Автореферат (Некоторые классы циклических модулей над алгебрами Ли), страница 2

 äàííîé äèññåðòàöèè èçó÷àþòñÿ ìîäóëè Âåéëÿ èíåêîòîðûå èõ îáîáùåíèÿ. Îíè îïðåäåëÿþòñÿ ñëåäóþùèì îáðàçîì. Ïóñòüg ïðîñòàÿ êîíå÷íîìåðíàÿ àëãåáðà. Ìû èçó÷àåì ïðåäñòàâëåíèÿ àëãåáðûòîêîâ íàä íåé, òî åñòü àëãåáðû g ⊗ K[t]. h ïîäàëãåáðà êàðòàíà êîíå÷íîìåðíîé àëãåáðû, ∆ ñèñòåìà êîðíåé g, ∆+ ìíîæåñòâî ïîëîæèòåëüíûõêîðíåé, eα , fα îáðàçóþùèå Øåâàëëå. Òîãäà îáîáùåííûé ìîäóëü Âåéëÿîïðåäåëÿåòñÿ ñëåäóþùèìè îáðàçóþùèìè è ñîîòíîøåíèÿìè.8Îïðåäåëåíèå 2.(1) h ⊗ tk v = 0 äëÿ ëþáîãî h ∈ h, k > 0;(eα ⊗ t)v = 0, α ∈ σ(∆+ ) ∩ ∆+ ;(fα ⊗ 1)v = 0, α ∈ σ(∆+ ) ∩ ∆− ;(eσ(α) ⊗ t)−hα∨ ,λi+1v = 0, α ∈ ∆− , σ(α) ∈ ∆+ ;(fσ(α) ⊗ 1)−hα∨ ,λi+1v = 0, α ∈ ∆− , σ(α) ∈ ∆− . ñëó÷àå òðèâèàëüíîãî ýëåìåíòà èç ãðóïïû Âåéëÿ ïîëó÷àåòñÿ îïðåäåëåíèå êëàññè÷åñêîãî ìîäóëÿ Âåéëÿ. Äëÿ ïðîèçâîëüíîãî ýëåìåíòà ãðóïïû Âåéëÿ ïîëó÷åííûé ìîäóëü íàçîâåì îáîáùåííûì ìîäóëåì Âåéëÿ.

Ýòèìîäóëè õîðîøè òåì, ÷òî äëÿ íèõ ïèøåòñÿ óäîáíàÿ ðåêóðñèÿ. Òî÷íåå ãîâîðÿ, êàæäûé òàêîé ìîäóëü ðàçáèðàåòñÿ íà ïîäôàêòîðû òîãî æå âèäà,íî ñîîòâåòñòâóþùèå ìåíüøåìó âåñó. Ñ ïîìîùüþ ýòîé ðàçáîðêè ïîëó÷àåòñÿ ðåêóððåíòíîå ñîîòíîøåíèå íà õàðàêòåðû. Àíàëîãè÷íîå ñîîòíîøåíèå ïîëó÷àåòñÿ äëÿ ñïåöèàëèçàöèé ìíîãî÷ëåíîâ Ìàêäîíàëüäà ñ ïîìîùüþ ïîäõîäà Îððà è Øèìîçîíî.

Îòñþäà ïîëó÷àåòñÿ ñîâïàäåíèå õàðàêòåðîâ íåêîòîðûõ îáîáùåííûõ ìîäóëåé Âåéëÿ è íåêîòîðûõ ñïåöèàëèçàöèéíåñèììåòðè÷åñêèõ ìíîãî÷ëåíîâ Ìàêäîíàëüäà. Òî÷íåå, âåðíà ñëåäóþùàÿòåîðåìà.Òåîðåìà 3.(i)(ii)(iii)(iv)Ïóñòü λ äîìèíàíòíûé âåñ, σ ∈ W . Òîãäàdim Wσ(λ) = dim Wλ .chWw0 λ = Ew0 λ (x, q −1 , ∞).chWλ = Ew0 λ (x, q, 0).Äëÿ ëþáîãî i = 1, .

. . , rk(g)-ìîäóëè Wσ(λ) ðàñêëàäûâàþòñÿ íàïîäôàêòîðû âèäà Wκ(λ−ωi ) , κ ∈ W . Ïîäôàêòîðû ñîîòâåòñòâóþò íåêîòîðûì àëüêîâíûì ïóòÿì è êîëè÷åñòâî ïîäôàêòîðîâðàâíî ðàçìåðíîñòè Wωi .Àíàëîãè÷íûå ðåçóëüòàòû ïîëó÷àþòñÿ äëÿ ïðîñòåéøåé ñóïåðàëãåáðûosp(1, 2).

Ñóïåðàëãåáðû osp(1, 2l) èìåþò ìíîãî îáùèõ ñâîéñòâ ñ îáû÷íûìè ïîëóïðîñòûìè àëãåáðàìè Ëè.  ÷àñòíîñòè, â îòëè÷èå îò äðóãèõ ïðîñòûõ ñóïåðàëãåáð ó íèõ åñòü ãðóïïà Âåéëÿ è îíà ïîçâîëÿåò îïðåäåëèòüîáîáùåííûå ìîäóëè Âåéëÿ. Òàêèì îáðàçîì äîêàçûâàåòñÿ, ÷òî õàðàêòåðûìîäóëåé Âåéëÿ äëÿ ñóïåðàëãåáðû osp(1, 2) ñîâïàäàþò ñî ñïåöèàëèçàöèÿ(2)ìè ìíîãî÷ëåíîâ Ìàêäîíàëüäà òèïà A2 .Äëÿ äîêàçàòåëüñòâà äàííîãî óòâåðæäåíèÿ ÿâíî íàõîäèòñÿ áàçèñ ìîäóëÿ Âåéëÿ íàä èññëåäóåìîé ñóïåðàëãåáðîé.

Ïåðâîèñòî÷íèê ïðèâåäåííûõíèæå îïðåäåëåíèé â ñòàòüÿõ [P, Mus1, Mus2]. Ñóïåðàëãåáðà Ëè osp(1, 2)èçîìîðôíà ïðÿìîé ñóììå sl2 ⊕ π1 , ãäå sl2 ÷åòíàÿ ÷àñòü è äâóìåðíûésl2 -ìîäóëü π1 â íå÷åòíîé ÷àñòè. Ïóñòü e, h, f ñòàíäàðòíûé áàçèñ sl2 è9ïóñòü g + , g − áàçèñ π1 . Èìååì ñëåäóþùèå íåòðèâèàëüíûå êîììóòàöèîííûå ñîîòíîøåíèÿ:[e, f ] = h, [h, e] = 2e, [h, f ] = −2f,[h, g + ] = g + , [h, g − ] = −g − , [g + , g − ]+ = h,[g + , g + ]+ = 2e, [g − , g − ]+ = −2f, [f, g + ] = g − , [e, g − ] = −g + .Èìååì ðàçëîæåíèå Êàðòàíàosp(1, 2) = n− ⊕ h ⊕ n+ , n− = span(f, g − ), n+ = span(e, g + ), h = span(h).Ðàññìîòðèì àëãåáðó òîêîâ osp(1, 2][t] = osp(1, 2) ⊗ K[t], åå ìîäóëü Âåéëÿ W−n îïðåäåëåí êàê öèêëè÷åñêèé ìîäóëü ñ îáðàçóþùåé w−n è ñîîòíîøåíèÿìè(n− ⊕ h) ⊗ tC[t].w−n = 0, (n− ⊗ 1).w−n = 0,h0 .w−n = −nw−n , (e ⊗ 1)n+1 .w−n = 0.Äëÿ x ∈ osp(1, 2) îïðåäåëèì êàê xi ∈ osp(1, 2)[t] ýëåìåíò x ⊗ ti .Òåîðåìà 4.ìè âèäà(2)Èìååì osp(1, 2) ⊗ tn K[t].w−n = 0 è W−n ïîðîæäåí ìîíîìàea1 .

. . eas gb+1 . . . gb+k w−n ,0 ≤ b1 < · · · < bk ≤ n − 1, 0 ≤ a1 ≤ · · · ≤ as ≤ n − k − s.Ýòîò áàçèñ äàåò ôîðìóëó äëÿ õàðàêòåðà. Äëÿ ýòîãî îïðåäåëèì ýëåìåíòû:(3)c1 (k22 , k12 , k11 ) = q2 +2kk1222k22 + k12 + k11k22 , k12 , k11,q2Òåïåðü ïîëó÷àåì, ÷òî õàðàêòåð ðàâåí:chW−m =Xxk11 −k22 c1 (k22 , k12 , k11 ).k11 ,k12 ,k22Äàëüøå, âû÷èñëÿÿ ìíîãî÷ëåíû Ìàêäîíàëüäà ñ ïîìîùüþ ôîðìóëû ÐàìàÉèï, ïîëó÷àåì ñëåäóþùóþ Òåîðåìó:Õàðàêòåðû ìîäóëåé Âåéëÿ íàä osp(1, 2) ÿâëÿþòñÿ ñïåöè(2)àëèçàöèÿìè íåñèììåòðè÷åñêèõ ìíîãî÷ëåíîâ Ìàêäîíàëüäà òèïà A2 âòî÷êå t = 0.Òåîðåìà 5.Òàêæå â äèññåðòàöèè äîêàçûâàåòñÿ ÷àñòíûé ñëó÷àé ãèïîòåçû ×åðåäíèêàÎððà, à èìåííî, äîêàçûâàåòñÿ, ÷òî äëÿ âåñîâ, êðàòíûõ îäíîìó ôóíäàìåíòàëüíîìó è äëÿ ëèíåéíûõ êîìáèíàöèé ïåðâîãî è ïîñëåäíåãî ôóíäàìåíòàëüíîãî äëÿ àëãåáð Ëè òèïîâ An ñïåöèàëèçàöèÿ íåñèììåòðè÷åñêèõ ìíîãî÷ëåíîâ Ìàêäîíàëüäà, ñîîòâåòñòâóþùèõ àíòèäîìèíàíòíîìó10âåñó, â òî÷êå t = ∞, ñîâïàäàåò ñ õàðàêòåðîì ýòèõ ìîäóëåé, ïîäêðó÷åííûõ íà ÏÁÂ-ôèëüòðàöèþ. Äëÿ ýòîé öåëè òàêæå íàõîäèòñÿ ÏÁÂ-áàçèñäëÿ èññëåäóåìûõ ïðåäñòàâëåíèé.

Íà îñíîâå ýòîãî áàçèñà âûïèñûâàåòñÿÏÁÂ-õàðàêòåð. Òàêæå ñ ïîìîùüþ êîìáèíàòîðíîé ôîðìóëû ÕàãëóíäàÕàèìàíà-Ëîåðà âû÷èñëÿåòñÿ ñïåöèàëèçàöèÿ ìíîãî÷ëåíîâ Ìàêäîíàëüäàè äîêàçûâàåòñÿ, ÷òî îíè ñîâïàäàþò. Òî÷íåå ãîâîðÿ, âåðíà ñëåäóþùàÿÒåîðåìà.Ïóñòü g ïðîñòàÿ àëãåáðà Ëè òèïà An è ïðåäïîëîæèì,÷òî λ = −mωi èëè λ = −m1 ω1 − mn ωn . ÒîãäàÒåîðåìà 6.Eλ (x, q −1 , ∞) = chP BW Wλ |p=q .Ñïèñîê ëèòåðàòóðû[1][2]E.Feigin, I.Makedonskyi, Nonsymmetric Macdonald polynomials, Demazure modulesand PBW ltration, Journal of Combinatorial Theory, Series A (2015), pp. 6084.Å. À.

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