𝐙_{𝐧}-graded polynomial identities of the full matrix algebra of order 𝐧

Vasilovsky, Sergei

doi:10.1090/S0002-9939-99-04986-2

$\mathbf {Z}_n$-graded polynomial identities of the full matrix algebra of order $n$
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by Sergei Yu. Vasilovsky PDF

Proc. Amer. Math. Soc. 127 (1999), 3517-3524 Request permission

Abstract:

The algebra $M_n(F)$ of all $n\times n$ matrices over a field $F$ has a natural $\mathbf {Z}_n$-grading $M_n(F)=\sum _{\alpha \in \mathbf {Z}_n}\bigoplus \mathcal {M}_n^{(\alpha )}$. In this paper graded identities of the $\mathbf {Z}_n$-graded algebra $M_n(F)$ over a field of characteristic zero are studied. It is shown that all the $\mathbf {Z}_n$-graded polynomial identities of $M_n(F)$ follow from the following: \[ x_1x_2-x_2x_1=0,~~~~\alpha (x_1)=\alpha (x_2)=\overline {0};\] \[ x_1xx_2-x_2xx_1=0,~~~~\alpha (x_1)=\alpha (x_2)=-\alpha (x).\]

References

Yu. A. Bakhturin, Tozhdestva v algebrakh Li, “Nauka”, Moscow, 1985 (Russian). MR 785004
Allan Berele, Trace identities and $\bf {Z}/2\bf {Z}$-graded invariants, Trans. Amer. Math. Soc. 309 (1988), no. 2, 581–589. MR 938917, DOI 10.1090/S0002-9947-1988-0938917-0
Onofrio Mario Di Vincenzo, On the graded identities of $M_{1,1}(E)$, Israel J. Math. 80 (1992), no. 3, 323–335. MR 1202575, DOI 10.1007/BF02808074
Onofrio Mario Di Vincenzo, Cocharacters of $G$-graded algebras, Comm. Algebra 24 (1996), no. 10, 3293–3310. MR 1402563, DOI 10.1080/00927879608825751
V. S. Drenski, A minimal basis for identities of a second-order matrix algebra over a field of characteristic $0$, Algebra i Logika 20 (1981), no. 3, 282–290, 361 (Russian). MR 648317
A. R. Kemer, Varieties and $Z_2$-graded algebras, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 5, 1042–1059 (Russian). MR 764308
A. R. Kemer, Finite basability of identities of associative algebras, Algebra i Logika 26 (1987), no. 5, 597–641, 650 (Russian). MR 985840
C. Procesi, The invariant theory of $n\times n$ matrices, Advances in Math. 19 (1976), no. 3, 306–381. MR 419491, DOI 10.1016/0001-8708(76)90027-X
Ju. P. Razmyslov, The existence of a finite basis for the identities of the matrix algebra of order two over a field of characteristic zero, Algebra i Logika 12 (1973), 83–113, 121 (Russian). MR 0340348
Ju. P. Razmyslov, Identities with trace in full matrix algebras over a field of characteristic zero, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 723–756 (Russian). MR 0506414
Yu. P. Razmyslov, Trace identities and central polynomials in matrix superalgebras $M_{n,k}$, Mat. Sb. (N.S.) 128(170) (1985), no. 2, 194–215, 287 (Russian). MR 809485