# On the graded identities of the Grassmann algebra

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

Keywords:
Graded polynomial identities, Grassmann algebra

### Abstract

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

### References

E. Aldjadeff, A. Kanel–Belov, Representability and Specht problem for G-graded algebras, Adv. Math. 225(5) (2010) 2391–2428.

N. Anisimov, Zp-codimension of Zp-identities of Grassmann algebra, Comm. Algebra 29(9) (2001) 4211–4230.

S. S. Azevedo, Graded identities for the matrix algebra of order n over an infinite field, Comm. Algebra 30(12) (2002) 5849–5860.

S. S. Azevedo, M. Fidelis, P. Koshlukov, Tensor product theorems in positive characteristic, J. Algebra 276(2) (2004) 836–845.

S. S. Azevedo, M. Fidelis, P. Koshlukov, Graded identities and PI equivalence of algebras in positive characteristic, Comm. Algebra 33(4) (2005) 1011–1022.

A. Berele, A. Regev, Exponential growth for codimensions of some p.i. algebras, J. Algebra 241(1) (2001) 118–145.

L. Centrone, Z2-graded identities of the Grassmann algebra in positive characteristic, Linear Algebra Appl. 435(12) (2011) 3297–3313.

L. Centrone, The G-graded identities of the Grassmann algebra, Arch. Math. (Brno) 52(3) (2016) 141–158.

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.

O. M. Di Vincenzo, A note on the identities of the Grassmann algebras, Boll. Un. Mat. Ital. A(7) 5(3) (1991) 307–315.

O. M. Di Vincenzo, On the graded identities of M_{1;1}(E)M

1;1

(E), Israel J. Math. 80(3) (1992) 323–335.

O. M. Di Vincenzo, Cocharacters of G–graded algebras, Comm. Algebra 24(10) (1996) 3293–3310.

O. M. Di Vincenzo, V. Nardozza, Graded polynomial identities for tensor products by the Grassmann algebra, Comm. Algebra 31(3) (2003) 1453–1474.

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.

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.

V. Drensky, Identities of representations of nilpotent Lie algebras, Comm. Algebra 25(7) (1997) 2115–2127.

V. Drensky, Free Algebras and PI–Algebras: Graduate Course in Algebra, Springer Series in Discrete Mathematics and Theoretical Computer Science, Springer, 2000.

A. Giambruno, P. Koshlukov, On the identities of the Grassmann algebras in characteristic p > 0, Isr. J. Math. 122(1) (2001) 305–316.

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.

A. Giambruno, M. V. Zaicev, Polynomial Identities and Asymptotic Methods, American Mathematical Society Mathematical Surveys and Monographs, Volume 122, 2005.

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.

A. R. Kemer, Ideals of Identities of Associative Algebras, AMS Trans. of Math. Monographs 87, 1991.

P. Koshlukov, Ideals of identities of representations of nilpotent Lie algebras, Comm. Algebra 28(7) (2000) 3095–3113.

P. Koshlukov, Basis of the identities of the matrix algebra of order two over a field of characteristic pneq2p≠2, J. Algebra 241(1) (2001) 410–434.

P. Koshlukov, S. S. Azevedo, Graded identities for T–prime algebras over fields of positive characteristic, Isr. J. Math. 128(1) (2002) 157–176.

D. Krakowski, A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc. 181 (1973) 429–438.

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.

V. N. Latyshev, Finite basis property of identities of certain rings (Russian), Usp. Mat. Nauk 32(4(196)) (1977) 259–260.

J. B. Olsson, A. Regev, Colength sequence of some T-ideals, J. Algebra 38(1) (1976) 100–111.

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.

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.

A. Regev, Existence of identities in A B, Israel J. Math. 11(2) (1972) 131–152.

A. Regev, Tensor products of matrix algebras over the Grassmann algebra, J. Algebra 133(2) (1990) 512–526.

A. Regev, Grassmann algebras over finite fields, Comm. Algebra 19(6) (1991) 1829–1849.

V. R. T. da Silva, Z2-codimensions of the Grassmann algebra, Comm. Algebra 37(9) (2009) 3342–3359.

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.

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.

S. Yu. Vasilovsky, Zn-graded polynomial identities of the full matrix algebra of order n, Proc. Amer. Math. Soc. 127(12) (1999) 3517–3524.

N. Anisimov, Zp-codimension of Zp-identities of Grassmann algebra, Comm. Algebra 29(9) (2001) 4211–4230.

S. S. Azevedo, Graded identities for the matrix algebra of order n over an infinite field, Comm. Algebra 30(12) (2002) 5849–5860.

S. S. Azevedo, M. Fidelis, P. Koshlukov, Tensor product theorems in positive characteristic, J. Algebra 276(2) (2004) 836–845.

S. S. Azevedo, M. Fidelis, P. Koshlukov, Graded identities and PI equivalence of algebras in positive characteristic, Comm. Algebra 33(4) (2005) 1011–1022.

A. Berele, A. Regev, Exponential growth for codimensions of some p.i. algebras, J. Algebra 241(1) (2001) 118–145.

L. Centrone, Z2-graded identities of the Grassmann algebra in positive characteristic, Linear Algebra Appl. 435(12) (2011) 3297–3313.

L. Centrone, The G-graded identities of the Grassmann algebra, Arch. Math. (Brno) 52(3) (2016) 141–158.

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.

O. M. Di Vincenzo, A note on the identities of the Grassmann algebras, Boll. Un. Mat. Ital. A(7) 5(3) (1991) 307–315.

O. M. Di Vincenzo, On the graded identities of M_{1;1}(E)M

1;1

(E), Israel J. Math. 80(3) (1992) 323–335.

O. M. Di Vincenzo, Cocharacters of G–graded algebras, Comm. Algebra 24(10) (1996) 3293–3310.

O. M. Di Vincenzo, V. Nardozza, Graded polynomial identities for tensor products by the Grassmann algebra, Comm. Algebra 31(3) (2003) 1453–1474.

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.

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.

V. Drensky, Identities of representations of nilpotent Lie algebras, Comm. Algebra 25(7) (1997) 2115–2127.

V. Drensky, Free Algebras and PI–Algebras: Graduate Course in Algebra, Springer Series in Discrete Mathematics and Theoretical Computer Science, Springer, 2000.

A. Giambruno, P. Koshlukov, On the identities of the Grassmann algebras in characteristic p > 0, Isr. J. Math. 122(1) (2001) 305–316.

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.

A. Giambruno, M. V. Zaicev, Polynomial Identities and Asymptotic Methods, American Mathematical Society Mathematical Surveys and Monographs, Volume 122, 2005.

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.

A. R. Kemer, Ideals of Identities of Associative Algebras, AMS Trans. of Math. Monographs 87, 1991.

P. Koshlukov, Ideals of identities of representations of nilpotent Lie algebras, Comm. Algebra 28(7) (2000) 3095–3113.

P. Koshlukov, Basis of the identities of the matrix algebra of order two over a field of characteristic pneq2p≠2, J. Algebra 241(1) (2001) 410–434.

P. Koshlukov, S. S. Azevedo, Graded identities for T–prime algebras over fields of positive characteristic, Isr. J. Math. 128(1) (2002) 157–176.

D. Krakowski, A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc. 181 (1973) 429–438.

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.

V. N. Latyshev, Finite basis property of identities of certain rings (Russian), Usp. Mat. Nauk 32(4(196)) (1977) 259–260.

J. B. Olsson, A. Regev, Colength sequence of some T-ideals, J. Algebra 38(1) (1976) 100–111.

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.

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.

A. Regev, Existence of identities in A B, Israel J. Math. 11(2) (1972) 131–152.

A. Regev, Tensor products of matrix algebras over the Grassmann algebra, J. Algebra 133(2) (1990) 512–526.

A. Regev, Grassmann algebras over finite fields, Comm. Algebra 19(6) (1991) 1829–1849.

V. R. T. da Silva, Z2-codimensions of the Grassmann algebra, Comm. Algebra 37(9) (2009) 3342–3359.

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.

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.

S. Yu. Vasilovsky, Zn-graded polynomial identities of the full matrix algebra of order n, Proc. Amer. Math. Soc. 127(12) (1999) 3517–3524.

Published

2017-05-15

Section

Articles