Rad HAZU, Matematičke znanosti, Vol. 18 (2014), 55-72.

ON KAC PARAMETERS AND SPECTRAL DECOMPOSITION OF A MATRIX OF SPECIALIZED ROOTS OF LIE ALGEBRA sln

Zlatko Drmač and Tomislav Šikić

Faculty of Sciences, Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: drmac@math.hr

Faculty of Electrical Engineering and Computing, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: tomislav.sikic@fer.hr

Abstract.   This paper presents interesting spectral properties of a particular integer skew-symmetric matrix, used to encode information on Z-gradation of type s~ for classical affine Lie algebra sln~. It is shown that the hidden Kac parameters can be revealed using an explicitly computed eigenvector in a Gram-Schmidt orthogonalization process.

2010 Mathematics Subject Classification.   17B67, 15A18, 15B57.

Key words and phrases.   Z-gradation for classical (affine) Lie algebra, Kac parameters of finite order automorphisms of simple Lie algebras, matrix of specialized roots, integer skew-symmetric matrix, spectral decomposition.

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