Asymptotics for Graded Capelli Polynomials

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• 作者：Francesca Benanti (1)
  
  1. Dipartimento di Matematica ed Informatica ; Universit脿 di Palermo ; via Archirafi ; 34 ; 90123 ; Palermo ; Italy
  
• 关键词：Superalgebras ; Polynomial identities ; Codimensions ; Growth ; 16R10 ; 16P90 ; 16W55
• 刊名：Algebras and Representation Theory
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：18
• 期：1
• 页码：221-233
• 全文大小：278 KB
• 参考文献：1. Benanti, F.: On the exponential growth of graded Capelli polynomials. Israel J. Math. 196, 51鈥?5 (2013) CrossRef
  2. Benanti, F., Giambruno, A., Pipitone, M.: Polynomial identities on superalgebras and exponential growth. J. Algebra 269, 422鈥?38 (2003) CrossRef
  3. Benanti, F., Sviridova, I.: Asymptotics for Amitsur鈥檚 Capelli-type polynomials and verbally prime PI-algebras. Israel. J. Math. 156, 73鈥?1 (2006)
  4. Giambruno, A., Regev, A.: Wreath products and P.I. algebras. J. Pure Applied Algebra 35, 133鈥?49 (1985) CrossRef
  5. Giambruno, A., Zaicev, M.: Exponential codimension growth of P.I. algebras: an exact estimate,. Adv. Math. 142, 221鈥?43 (1999) CrossRef
  6. Giambruno, A., Zaicev, M.: Asymptotics for the standard and the Capelli identities. Israel J. Math 135, 125鈥?45 (2003) CrossRef
  7. Giambruno, A., Zaicev, M.: Polynomial identities and asymptotic methods. Amer. Math. Soc. Mathematical Surveys and Monographs, vol, 122 (2005)
  8. Kemer, A.R.: / Varieties of \(\mathbb {Z}_{2}\) / -graded algebras,(Russian). Izv. Akad. Nauk SSSR Ser. Math 48 (5), 1042鈥?059 (1984)
  9. Kemer A.R.: Ideals of identities of associative algebras. Amer. Math. Soc. Translations of Math. Monographs 87, Providence, RI, (1991)
  
• 刊物主题：Commutative Rings and Algebras; Associative Rings and Algebras; Non-associative Rings and Algebras;
• 出版者：Springer Netherlands
• ISSN：1572-9079

文摘

The finite dimensional simple superalgebras play an important role in the theory of PI-algebras in characteristic zero. The main goal of this paper is to characterize the T 2-ideal of graded identities of any such algebra by considering the growth of the corresponding supervariety. We consider the T 2-ideal 螕 M+1,L+1 generated by the graded Capelli polynomials C a p M+1[Y,X] and C a p L+1[Z,X] alternanting on M+1 even variables and L+1 odd variables, respectively. We prove that the graded codimensions of a simple finite dimensional superalgebra are asymptotically equal to the graded codimensions of the T 2-ideal 螕 M+1,L+1, for some fixed natural numbers M and L. In particular $$c^{sup}_{n}(\Gamma_{k^{2}+l^{2}+1,2kl+1})\simeq c^{sup}_{n}(M_{k,l}(F))$$ and $$c^{sup}_{n}(\Gamma_{s^{2}+1,s^{2}+1})\simeq c^{sup}_{n}(M_{s}(F\oplus tF)).$$ These results extend to finite dimensional superalgebras a theorem of Giambruno and Zaicev [6] giving in the ordinary case the asymptotic equality $$c^{sup}_{n}(\Gamma_{k^{2}+1,1})\simeq c^{sup}_{n}(M_{k}(F)) $$ between the codimensions of the Capelli polynomials and the codimensions of the matrix algebra M k (F).