Invariant tensors for simple groups

Abstract

The forms of the invariant primitive tensors for the simple Lie algebras A_l, B_l, C_l and D_l are investigated. A new family of symmetric invariant tensors is introduced using the non-trivial cocycles for the Lie algebra cohomology. For the A_l algebra it is explicitly shown that the generic forms of these tensors become zero except for the l primitive ones and that they give rise to the l primitive Casimir operators. Some recurrence and duality relations are given for the Lie algebra cocycles. Tables for the 3- and 5-cocycles for su(3) and su(4) are also provided. Finally, new relations involving the d and f su(n) tensors are given.Comment: Latex file. 34 pages. (Trivial) misprints corrected. To appear in Nucl. Phys.

LIE ALGEBRA AND INVARIANT TENSOR TECHNOLOGY FOR g2

“…Also, the tensor T ijklpq , while totally symmetric, is not traceless. For the optimal construction a sixth order scalar, we should construct [29] …”

Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups

“…Evans, Hassan, MacKay, Mountain (see [24] and references therein) constructed local invariant chiral currents as polynomials of the initial chiral currents of SU (n), SO(n), SP (n) principal chiral models and they found such combination of them that the corresponding charges are Casimir operators of these dynamical systems. Their paper was based on the paper of de Azcarraga, Macfarlane, MacKay, Perez Bueno (see [25] and references therein) about invariant tensors for simple Lie algebras. Let t µ be n ⊗ n traceless hermitian matrix representations of generators Lie algebra…”

“…, ∞. Another family of invariant symmetric tensors [29,30] (see also [25]) called D-family based on the product of the symmetric structure constant d µνλ of the SU (n) algebra is as follows: Here are n − 1 primitive invariant tensors on SU (n). The invariant tensors for M ≥ n are functions of primitive tensors.…”

“…The similar method of construction of chiral currents for SO(2l + 1) = B l , SP (2l) = C l groups was used by Evans et al [24] on the base of symmetric invariant tensors of de Azcarraga et al [25]. In the defining representation these group generators corresponding to algebras t µ satisfy the rules…”

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“…On Racah-Casimir Polynomials in g A systematic study of primitive invariant tensors of a compact, simple Lie algebra g has been the subject of a number of monographies and papers along the years; here, we will give a concise résumé, mainly based on [74,75,76,77], to which we address the reader for further elucidation and a list of references.…”