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。它在环论、表示论、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基类计算，人们可以利用最近发展的理想之间的字母位置对应，并在一个大的交换多项式环上进行计算。这样的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，非常有效的数据结构是已知的，基本算法已经被优化并在计算机代数系统中实现。对于非交换环，相应的情形就不那么发达了。字母对应的系统使用为自由代数及其因子环中具有挑战性的计算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奇异的扩展LETTERPLACE。这将首次提供Gröbner基的有效实现，以及涉及这些代数中的Gröbner基的所有基本算法，例如合子模、消去、环的核和模同态。在奇异核上初步实现了Letterplace Gröbner基算法，并且已经表现出了很好的性能。下一步将是使用这些实现来解决迄今无法访问的问题。例如，我们打算计算非交换K-代数的K-维显式K-基和(截断)Hilbert级数。另一个应用领域是么半群和群环中的计算，我们计划在其中加入一些问题，如有限表示群的有限性、广义字问题、共轭子搜索问题、群的自由性检验以及群代数的扭元群的结构。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.