The Graded Structure of Nondegenerate Meson Algebras

Abstract

Meson algebras are involved in the wave equation of meson particles in the same way as Clifford algebras are involved in the Dirac wave equation of electrons.Here we improve and generalize the information already obtained about their structure and their representations, when the symmetric bilinear form under consideration is nondegenerate. We emphasize their parity grading. We calculate the center of these meson algebras, and the center of their even subalgebra. Finally we show that every nondegenerate meson algebra over a field contains a group isomorphic to the group of automorphisms of the symmetric bilinear form.

Interior Multiplications and Deformations with Meson Algebras

“…After comparison with the formula (12.2), we realize that the Lie algebra morphism L 4 → L 4 is an isomorphism. In [HM2] it is explained that over this Lie algebra isomorphism L 4 → L 4 there is a group isomorphism G → Aut(M, ϕ). The group G is the multiplicative group in B 0 (M, ϕ) generated by all elements 2d 2 ϕ(d, d) −1 − 1 such that d ∈ M and ϕ(d, d) = 0 ; since these generators have an odd subparity, the elements of G are homogeneous for the subgrading of B 0 (M, ϕ), and we get an isomorphism G → Aut(M, ϕ) if we map each x ∈ G to the orthogonal transformation a −→ (−1) ℘x xax −1 ; moreover x −1 = ρ(x) .…”

“…Mind that usual Lie theory is not relevant here, because in characteristic 2 the derivatives of a polynomial of degree ≤ 2 may all vanish without this polynomial being constant. To understand how Aut(M, ϕ) looks like, we must look in another direction, as it is explained in [HM2]. The quadratic form a −→ ϕ(a, a) is now a semi-linear function M → K :…”

“…When n 0 = 0, then ϕ is called anisotropic, the mapping a −→ ϕ(a, a) is injective, and consequently the group Aut(M, ϕ) is reduced to the only element id M . In [HM2] it is proved that the dimension of Aut(M, ϕ) (which can be defined at least when K is infinite) is n 0 (n 0 + 1)/2 , and that this group is generated by all transformations a −→ a+ λ ϕ(a, d) d associated with an isotropic vector d ∈ M 0 and an scalar λ ∈ K. Moreover in B 0 (M, ϕ) there is again a group G isomorphic to Aut(M, ϕ) by an isomorphism that maps every x ∈ G to the transformation a −→ xax −1 . This group G is generated by the elements 1 + λd 2 which correspond to the generators of Aut(M, ϕ) mentioned above:…”

“…Remember that the universal algebra associated with Dirac's relations is a Clifford algebra. I refer to [HM2] for more information about the history of the subject (in particular the bibliography). I also refer to [HM2] for the description of nondegenerate meson algebras (meson algebras associated with nondegenerate symmetric bilinear forms) and their representations; up to now, almost all works involving meson algebras were devoted to this topic.…”

About the Structure of Meson Algebras

“…Therefore it suffices to prove the sur- Unlike previous authors (for instance Kemmer and Littlewood) we have discovered the structure of B(M, f ) without a preliminary study of its center. On the contrary we can now easily describe this center, and even calculate its idempotents when a basis of M is given; we refer to [6] for more details. From Theorem 3.1 we deduce that the center Z(B 0 (M, f )) of the even subalgebra has dimension n + 1, and admits a basis (ε 0 , ε 1 , ..., ε n ) made of idempotents; the operation of In all cases, ε 0 (that is ε k +ε n+1−k with k = 0) generates an ideal of dimension 1 which is supplementary to the ideal generated by M , because aε 0 = ε 0 a = 0 for all a ∈ M .…”