This is a joint work with Viktor Levandovskyy (RWTH Aachen University,
Germany) and. Roberto ... algebra, especially Gröbner basis and dimension
computations, and in addition ... elements of the product set X ×P are denoted by
(xi|j)=(xi,j).
SINGULAR: LETTERPLACE, COMPUTATIONS IN NON-COMMUTATIVE, ASSOCIATIVE ALGEBRAS GRISCHA STUDZINSKI
This is a joint work with Viktor Levandovskyy (RWTH Aachen University, Germany) and Roberto La Scala (Universita degli studi di Bari, Italy). With this poster we present recently developed methods to deal with the free associative algebra, especially Gröbner basis and dimension computations, and in addition applications to group algebras. Moreover, we like to give an insight in our current development. The letterplace paradigm uses special computations in commutative polynomial rings and allows to compute non-commutative Gröbner bases via the Letterplace Gröbner basis algorithm. It is possible to build the advanced Gröbner basis-based functionality, using the paradigm. If X = {x1 , . . . , xn } is the set of “letters” and P = N = {0, 1, . . .} the set of “places”, the elements of the product set X × P are denoted by (xi |j) = (xi , j). Then K[X|P ] is the infinitely generated polynomial ring in the commutative variables (xi |j). There is a one-to-one correspondence between the elements hXi 3 xi1 · · · xin ↔ (xi1 |0) · · · (xin |n − 1) ∈ K[X|P ], which, however, does not raise immediately to the correspondence of ideals. The monomial K-basis of a given factor algebra can be represented as a rooted tree, which consists of words, normal with respect to the given Gröbner basis. In the non-commutative case this tree has a special property: every vertex is uniquely determined by its path. This suggests a new and effective way to store the information on the K-basis of a factor algebra: by storing only the endpoints (so-called mistletoes) of this tree. All other elements in a K-basis can be recursively recovered from mistletoes in linear time. It is common to work with a degree bound, while computing non-commutative Gröbner bases, since the free associative algebra is not Noetherian and therefore a Gröbner basis is not necessarily finite. We are going to show adaptive methods which will lead to the information, whether the Gröbner basis is finite or not, and their functionality. Beyond that we point out several possibilities to improve existing methods. We also address the preprocessing of the computation of Gel’fand-Kirillov dimension. We present one of the many applications: Consider a finitely presented group G. Each group can be embedded in its group algebra KG, which again can be considered as a factor of the free associative algebra in finitely many generators. Now, since G is a K-basis for KG, one can apply the methods mentioned above to KG to obtain important information about G (for instance finiteness).
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GRISCHA STUDZINSKI
References [BB98] B. Buchberger. The Main Theorem and Elementary Applications of Groebner Bases Theory. (8 hours), January 12-17 1998b. Contributed talk at Intensive Course on Groebner Bases, RISC, Hagenberg, Austria. [DGPS10] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann. Singular 3-1-2. A computer algebra system for polynomial computations. Centre for Computer Algebra, University of Kaiserslautern, 2010. http://www.singular.uni-kl.de. [EGR00] E. Green. Multiplicative Bases, Gröbner Bases, and Right Gröbner Bases. J. Symbolic Computation, 29(4/5), 2000. [KMRU06] M. Kreuzer, A. Myasnikov, G. Rosenberger, and A. Ushakov. Quotient tests and Gröbner bases. Contemp. Math., 421:187–200, 2006. [LSL09] R. L. Scala and V. Levandovskyy. Letterplace ideals and non-commutative Gröbner bases. J. Symbolic Computation, 44(10):1374–1393, 2009. [MORA89] T. Mora. Gröbner bases in non-commutative algebras. In Proc. of the International Symposium on Symbolic and Algebraic Computation (ISSAC’88), pages 150–161. LNCS 358, 1989. [NBK98] P. Nordbeck. On some basic applications of Gröbner bases in non-commutative polynomial rings. London mathematical society lecture note series, 251:323 – 338, 1998. [STUD10] G. Studzinski. Algorithmic computations for factor algebras. Diploma thesis, RWTH Aachen, 2010. [UFN98] V. Ufnarovskij. Introduction to noncommutative Gröbner bases theory. In B. Buchberger and F. Winkler, editors, Gröbner bases and applications, pages 259–280. Cambridge University Press, 1998.
Lehrstuhl D für Mathematik, RWTH Aachen University E-mail address:
[email protected]
