Development, implementation and applications of fundamental algorithms, relying on Gröbner bases in free associative algebras

这个项目是数学计算机代数的一部分。它在环理论、表示理论、计算机科学等学科中都有应用。为了在自由结合代数中进行Gröbner类基计算，可以使用最近发展的理想间的字母对应，并在一个大的可交换多项式环中进行计算。这种环以前已经被深入研究过，特别是从计算机代数的角度。因此，已知非常有效的数据结构和基本算法已在计算机代数系统中得到优化和实现。非交换环的相应情况还不太发达。系统地使用字母通信为自由代数及其因子环计算的挑战性领域提供了新的见解和新的效率水平。我们的目标是创建一个著名的计算机代数系统SINGULAR的扩展LETTERPLACE。这将首次为这些代数中Gröbner基和所有涉及Gröbner基的基本算法提供有效的实现，例如协同模、消去、环核和模同态。Letterplace Gröbner基算法已在SINGULAR内核中初步实现，并已显示出良好的性能。下一步将是使用这些实现来解决迄今为止无法解决的问题。例如，我们打算计算非交换k代数的k维，显式k基和（截断的）希尔伯特级数。另一个应用领域是在群环和群环中的计算，我们计划在其中解决诸如有限表示群的有限性、广义词问题、共轭子搜索问题、群的自由检验以及群代数中扭转元素群的结构等问题。为了指导这些应用，我们打算与德国和世界各地的几个研究小组合作。

This project is part of mathematical Computer Algebra. It has applications in Ring Theory, Representation Theory, Computer Science and other disciplines. In order to perform Gröbner basis-like computations in a free associative algebra, one can use the recently developed letterplace correspondence between ideals and perform the computations in a large commutative polynomial ring. Such rings have been intensively studied before, in particular from a computer algebraic point of view. As a result, very effective data structures are known and fundamental algorithms have been optimized and implemented in computer algebra systems. The corresponding situation for non-commutative rings is less developed. The systematic use of the letterplace correspondence provides new insights and new levels of efficiency into the challenging realm of computations in free algebras and their factor rings. We aim at the creation of an extension LETTERPLACE of the well-known computer algebra system SINGULAR. This will for the first time provide efficient implementations of Gröbner bases and all the basic algorithms involving Gröbner bases in these algebras, for instance syzygy modules, elimination, kernels of ring and module homomorphisms. A preliminary implementation of the Letterplace Gröbner basis algorithm is available in the kernel of SINGULAR and it has already demonstrated very good performance. The next step will be to use these implementations to tackle hitherto unaccessible problems. For instance, we intend compute the K-dimension, explicit K-bases and (truncated) Hilbert series for non-commutative K-algebras. Another area of applications are computations in monoid and group rings where we plan to adress questions such as finiteness of a finitely presented group, the generalized word problem, the conjugator search problem, freeness tests for groups and the structure of the group of torsion elements of a group algebra. To guide these applications, we intend to collaborate with several research groups in Germany and across the world.