Decomposition of Finite - Dimensional Matrix Algebras over F q

Computing the structure of a finite-dimensional algebra is a classical mathematical problem in symbolic computation with many applications such as polynomial factorization, computational group theory and differential factorization. We will investigate the computational complexity and exhibit new algorithms for this problem over the field Fq(y), where Fq is the finite field with q elements. A finite-dimensional vector space A over a field F is called a finite-dimensional associative algebra over F , if A is equipped with a binary associative F -bilinear operation (which is always called multiplication and not necessarily commutative) and the distributive law holds with respect to the addition of linear space and the multiplication. The matrix algebra is the subalgebra of the matrix ring Fm×m with the identity matrix. For an algebra A, there exists a largest nilpotent ideal Rad(A), called the radical of A in every finite dimensional algebra A. Rad(A) is the set of all strongly nilpotent elements (where an element α is said to be strongly nilpotent if for any β ∈ A, αβ is nilpotent). If Rad(A) = (0) we call the algebra A semisimple. So the factor algebra A/Rad(A) is semisimple. A is called simple if A has no proper nonzero ideal. Semisimple algebras admit a very nice structure theorem which is also due to Wedderburn [34]. Theorem 1. [Wedderburn] Suppose that A is a finite-dimensional semisimple algebra over the field F . Then A can be expressed as a direct sum of simple algebras. A = A1 ⊕ A2 ⊕ ...⊕ At, where A1, A2, . . . , At are the minimal nontrivial ideals of A. Each Ai is isomorphic to some full matrix algebra Mni(Fi), where Fi is an extension division ring of F for 1 ≤ i ≤ t. Such decomposition is called Wedderburn decomposition. In this thesis we will first present a new probabilistic algorithm for Wedderburn decomposition. The Wedderburn decomposition of separable algebra is solved with almost nearly optimal algorithms by Eberly and Giesbrecht [8, 7]. However, when it comes to the algebra over the field Fq(y), it becomes non-separable. Ivanyos et al. present a polynomial-time algorithm for Wedderburn decomposition over Fq(y), but it is not acceptable because of large exponent [24]. We will exhibit a new probabilistic algorithm of Monte Carlo type for decomposition of general semisimple matrix algebras. The idea is inspired by Eberly and Giesbrecht [8, 7]: Demonstrate the large probability to pick up a “good” element randomly, use it to compute the “good” idempotents and then decompose the algebra. Our algorithm is more efficient and easier to implement than the algorithm of Ivanyos et al. [24]. The second part of this thesis is a new probabilistic algorithm for computing the radical of a finite-dimensional algebra. Fröhlich and Shepherdson [10] proved that in general this is algorithmically undecidable over a general computable field. But there are some efficient results over specific fields such as the finite field Fq. The latest paper about computing the radical of the finite-dimensional algebras over Fq(y) is also developed by Ivanyos et al. [24], which is polynomial-time but