U linijnij algebri i teoriyi kilec simetrichna algebra algebra nad polem chi nad kilcem sho ye pevnoyu miroyu uzagalnennyam algebri mnogochleniv Simetrichna algebra ye pidalgebroyu tenzornoyi algebri i maye bagato spilnih vlastivostej iz zovnishnoyu algebroyu Zmist 1 Oznachennya 2 Vlastivosti 3 Div takozh 4 Posilannya 5 LiteraturaOznachennya red Yaksho M displaystyle M nbsp modul nad kommutativno asociativnim kilcem A displaystyle A nbsp z odiniceyu T M k 0 T k M displaystyle T M bigoplus k 0 infty T k M nbsp de T k M M M displaystyle T k M M otimes cdots otimes M nbsp tenzorna algebra modulya M displaystyle M nbsp Vvedemo takozh ideal I V T V displaystyle I V subseteq T V nbsp vidu I M s p a n v w w v v w V displaystyle I M mathrm span left v otimes w w otimes v Big v w in V right nbsp Simetrichnoyu algebroyu modulya M displaystyle M nbsp nazivayetsya algebra S M T M I M displaystyle S M T M I M nbsp Vlastivosti red Simetrichna algebra ye komutativnoyu i asociativnoyu A displaystyle A nbsp algebroyu z odiniceyu Simetrichna algebra ye gradujovanoyu S M k 0 S k M displaystyle S M bigoplus k 0 infty S k M nbsp de S k M T k M I M T k M displaystyle S k M T k M I M cap T k M nbsp Zokrema S 0 M A S 1 M M displaystyle S 0 M A S 1 M M nbsp Modul S k M displaystyle S k M nbsp nazivayetsya k im simetrichnim stepenem modulya M displaystyle M nbsp Yaksho M displaystyle M nbsp vilnij modul iz skinchennim bazisom x 1 x n displaystyle x 1 ldots x n nbsp to vidpovidnist x i X i displaystyle x i to X i nbsp prodovzhuyetsya do izomorfizmu algebri S M displaystyle S M nbsp i algebri mnogochleniv A X 1 X n displaystyle A X 1 ldots X n nbsp Takim chinom simetrichnu algebra ye uzagalnennyam algebri mnogochleniv Dlya bud yakogo gomomorfizmu A moduliv f M N displaystyle f M to N nbsp k ij tenzornij stepin T k f T k M T k N displaystyle T k f T k M to T k N nbsp indukuye gomomorfizm S k f S k M S k N displaystyle S k f S k M to S k N nbsp k ij simetrichnij stepin gomomorfizmu f displaystyle f nbsp Ci gomomorfizmi razom zadayut gomomorfizm A algebr S f S M S N displaystyle S f S M to S N nbsp Vidpovidnosti f S k f displaystyle f to S k f nbsp i f S k f displaystyle f to S k f nbsp ye vidpovidno kovariantnimi funktorami z kategoriyi A displaystyle A nbsp moduliv v sebe i v kategoriyu A algebr Dlya bud yakih dvoh A moduliv M i N isnuye prirodnij izomorfizm S M N S M S M displaystyle S M oplus N cong S M otimes cong S M nbsp Yaksho V displaystyle V nbsp vektornij prostir nad polem K displaystyle mathbb K nbsp harakteristiki 0 to simetrichna algebra S V displaystyle S V nbsp ye izomorfnoyu algebri simetrichnih kontravariantnih tenzoriv T S displaystyle T S nbsp tobto algebri polilinijnih vidobrazhen P V V K displaystyle P underbrace V otimes cdots otimes V rightarrow mathbb K nbsp na V displaystyle V nbsp razom z operaciyeyu simetrichnogo mnozhennya Yaksho P T k S Q T l S displaystyle P in T k S Q in T l S nbsp dva kontravariantni tenzori vidpovidnih poryadkiv to yih simetrichnij dobutok P Q T k l S displaystyle PQ in T k l S nbsp za oznachennyam zadayetsya yak P Q v 1 v k l 1 k l s S k l P v s 1 v s k Q v s k 1 v s k l displaystyle PQ v 1 ldots v k l frac 1 k l sum sigma in S k l P v sigma 1 ldots v sigma k Q v sigma k 1 ldots v sigma k l nbsp Yaksho V displaystyle V nbsp vektornij prostir rozmirnosti n to rozmirnist k ogo simetrichnogo stepenya rivnadim S k V n k 1 k displaystyle operatorname dim S k V binom n k 1 k nbsp Yak naslidok rozmirnist usiyeyi simetrichnoyi algebri ye neskinchennoyu na vidminu vid vipadku zovnishnoyi algebri Simetrichna algebra na vektornomu prostori ye vilnim ob yektom kategoriyi komutativnih asociativnih algebr z odiniceyu Div takozh red Zovnishnya algebra Kilce mnogochleniv Tenzorna algebraPosilannya red Hazewinkel Michiel red 2001 algebra Symmetric algebra Matematichna enciklopediya Springer ISBN 978 1 55608 010 4 Literatura red Bourbaki Nicolas 1989 Elements of mathematics Algebra I Springer Verlag ISBN 3 540 64243 9 Johan L Dupont Curvature and characteristic classes Lecture Notes in Mathematics Vol 640 Springer Berlin New York 1978 ISBN 3 540 08663 3 Otrimano z https uk wikipedia org w index php title Simetrichna algebra amp oldid 34227618
