Lie Rings and Groups Defined by Carpets of Symplectic Type

Earlier the author found necessary and sufficient conditions for the invariance of a carpet Lie ring with respect to the carpet subgroup corresponding to the same carpet of additive subgroups over an arbitrary commutative ring (Trudy Inst. Mat. i Mekh. UrO RAN, 2012). These invariance conditions, denoted by $ (*) $ , are formulated in terms of pairs of opposite additive subgroups of the initial carpet. In 2023 the author established that the conditions $ (*) $ are sufficient for the closedness of carpets of every type, except for the symplectic one (J. SFU Math. Phys.). In the present article, the sufficiency of the conditions $ (*) $ for the closedness of a carpet of symplectic type is proved. Thus, a complete positive answer to Question 19.63 from the Kourovka Notebook is obtained and, in particular, the hypothesis of Levchuk that assumptions stronger than the conditions $ (*) $ are sufficient for the closedness of a carpet is confirmed.