RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

 Mat. Sb. (N.S.), 1985, Volume 128(170), Number 2(10), Pages 194–215 (Mi msb2123)

Trace identities and central polynomials in the matrix superalgebras $M_{n,k}$

Yu. P. Razmyslov

Abstract: A complete description is given of trace identities for matrix superalgebras $M_{n,k}=\{\begin{pmatrix} a_{11} & a_{12} a_{21} & a_{22} \end{pmatrix}\}$, where $a_{11}$ and $a_{22}$ are square matrices of orders $n$ and $k$ respectively over the even elements of a Grassmann algebra $G$ with countably many generators, while $a_{12}$ and $a_{21}$ are $n\times k$ and $k\times n$ rectangular matrices respectively over the odd elements of $G$. A relation is found between multilinear trace identities of degree $l$ in the algebra $M_{n,k}$ and irreducible representations of a symmetric group of order $(l+1)!$. It is proved that over a field of characteristic zero all trace identities of $M_{n,k}$ follow from identities of degree $nk+n+k$ that hold in that algebra. For every algebra $M_{n,k}$ over a field of arbitrary characteristic a central polynomial is given explicitly.
Bibliography: 7 titles.

Full text: PDF file (1302 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1987, 56:1, 187–206

Bibliographic databases:

UDC: 512
MSC: 16A38

Citation: Yu. P. Razmyslov, “Trace identities and central polynomials in the matrix superalgebras $M_{n,k}$”, Mat. Sb. (N.S.), 128(170):2(10) (1985), 194–215; Math. USSR-Sb., 56:1 (1987), 187–206

Citation in format AMSBIB
\Bibitem{Raz85} \by Yu.~P.~Razmyslov \paper Trace identities and central polynomials in the matrix superalgebras~$M_{n,k}$ \jour Mat. Sb. (N.S.) \yr 1985 \vol 128(170) \issue 2(10) \pages 194--215 \mathnet{http://mi.mathnet.ru/msb2123} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=809485} \zmath{https://zbmath.org/?q=an:0604.16019|0601.16016} \transl \jour Math. USSR-Sb. \yr 1987 \vol 56 \issue 1 \pages 187--206 \crossref{https://doi.org/10.1070/SM1987v056n01ABEH003031} 

• http://mi.mathnet.ru/eng/msb2123
• http://mi.mathnet.ru/eng/msb/v170/i2/p194

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Berele A., “Trace Identities and Z/2Z-Graded Invariants”, Trans. Am. Math. Soc., 309:2 (1988), 581–589
2. Okhitin S., “Central Polynomials of 2nd-Order Matrices Algebra”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1988, no. 4, 61–63
3. A. A. Zolotykh, “Trace identities of matrix superalgebras with an involution”, Math. USSR-Izv., 37:1 (1991), 37–68
4. Allan Berele, “Azumaya-like properties of verbally prime algebras”, Journal of Algebra, 133:2 (1990), 272
5. Kantor I., Trishin I., “On a Concept of Determinant in the Supercase”, Commun. Algebr., 22:10 (1994), 3679–3739
6. Allan Berele, Amitai Regev, “On the codimensions of the verbally prime P.I. algebras”, Isr J Math, 91:1-3 (1995), 239
7. Vasilovsky S., “Graded Polynomial Identities of the Jordan Superalgebra of a Bilinear Form”, J. Algebra, 184:1 (1996), 255–296
8. Szigeti J., “On the Characteristic Polynomial of Supermatrices”, Isr. J. Math., 107 (1998), 229–235
9. Vishne U., “Polynomial Identities of M-2(G)”, Commun. Algebr., 30:1 (2002), 443–454
10. Kemer A., “On Some Problems in Pi-Theory in Characteristic P Connected with Dividing by P”, Proceedings of the Third International Algebra Conference, eds. Fong Y., Shiao L., Zelmanov E., Springer, 2003, 53–66
11. Carini L., “Computing Cocharacters of Sign Trace Identities in Reduced Notation”, Linear Multilinear Algebra, 54:2 (2006), 147–156
12. L. M. Samoǐlov, “An analog of the Amitsur–Levitzki theorem for matrix superalgebras”, Siberian Math. J., 51:3 (2010), 491–495
13. Vishne U., “Polynomial Identities of M-2,M-1(G)”, Commun. Algebr., 39:6 (2011), 2044–2050
14. L. M. Samoilov, “On the Primality Property of Central Polynomials of Prime Varieties of Associative Algebras”, Math. Notes, 99:3 (2016), 413–416
15. KanelBelov A. Karasik Y. Rowen L., “Computational Aspects of Polynomial Identities: Vol 1, Kemer'S Theorems, 2Nd Edition”, Computational Aspects of Polynomial Identities: Vol 1, Kemer'S Theorems, 2Nd Edition, Monographs and Research Notes in Mathematics, 16, Crc Press-Taylor & Francis Group, 2016, 1–407
16. Diniz D. Bezerra Junior C.F., “Primeness Property For Graded Central Polynomials of Verbally Prime Algebras”, J. Pure Appl. Algebr., 222:6 (2018), 1388–1404
17. Fidelis C. Diniz D. Koshlukov P., “Embeddings For the Jordan Algebra of a Bilinear Form”, Adv. Math., 337 (2018), 294–316
•  Number of views: This page: 309 Full text: 101 References: 25