Noncommutative algebraic equations and the noncommutative eigenvalue problem. (English) Zbl 0972.15001

Schwarz, Albert

Lett. Math. Phys. 52, No. 2, 177-184 (2000).

For the noncommutative algebraic equations \[ x^n= a,\quad x^{n-1}+ a_2x^{n-2}+\cdots+ a_n\tag{1} \] the noncommutative eigenvalue \((\lambda)\) problem: \(AX= \lambda X\) \((A= (a_{ij}), \widetilde X= (x_j))\), i.e. the generalized Vieta theorem [cf. D. Fuchs and A. Schwarz, Am. Math. Soc. Transl., Ser. 2, 169, 15-22 (1995; Zbl 0837.15011); A. Connes and A. Schwarz, Lett. Math. Phys. 39, No. 4, 349-353 (1997; Zbl 0874.15010)] is analyzed. As a result, the theorem is proved about the structure of perturbation series for the traces Tr\(x^r\), where \(x\) is solution of the equation (1). This theorem generalizes to the case of arbitrary number \(r\) the recently proposed \((r=1)\) perturbative approach applied to the gauge \(U(1)^k\)-invariant Born-Infeld Lagrangian [cf. P. Aschieri, D. Brace, B. Morariu and B. Zumino, Nucl. Phys. B 588, No. 1-2, 521-527 (2000; Zbl 0972.81092)]. The noncommutative generalization of the Vieta theorem is considered as a part of the general theory of noncommutative functions [cf. I. Gelfand and V. Retakh, The Gelfand Mathematical Seminars, 93-100 (1996; Zbl 0865.05074); Sel. Math., New Ser. 3, No. 4, 517-546 (1997; Zbl 0919.16011)].

Reviewer: A.A.Bogush (Minsk)

MSC:

15A18	Eigenvalues, singular values, and eigenvectors
81Q15	Perturbation theories for operators and differential equations in quantum theory
15A90	Applications of matrix theory to physics (MSC2000)
81R60	Noncommutative geometry in quantum theory

Keywords:

gauge invariant Born-Infeld Lagrangian; noncommutative eigenvalue problem; noncommutative algebraic equations; generalized Vieta theorem; perturbation series; noncommutative functions

Citations:

Zbl 0837.15011; Zbl 0874.15010; Zbl 0972.81092; Zbl 0865.05074; Zbl 0919.16011