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Coxeter element and particle masses

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Abstract

Let \({\mathfrak {g}}\) be a simple Lie algebra of rank r over \(\mathbb {C}, {\mathfrak {h}}\subset {\mathfrak {g}}\) a Cartan subalgebra. We construct a family of r commuting Hermitian operators acting on \({\mathfrak {h}}\) whose eigenvalues are equal to the coordinates of the eigenvectors of the Cartan matrix of \({\mathfrak {g}}\).

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Notes

  1. Note that A is (close to) a symmetric matrix, whereas c is an orthogonal matrix, the passage from one to another is somewhat similar to the classical Cayley transform.

  2. The couple of Cartan subalgebras \(({\mathfrak {h}}, {\mathfrak {h}}^{\prime })\) from [14], § 6 becomes \(({\tilde{{\mathfrak {h}}}}, {\mathfrak {h}})\) in op. cit., 8.6. Two occurrences of \({\mathfrak {h}}\) in [14], p. 1023, 2nd line, should be replaced by \(\tilde{\mathfrak h}\).

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062, Toulouse, France

    Laura Brillon & Vadim Schechtman

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  1. Laura Brillon
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  2. Vadim Schechtman
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Correspondence to Vadim Schechtman.

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To Joseph Bernstein on his 70th birthday

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Brillon, L., Schechtman, V. Coxeter element and particle masses. Sel. Math. New Ser. 22, 2591–2609 (2016). https://doi.org/10.1007/s00029-016-0283-5

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