THE NOWICKI CONJECTURE FOR BICOMMUTATIVE ALGEBRAS

Year 2023, Volume: 11 Issue: 2, 104 - 108, 28.08.2023

Şehmus Fındık

https://doi.org/10.20290/estubtdb.1200175

Abstract

Let K be a field of characteristic zero, and K[Xn,Yn ] be the commutative associative unitary polynomial algebra of rank 2n generated by the set Xn∪Yn={x1,…,xn,y1,…,yn }. It is well known that the algebra K[Xn,Yn ]^δ of constants of the locally nilpotent linear derivation δ of K[Xn,Yn ] sending yi to xi, and xi to 0, is generated by x1,…,xn and the determinants of the form xi yj-xj yi; that was first conjectured by Nowicki in 1994, and later proved by several authors. Bicommutative algebras are nonassociative noncommutative algebras satisfying the identities (xy)z=(xz)y and x(yz)=y(xz). In this study, we work in the 2n generated free bicommutative algebra as a noncommutative nonassociative analogue of the Nowicki conjecture, and find the generators of the algebra of constants in this algebra.

Keywords

Algebra of constants, bicommutative algebra, the Nowicki conjecture

BİKOMÜTATİF CEBİRLER İÇİN NOWICKI SANISI

Abstract

K, karakteristiği sıfır olan bir cisim ve K[Xn,Yn ], Xn∪Yn={x1,…,xn,y1,…,yn } kümesi tarafından üretilen rankı 2n olan değişmeli birleşmeli birimli polinom cebiri olsun. K[Xn,Yn ]'nin yi'yi xi'ye ve xi'yi 0'a gönderen yerel nilpotent doğrusal türevi δ nin sabitler cebiri K[Xn,Yn ]^δ nin x1,…,xn ve xi yj-xj yi formundaki determinantlar tarafından üretildiği iyi bilinmekte olup bu ilk olarak 1994 yılında Nowicki tarafından tahmin edildi ve daha sonra birkaç yazar tarafından kanıtlandı. Bikomütatif cebirler, (xy)z=(xz)y ve x(yz)=y(xz) özdeşliklerini sağlayan birleşmeli olmayan ve değişmeli olmayan cebirlerdir. Bu çalışmada, Nowicki sanısının değişmeli olmayan ve birleşmeli olmayan bir analogu olarak 2n ranklı serbest bikomütatif cebir içinde çalışarak ve bu cebirin sabitler cebirinin üreteçlerini bulmaktayız.

Keywords

Bikomütatif cebir, Nowicki sanısı, sabitler cebiri

References

• [1] Hilbert D. Mathematische probleme. Göttinger Nachrichten 1900; 253-297, Arch. Math. u. Phys. 1901; 3(1): 44-63, Bull. Amer. Math. Soc. 1902; 8(10): 437-479.
• [2] Nagata M. On the 14-th problem of Hilbert. Amer J Math 1959; 81: 766-772.
• [3] Noether E. Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math Ann 1916; 77: 89-92.
• [4] Weitzenböck R. Über die Invarianten von linearen Gruppen. Acta Mathematica 1932; 58: 231-293.
• [5] Nowicki A. Polynomial derivations and their rings of constants. Toruń: Uniwersytet Mikolaja Kopernika, 1994.
• [6] Khoury J. A Groebner basis approach to solve a conjecture of Nowicki. Journal of Symbolic Computation 2008; 43(12): 908-922.
• [7] Drensky V, Makar-Limanov L. The conjecture of Nowicki on Weitzenböck derivations of polynomial algebras. J Algebra Appl 2009; 8(01): 41-51. doi: 10.1142/S0219498809003217
• [8] Kuroda S. A Simple Proof of Nowicki's Conjecture on the Kernel of an Elementary Derivation. Tokyo Journal of Mathematics 2009; 32(1): 247-251.
• [9] Drensky V. Another proof of the Nowicki conjecture. Tokyo Journal of Mathematics 2020; 43 (2): 537-542. doi: 10.3836/tjm/1502179320
• [10] Drensky V, Fındık Ş. The Nowicki conjecture for free metabelian Lie algebras. International Journal of Algebra and Computation 2020; 19 (5): 2050095. doi: 10.1142/S0219498820500954
• [11] Centrone L, Fındık Ş. The Nowicki conjecture for relatively free algebras. Journal of Algebra 2020; 552: 68-85.
• [12] Centrone L, Dushimirimana A, Fındık Ş. On Nowicki's conjecture: a survey and a new result. Turkish Journal of Mathematics 2022; 46(5): 1709-1734. doi: 10.55730/1300-0098.3228
• [13] Drensky V. Invariant theory of free bicommutative algebras. arXiv 2022, 2210.08317.