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

This article is cited in 17 scientific papers (total in 17 papers)

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.

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

Bibliographic databases:

UDC: 512
MSC: 16A38
Received: 15.02.1984

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}


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    Citing articles on Google Scholar: Russian citations, English citations
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    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  crossref  mathscinet  zmath  isi
    2. Okhitin S., “Central Polynomials of 2nd-Order Matrices Algebra”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1988, no. 4, 61–63  mathscinet  zmath  isi
    3. A. A. Zolotykh, “Trace identities of matrix superalgebras with an involution”, Math. USSR-Izv., 37:1 (1991), 37–68  mathnet  crossref  mathscinet  zmath  adsnasa
    4. Allan Berele, “Azumaya-like properties of verbally prime algebras”, Journal of Algebra, 133:2 (1990), 272  crossref
    5. Kantor I., Trishin I., “On a Concept of Determinant in the Supercase”, Commun. Algebr., 22:10 (1994), 3679–3739  crossref  mathscinet  zmath  isi
    6. Allan Berele, Amitai Regev, “On the codimensions of the verbally prime P.I. algebras”, Isr J Math, 91:1-3 (1995), 239  crossref  mathscinet  zmath  isi
    7. Vasilovsky S., “Graded Polynomial Identities of the Jordan Superalgebra of a Bilinear Form”, J. Algebra, 184:1 (1996), 255–296  crossref  mathscinet  zmath  isi
    8. Szigeti J., “On the Characteristic Polynomial of Supermatrices”, Isr. J. Math., 107 (1998), 229–235  crossref  mathscinet  zmath  isi
    9. Vishne U., “Polynomial Identities of M-2(G)”, Commun. Algebr., 30:1 (2002), 443–454  crossref  mathscinet  zmath  isi
    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  mathscinet  isi
    11. Carini L., “Computing Cocharacters of Sign Trace Identities in Reduced Notation”, Linear Multilinear Algebra, 54:2 (2006), 147–156  crossref  mathscinet  zmath  isi
    12. L. M. Samoǐlov, “An analog of the Amitsur–Levitzki theorem for matrix superalgebras”, Siberian Math. J., 51:3 (2010), 491–495  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    13. Vishne U., “Polynomial Identities of M-2,M-1(G)”, Commun. Algebr., 39:6 (2011), 2044–2050  crossref  mathscinet  zmath  isi  elib
    14. L. M. Samoilov, “On the Primality Property of Central Polynomials of Prime Varieties of Associative Algebras”, Math. Notes, 99:3 (2016), 413–416  mathnet  crossref  crossref  mathscinet  isi  elib
    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  mathscinet  isi
    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  crossref  isi
    17. Fidelis C. Diniz D. Koshlukov P., “Embeddings For the Jordan Algebra of a Bilinear Form”, Adv. Math., 337 (2018), 294–316  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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