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Register a new userLorentz and SU(3) groups derived from cubic quark algebra
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Richard Kerner
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We show that the Lorentz and the SU(3) groups can be derived from the covariance principle conserving a $Z_3$-graded three-form on a $Z_3$-graded cubic algebra representing quarks endowed with non-standard commutation laws.
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In the current version of QCD the quarks are described by ordinary Dirac fields, organized in the following internal symmetry multiplets: the $SU(3)$ colour, the $SU(2)$ flavour, and broken $SU(3)$ providing the family triplets. oindent In this paper we argue that internal and external (i.e. space-time) symmetries are entangled at least in the colour sector in order to introduce the spinorial quark fields in a way providing all the internal quarks degrees of freedom which do appear in the Standard Model. Because the $SU(3)$ colour algebra is endowed with natural $Z_3$-graded discrete automorphisms, in order to introduce entanglement the $Z_3$-graded version of Lorentz and Poincare algebras with their realizations are considered. The colour multiplets of quarks are described by $12$-component colour Dirac equations, with a $Z_3$-graded triplet of masses (one real and a Lee-Wick complex conjugate pair). We argue that all quarks in the Standard Model can be described by the $72$-component master quark sextet of $12$-component coloured Dirac fields.
An algebraic structure, Quotient Algebra Partition or QAP, is introduced in a serial of articles. The structure QAP is universal to Lie Algebras and enables algorithmic and exhaustive Cartan decompositions. The first episode draws the simplest form of the structure in terms of the spinor representation.
Else from the quotient algebra partition considered in the preceding episodes, two kinds of partitions on unitary Lie algebras are created by nonabelian bi-subalgebras. It is of interest that there exists a partition duality between the two kinds of partitions. With an application of an appropriate coset rule, the two partitions return to a quotient algebra partition when the generating bi-subalgebra is abelian. Procedures are proposed to merge or detach a co-quotient algebra, which help deliver type-AIII Cartan decompositions of more varieties. In addition, every Cartan decomposition is obtainable from the quotient algebra partition of the highest rank. Of significance is the universality of the quotient algebra partition to classical and exceptional Lie algebras.
In the 3rd episode of the serial exposition, quotient algebra partitions of rank zero earlier introduced undergo further partitions generated by bi-subalgebras of higher ranks. The refin
This is the sequel exposition following [1]. The framework quotient algebra partition is rephrased in the language of the s-representation. Thanks to this language, a quotient algebra partition of the simplest form is established under a minimum number of conditions governed by a bi-subalgebra of rank zero, i.e., a Cartan subalgebra. Within the framework, all Cartan subalgebras of su(N) are classified and generated recursively through the process of the subalgebra extension.
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