Abstract
The paper is survey of recent results of investigations on varieties of Leibniz-Poisson algebras. We show that a variety of Leibniz-Poisson algebras has either polynomial growth or growth with exponential not less than 2, the eld being arbitrary. We show that every variety of Leibniz-Poisson algebras of polynomial growth over a eld of characteristic zero has a nite basis for its polynomial identities.We construct a variety of Leibniz-Poisson algebras with almost polynomial growth. We give equivalent conditions of the polynomial codimension growth of a variety of Leibniz-Poisson algebras over a eld of characteristic zero. We show all varieties of Leibniz-Poisson algebras with almost polynomial growth in one class of varieties. We study varieties of Leibniz-Poisson algebras, whose ideals of identities contain the identity fx; yg fz; tg = 0, we study an interrelation between such varieties and varieties of Leibniz algebras. We show that from any Leibniz algebra L one can construct the Leibniz-Poisson algebra A and the properties of L are close to the properties of A. We show that if the ideal of identities of a Leibniz-Poisson variety V does not contain any Leibniz polynomial identity then V has overexponential growth of the codimensions. We construct a variety of Leibniz-Poisson algebras with almost exponential growth. Let f n(V)gn1 be the sequence of proper codimension growth of a variety of Leibniz-Poisson algebras V. We give one class of minimal varieties of Leibniz-Poisson algebras of polynomial growth of the sequence f n(V)gn1, i.e. the sequence of proper codimensions of any such variety grows as a polynomial of some degree k, but the sequence of proper codimensions of any proper subvariety grows as a polynomial of degree strictly less than k.
Highlights
F Leibniz-Poisson algebras over a eld of characteristic zero
Of Leibniz-Poisson algebras, whose ideals of identities contain the identity {x, y} · {z, t} = 0, we study an interrelation between such varieties and varieties of Leibniz algebras
We show that from any Leibniz algebra L one can construct the Leibniz-Poisson algebra A and the properties of L are close to the properties of A
Summary
Õîðîøî èçâåñòíî, ÷òî â ñëó÷àå îñíîâíîãî ïîëÿ íóëåâîé õàðàêòåðèñòèêè èäåàë òîæäåñòâ ïðîèçâîëüíîãî ìíîãîîáðàçèÿ V ïîðîæäàåòñÿ ñîâîêóïíîñòüþ ïîëèëèíåéíûõ òîæäåñòâ äàííîãî ìíîãîîáðàçèÿ. < jt , îáðàçóþò áàçèñ ïðîñòðàíñòâà Pn (U2 ); 4) ðîñò ìíîãîîáðàçèÿ var(U2 ), ïîðîæäåííîãî àëãåáðîé U2 , ÿâëÿåòñÿ ïî÷òè ïîëèíîìèàëüíûì, ïðè÷åì äëÿ ëþáîãî n ≥ 2 âûïîëíåíî ðàâåíñòâî cn (U2 ) = 2n−1 (n − 2) + 2; Ïðîñòðàíñòâî Pn (V) íàäåëåíî ñòðóêòóðîé ëåâîãî Sn -ìîäóëÿ, ãäå Sn ñèììåòðè÷åñêàÿ ãðóïïà ñòåïåíè n.  ñëó÷àå îñíîâíîãî ïîëÿ íóëåâîé õàðàêòåðèñòèêè äëÿ ìíîãîîáðàçèÿ àëãåáð ËåéáíèöàÏóàññîíà V ñëåäóþùèå óñëîâèÿ ýêâèâàëåíòíû:. Ïóñòü V ìíîãîîáðàçèå àëãåáð ËåéáíèöàÏóàññîíà íàä ïðîèçâîëüíûì ïîëåì, èäåàë òîæäåñòâ êîòîðîãî ñîäåðæèò ïîëèëèíåéíûå òîæäåñòâà (4). Ïîñëå âûÿñíåíèÿ òîãî ôàêòà, ÷åì àïïðîêñèìèðóåòñÿ ïîñëåäîâàòåëüíîñòü êîðàçìåðíîñòåé ïðîèçâîëüíîãî ìíîãîîáðàçèÿ àëãåáð ËåéáíèöàÏóàññîíà, èäåàë òîæäåñòâ êîòîðîãî ñîäåðæèò ïîëèëèíåéíûå òîæäåñòâà âèäà (4), çàêîíîìåðíî âîçíèêàåò âîïðîñ, ñ ïîìîùüþ êàêèõ óñëîâèé (êðèòåðèåâ) èñêàòü çíà÷åíèÿ ýêñïîíåíò d, ôèãóðèðóåìûõ â òåîðåìå 6. Ïóñòü V ìíîãîîáðàçèå àëãåáð ËåéáíèöàÏóàññîíà íàä ïîëåì íóëåâîé õàðàêòåðèñòèêè, èäåàë òîæäåñòâ êîòîðîãî ñîäåðæèò òîæäåñòâà âèäà (4) äëÿ íåêîòîðîãî m.
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