On commutativity and lie nilpoten y in matrix algebras

Abstract

ENGLISH ABSTRACT : In this thesis we first discuss the proof by Mirzakhani [9] of Schur's Theorem which gives the maximum number of linearly independent matrices in a commutative algebra of n×n matrices over a field F. An example illustrating the application of Schur's Theorem is given. Secondly, we discuss the Cayley-Hamilton Theorem which asserts that any n×n matrix A satisfies its characteristic polynomial. A deduction of a Cayley-Hamilton trace identity for a 2 × 2 matrix A over a commutative ring from the Cayley-Hamilton Theorem is shown. We then discuss the Cayley-Hamilton trace identity for any matrix A ∈ M2(R) when (i) R is commutative, (ii) R is not necessarily commutative, (iii) R is not necessarily commutative and sp(A) = 0, (iv) R is not necessarily commutative and satisfies the identity [[x, y], [x, z]] = 0. Lastly, we discuss the matrix algebras U∗n(R), in particular the matrix algebras U∗3 (R) and U∗4 (R), in relation to polynomial identities [[. . . [[x1, x2], x3], . . .], xn] = 0, [x, y][w, z] = 0 and [[x, y], [w, z]] = 0.

AFRIKAANSE OPSOMMING : In hierdie tesis beskryf ons eerstens die bewys deur Mirzakhani [9] van Schur se Stelling wat die maksimum aantal lineêr onafhanklike matrikse in 'n kommutatiewe algebra van n × n matrikse oor 'n liggaam F gee. 'n Voorbeeld word gegee wat die toepassing van Schur se Stelling illustreer. Tweedens bespreek ons die Cayley-Hamilton Stelling wat beweer dat elke n×n matriks A sy karakteristieke polinoom bevredig. 'n Afleiding van 'n Cayley-Hamilton spoor identiteit vir 'n 2 × 2 matriks A oor 'n kommutatiewe ring vanuit die Cayley-Hamilton Stelling word gegee. Ons bespreek dan die Cayley-Hamilton spoor identiteit vir enige matriks A ∈ M2(R) wanneer (i) R kommutatief is, (ii) R nie noodwendig kommutatief is nie, (iii) R nie noodwendig kommutatief is nie en sp(A) = 0, (iv) R nie noodwendig kommutatief is nie en die identiteit [[x, y], [x, z]] = 0 bevredig. Laastens bespreek ons die matriksalgebras U∗n(R), in besonder die matriksalgebras U∗3 (R)en U∗ 4 (R), met betrekking tot die polinoom identiteite [[. . . [[x1, x2], x3], . . .], xn] = 0, [x, y][w, z] = 0 en [[x, y], [w, z]] = 0.