Geodesic
On the multiplicative and $T$-space structure of the relatively free Grassmann algebra
Sbornik. Mathematics, Tome 200 (2009) no. 9, pp. 1299-1338 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We investigate the multiplicative and $T$-space structure of the relatively free algebra $F^{(3)}$ with unity corresponding to the identity $[[x_1,x_2],x_3]=0$ over an infinite field of characteristic $p>0$. One of the basic results is the decomposition of quotient $T$-spaces connected with $F^{(3)}$ into a direct sum of simple components. Also, the $T$-spaces under consideration are commutative subalgebras of $F^{(3)}$; thus, the structure of $F^{(3)}$ and its subalgebras is described as modules over these commutative subalgebras. Finally, we consider the specifics of the case $p=2$. Bibliography: 15 titles.
Keywords: $T$-space, $T$-ideal, $n$-word, canonical basis, relatively free Grassmann algebra. 
@article{SM_2009_200_9_a1,
     author = {A. V. Grishin and L. M. Tsybulya},
     title = {On the multiplicative and $T$-space structure of the relatively free {Grassmann} algebra},
     journal = {Sbornik. Mathematics},
     pages = {1299--1338},
     year = {2009},
     volume = {200},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2009_200_9_a1/}
}
TY  - JOUR
AU  - A. V. Grishin
AU  - L. M. Tsybulya
TI  - On the multiplicative and $T$-space structure of the relatively free Grassmann algebra
JO  - Sbornik. Mathematics
PY  - 2009
SP  - 1299
EP  - 1338
VL  - 200
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_2009_200_9_a1/
LA  - en
ID  - SM_2009_200_9_a1
ER  - 
%0 Journal Article
%A A. V. Grishin
%A L. M. Tsybulya
%T On the multiplicative and $T$-space structure of the relatively free Grassmann algebra
%J Sbornik. Mathematics
%D 2009
%P 1299-1338
%V 200
%N 9
%U http://geodesic.mathdoc.fr/item/SM_2009_200_9_a1/
%G en
%F SM_2009_200_9_a1
A. V. Grishin; L. M. Tsybulya. On the multiplicative and $T$-space structure of the relatively free Grassmann algebra. Sbornik. Mathematics, Tome 200 (2009) no. 9, pp. 1299-1338. http://geodesic.mathdoc.fr/item/SM_2009_200_9_a1/

[1] A. R. Kemer, “Finite basis property of identities of associative algebras”, Algebra and Logic, 26:5 (1987), 362–397 | DOI | MR | Zbl | Zbl

[2] A. V. Grishin, “On finitely based systems of generalized polynomials”, Math. USSR-Izv., 37:2 (1991), 243–272 | DOI | MR | Zbl | Zbl

[3] A. V. Grishin, “O konechnoi baziruemosti abstraktnykh $T$-prostranstv”, Fundament. i prikl. matem., 1:3 (1995), 669–700 | MR | Zbl

[4] A. V. Grishin, V. V. Shchigolev, “$T$-spaces and their applications”, J. Math. Sci. (N. Y.), 134:1 (2006), 1799–1878 | DOI | MR | Zbl

[5] V. V. Shchigolev, “The finite basis property of $T$-spaces over fields of characteristic zero”, Izv. Math., 65:5 (2001), 1041–1071 | DOI | MR | Zbl

[6] A. V. Grishin, “Primery ne konechnoi baziruemosti T-prostranstv i T-idealov v kharakteristike 2”, Fundament. i prikl. matem., 5:1 (1999), 101–118 | MR | Zbl

[7] V. V. Schigolev, “Primery beskonechno baziruemykh T-idealov”, Fundament. i prikl. matem., 5:1 (1999), 307–312 | MR | Zbl

[8] A. Ya. Belov, “O neshpekhtovykh mnogoobraziyakh”, Fundament. i prikl. matem., 5:1 (1999), 47–66 | MR | Zbl

[9] C. K. Gupta, A. N. Krasilnikov, “A non-finitely based system of polynomial identities which contains the identity $x^6 = 0$”, Q. J. Math., 53:2 (2002), 173–183 | DOI | MR | Zbl

[10] P. Zh. Chiripov, P. N. Siderov, “O bazisakh tozhdestv nekotorykh mnogoobrazii assotsiativnykh algebr”, PLISKA Stud. Math. Bulgar., 2 (1981), 103–115 | MR | Zbl

[11] A. V. Grishin, “The variety of associative rings, which satisfy the identity $x^{32}=0$, is not Specht”, Formal power series and algebraic combinatorics (Moscow, 2000), Springer-Verlag, Berlin, 686–691 | MR | Zbl

[12] A. V. Grishin, L. M. Surmina, “On $T$-spaces of $n$-words over a field of characteristic $p>0$”, Russian Math. Surveys, 62:4 (2007), 802–803 | DOI | MR | Zbl

[13] A. V. Grishin, L. M. Tsybulya, “O $(p,n)$-probleme”, Vestn. SamGU. Estestv. ser. Matem., 7:57 (2007), 35–55

[14] A. V. Grishin, L. M. Surmina, “O $T$-algebre $n$-slov”, Mezhdunarodnaya konferentsiya po algebre i teorii chisel, posvyaschennaya 80-letiyu V. E. Voskresenskogo. Tezisy dokladov (Samara, 2007), Univers grupp, Samara, 2007, 15–16

[15] A. V. Grishin, “On center of a relatively free Grassmann algebra”, Mezhdunarodnaya konferentsiya “Sovremennye problemy matematiki, mekhaniki i ikh prilozheniya”, posvyaschennaya 70-letiyu rektora MGU akademiku V. A. Sadovnichemu. Tezisy dokladov (Moskva, 2009), Universitetskaya kniga, M., 2009, 406–407