×

zbMATH — the first resource for mathematics

Quasi-identities on matrices and the Cayley-Hamilton polynomial. (English) Zbl 1332.16019
The trace identities of the full matrix algebra of order \(n\) over a field of characteristic 0 were described by Procesi and independently by Razmyslov. These all follow from the Cayley-Hamilton polynomial. Recall that one can write, using Newton’s formulae, the coefficients of the characteristic polynomial of a matrix in terms of traces of powers of the matrix.
The paper under review deals with quasi-identities, so let us recall what this is. Take a set \(X=\{x_k\}\) of noncommuting variables and let \(\{x_{ij}^{(k)}\mid 1\leq i,j\leq n\}\) be commuting ones. Form the free associative algebra \(C(X)\) on the variables \(X\) over the polynomial algebra of all \(x_{ij}^{(k)}\); its elements are the quasi-polynomials. (One may consider the ring of scalars of \(C(X)\) as the algebra of polynomial functions on the matrix algebra of order \(n\).)
The authors define in a natural way a trace in \(C(X)\), and study whether the trace quasi-identities for the \(n\times n\) matrices are generated by the Cayley-Hamilton polynomial. They describe large classes of trace quasi-identities that follow from the Cayley-Hamilton polynomial. Nevertheless there exist classes of (antisymmetric) quasi-identities which do not follow from the Cayley-Hamilton polynomial, and the authors describe one such class.

MSC:
16R60 Functional identities (associative rings and algebras)
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16R30 Trace rings and invariant theory (associative rings and algebras)
Software:
NCAlgebra
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Artin, M., On Azumaya algebras and finite-dimensional representations of rings, J. Algebra, 11, 532-563, (1969) · Zbl 0222.16007
[2] Auslander, M.; Goldman, O., Maximal orders, Trans. Amer. Math. Soc., 97, 1-24, (1960) · Zbl 0117.02506
[3] Auslander, M.; Goldman, O., The Brauer group of a commutative ring, Trans. Amer. Math. Soc., 97, 367-409, (1960) · Zbl 0100.26304
[4] Bahturin, Yu.; Brešar, M., Lie gradings on associative algebras, J. Algebra, 321, 264-283, (2009) · Zbl 1207.17037
[5] Bahturin, Yu.; Brešar, M.; Shestakov, I., Jordan gradings on associative algebras, Algebr. Represent. Theory, 14, 113-129, (2011) · Zbl 1237.16040
[6] Beidar, K. I.; Brešar, M.; Chebotar, M. A.; Martindale, W. S., On Herstein’s Lie map conjectures, I, Trans. Amer. Math. Soc., 353, 4235-4260, (2001) · Zbl 1019.16019
[7] Beidar, K. I.; Chebotar, M. A., On functional identities and d-free subsets of rings II, Comm. Algebra, 28, 3953-3972, (2000) · Zbl 0991.16018
[8] Beidar, K. I.; Chebotar, M. A., On Lie-admissible algebras whose commutator Lie algebras are Lie subalgebras of prime associative algebras, J. Algebra, 233, 675-703, (2000) · Zbl 0998.16024
[9] Beidar, K. I.; Fong, Y., On additive isomorphisms of prime rings preserving polynomials, J. Algebra, 217, 650-667, (1999) · Zbl 0934.16030
[10] Brešar, M., Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc., 335, 525-546, (1993) · Zbl 0791.16028
[11] Brešar, M.; Chebotar, M. A.; Martindale, W. S., Functional identities, (2007), Birkhäuser Verlag · Zbl 1132.16001
[12] Brešar, M.; Klep, I., A local-global principle for linear dependence of noncommutative polynomials, Israel J. Math., 193, 71-82, (2013) · Zbl 1293.16021
[13] Brešar, M.; Špenko, Š., Functional identities in one variable, J. Algebra, 401, 234-244, (2014) · Zbl 1303.16025
[14] Camino, J. F.; Helton, J. W.; Skelton, R. E.; Ye, J., Matrix inequalities: a symbolic procedure to determine convexity automatically, Integral Equations Operator Theory, 46, 399-454, (2003) · Zbl 1046.68139
[15] C. Chevalley, The Betti numbers of the exceptional Lie groups, in: Proc. International Congress of Mathematicians 1950, vol. II, pp. 21-24.
[16] De Concini, C.; Papi, P.; Procesi, C., The adjoint representation inside the exterior algebra of a simple Lie algebra · Zbl 1370.17014
[17] Drensky, V.; Formanek, E., Polynomial identity rings, Advanced Courses in Mathematics, CRM Barcelona, (2004), Birkhäuser Verlag Basel · Zbl 1077.16025
[18] Dynkin, E. B., Homologies of compact Lie groups, Amer. Math. Soc. Transl., 12, 251-300, (1959) · Zbl 0087.02502
[19] Formanek, E., Central polynomials for matrix rings, J. Algebra, 32, 129-132, (1972) · Zbl 0242.15004
[20] Formanek, E., The polynomial identities and invariants of \(n \times n\) matrices, (1991), Amer. Math. Soc. · Zbl 0714.16001
[21] Kanel-Belov, A.; Malev, S.; Rowen, L. H., The images of non-commutative polynomials evaluated on \(2 \times 2\) matrices, Proc. Amer. Math. Soc., 140, 465-478, (2012) · Zbl 1241.16017
[22] Kemer, A. R., Finite basability of identities of associative algebras, Algebra Logic, 26, 362-397, (1987) · Zbl 0664.16017
[23] Kostant, B., Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the ρ-decomposition \(C(\mathfrak{g}) = \mathit{End} V_\rho \otimes C(P)\), and the \(\mathfrak{g}\)-module structure of \(\bigwedge \mathfrak{g}\), Adv. Math., 125, 275-350, (1997)
[24] Le Bruyn, L.; Procesi, C., Etale local structure of matrix invariants and concomitants, (Algebraic Groups, Utrecht 1986, Springer Lecture Notes, vol. 1271, (1987)), 143-175
[25] Procesi, C., The invariant theory of \(n \times n\) matrices, Adv. Math., 19, 306-381, (1976) · Zbl 0331.15021
[26] Procesi, C., A formal inverse to the Cayley-Hamilton theorem, J. Algebra, 107, 63-74, (1987) · Zbl 0618.16014
[27] Procesi, C., Lie groups: an approach through invariants and representations, Universitext, (2007), Springer · Zbl 1154.22001
[28] Procesi, C., On the theorem of amitsur-levitzki, Israel J. Math., (2015), in press · Zbl 1333.16025
[29] Razmyslov, Yu. P., A certain problem of Kaplansky, Izv. Akad. Nauk SSSR, Ser. Mat., 37, 483-501, (1973)
[30] Razmyslov, Yu. P., Identities with trace in full matrix algebras over a field of characteristic zero, Izv. Akad. Nauk SSSR, Ser. Mat., 38, 723-756, (1974)
[31] Reeder, M., Exterior powers of the adjoint representation, Canad. J. Math., 49, 133-159, (1997) · Zbl 0878.20028
[32] Rosset, S., A new proof of the amitsur-levitski identity, Israel J. Math., 23, 187-188, (1976) · Zbl 0322.15020
[33] Rowen, L. H., Polynomial identities in ring theory, (1980), Academic Press · Zbl 0461.16001
[34] Saltman, D. J., Lectures on division algebras, (1999), Amer. Math. Soc. · Zbl 0934.16013
[35] Špenko, Š., On the image of a noncommutative polynomial, J. Algebra, 377, 298-311, (2013) · Zbl 1292.16016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.