On the graded identities of the Grassmann algebra

Yıl 2017, Cilt: 4 Sayı: 2 (Special Issue: Noncommutative rings and their applications), 165 - 180, 10.01.2017

Lucio Centrone

https://doi.org/10.13069/jacodesmath.284959

Öz

We survey the results concerning the graded identities of the infinite dimensional Grassmann algebra.

Anahtar Kelimeler

Graded polynomial identities, Grassmann algebra

Kaynakça

• [1] E. Aldjadeff, A. Kanel–Belov, Representability and Specht problem for G-graded algebras, Adv. Math. 225(5) (2010) 2391–2428.
• [2] N. Anisimov, Zp-codimension of Zp-identities of Grassmann algebra, Comm. Algebra 29(9) (2001) 4211–4230.
• [3] S. S. Azevedo, Graded identities for the matrix algebra of order n over an infinite field, Comm. Algebra 30(12) (2002) 5849–5860.
• [4] S. S. Azevedo, M. Fidelis, P. Koshlukov, Tensor product theorems in positive characteristic, J. Algebra 276(2) (2004) 836–845.
• [5] S. S. Azevedo, M. Fidelis, P. Koshlukov, Graded identities and PI equivalence of algebras in positive characteristic, Comm. Algebra 33(4) (2005) 1011–1022.
• [6] A. Berele, A. Regev, Exponential growth for codimensions of some p.i. algebras, J. Algebra 241(1) (2001) 118–145.
• [7] L. Centrone, Z2-graded identities of the Grassmann algebra in positive characteristic, Linear Algebra Appl. 435(12) (2011) 3297–3313.
• [8] L. Centrone, The G-graded identities of the Grassmann algebra, Arch. Math. (Brno) 52(3) (2016) 141–158.
• [9] L. Centrone, V. R. T. da Silva, On Z2-graded identities of UT_2(E) and their growth, Linear Algebra Appl. 471 (2015) 469–499.
• [10] O. M. Di Vincenzo, A note on the identities of the Grassmann algebras, Boll. Un. Mat. Ital. A(7) 5(3) (1991) 307–315.
• [11] O. M. Di Vincenzo, On the graded identities of $M_{1;1}(E)$, Israel J. Math. 80(3) (1992) 323–335.
• [12] O. M. Di Vincenzo, Cocharacters of G–graded algebras, Comm. Algebra 24(10) (1996) 3293–3310.
• [13] O. M. Di Vincenzo, V. Nardozza, Graded polynomial identities for tensor products by the Grassmann algebra, Comm. Algebra 31(3) (2003) 1453–1474.
• [14] O. M. Di Vincenzo, V. R. T. da Silva, On Z2–graded polynomial identities of the Grassmann algebra, Linear Algebra Appl. 43(1–2) (2009) 56–72.
• [15] V. Drensky, A minimal basis for identities of a second–order matrix algebra over a field of characteristic zero (Russian), Algebra i Logika 20 (1981) 282–290. Translation: Algebra and Logic 20 (1981) 188–194.
• [16] V. Drensky, Identities of representations of nilpotent Lie algebras, Comm. Algebra 25(7) (1997) 2115–2127.
• [17] V. Drensky, Free Algebras and PI–Algebras: Graduate Course in Algebra, Springer Series in Discrete Mathematics and Theoretical Computer Science, Springer, 2000.
• [18] A. Giambruno, P. Koshlukov, On the identities of the Grassmann algebras in characteristic p > 0, Isr. J. Math. 122(1) (2001) 305–316.
• [19] A. Giambruno, S. Mischenko, M. V. Zaicev, Polynomial identities on superalgebras and almost polynomial growth identities of Grassmann algebra, Comm. Algebra 29(9) (2001) 3787–3800.
• [20] A. Giambruno, M. V. Zaicev, Polynomial Identities and Asymptotic Methods, American Mathematical Society Mathematical Surveys and Monographs, Volume 122, 2005.
• [21] A. R. Kemer, Varieties and Z2–graded algebras (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 48 (1984) 1042–1059. Translation: Math. USSR, Izv. 25 (1985) 359–374.
• [22] A. R. Kemer, Ideals of Identities of Associative Algebras, AMS Trans. of Math. Monographs 87, 1991.
• [23] P. Koshlukov, Ideals of identities of representations of nilpotent Lie algebras, Comm. Algebra 28(7) (2000) 3095–3113.
• [24] P. Koshlukov, Basis of the identities of the matrix algebra of order two over a field of characteristic $p\neq2$, J. Algebra 241(1) (2001) 410–434.
• [25] P. Koshlukov, S. S. Azevedo, Graded identities for T–prime algebras over fields of positive characteristic, Isr. J. Math. 128(1) (2002) 157–176.
• [26] D. Krakowski, A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc. 181 (1973) 429–438.
• [27] V. N. Latyshev, Partially ordered sets and nonmatrix identities of associative algebras (Russian), Algebra i Logika 15 (1976) 53–70. Translation: Algebr. Log. 15 (1976) 34–45.
• [28] V. N. Latyshev, Finite basis property of identities of certain rings (Russian), Usp. Mat. Nauk 32(4(196)) (1977) 259–260.
• [29] J. B. Olsson, A. Regev, Colength sequence of some T-ideals, J. Algebra 38(1) (1976) 100–111.
• [30] A. P. Popov, Identities of the tensor square of the Grassmann algebra, Algebra i Logika 21(4) (1982) 442–471. Translation: Algebra and Logic 21(4) (1982) 296–316.
• [31] Yu. P. Razmyslov, Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero (Russian), Algebra i Logika 12 (1973) 83–113. Translation: Algebra and Logic 12 (1973) 47–63.
• [32] A. Regev, Existence of identities in A B, Israel J. Math. 11(2) (1972) 131–152.
• [33] A. Regev, Tensor products of matrix algebras over the Grassmann algebra, J. Algebra 133(2) (1990) 512–526.
• [34] A. Regev, Grassmann algebras over finite fields, Comm. Algebra 19(6) (1991) 1829–1849.
• [35] V. R. T. da Silva, Z2-codimensions of the Grassmann algebra, Comm. Algebra 37(9) (2009) 3342–3359.
• [36] V. R. T. da Silva, On ordinary and Z2-graded polinomial identities of the Grassmann algebra, Serdica Math. J. 38(1–3) (2012) 417–432.
• [37] N. Yu Anisimov, Codimensions of identities with the Grassmann algebra involution. (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2001, no. 3, 25–29, 77; translation in Moscow Univ. Math. Bull. 56(3) (2001) 25–29.
• [38] S. Yu. Vasilovsky, Zn-graded polynomial identities of the full matrix algebra of order n, Proc. Amer. Math. Soc. 127(12) (1999) 3517–3524.