmetric polynomials of algebras related with 2 x 2 generic traceless matrices
    (World Scientific Publ Co Pte Ltd, 2021) Findik, Sehmus; Kelekci, Osman
    Let X and Y be two generic traceless matrices of size 2 x 2 with entries from a commutative associative polynomial algebra over a field K of characteristic zero. Consider the associative unitary algebra W, and its Lie subalgebra L generated by X and Y over the field K. It is well known that the center C(W) = K[t,u,v] of W is the polynomial algebra generated by the algebraically independent commuting elements t = tr(X-2)I-2, u = tr(Y-2)I-2, v = tr(XY)I-2. We call a polynomial p is an element of W symmetric, if p(X,Y ) = p(Y,X). The set of symmetric polynomials is equal to the algebra W-S2 of invariants of symmetric group S-2. Similarly, we define the Lie algebra L-S2 of symmetric polynomials in the Lie algebra L. We give the description of the algebras W-S2 and L-S2, and we provide finite sets of free generators for W-S2, and [L,L](S2) as K[t + u,tu,v]-modules.