Narrow Results

From an octonion algebra $𝕆$ over a field $k$ of characteristic not two or three, we show that the fundamental representation $\mathrm{\text{Im}}(𝕆)$ of the derivation algebra $\mathrm{\text{Der}}(𝕆)$ and the spinor representation $𝕆$ of $𝔰𝔬(\mathrm{\text{Im}}(𝕆))$ are special orthogonal representations. They have particular geometric properties coming from their similarities with binary cubics and we show that the covariants of these representations and their Mathews identities are related to the Fano plane and the affine space ${({\mathbb{Z}}_2)}^3$. This also permits to give constructions of exceptional Lie superalgebras.

A consistent, local coordinate formulation of covariant Hamiltonian field theory is presented. Whereas the covariant canonical field equations are equivalent to the Euler–Lagrange field equations, the covariant canonical transformation theory offers more general means for defining mappings that preserve the form of the field equations than the usual Lagrangian description. It is proven that Poisson brackets, Lagrange brackets, and canonical 2-forms exist that are invariant under canonical transformations of the fields. The technique to derive transformation rules for the fields from generating functions is demonstrated by means of various examples. In particular, it is shown that the infinitesimal canonical transformation furnishes the most general form of Noether's theorem. Furthermore, we specify the generating function of an infinitesimal space-time step that conforms to the field equations.

After briefly describing Hamiltonian BRST methods and the multisymplectic approach to field theory, a symmetric geometric Lagrangian is studied by extending the BRST method to the multisymplectic setting. This work uses ideas first introduced by Hrabak [On the multisymplectic origin of the nonabelian deformation algebra of pseudoholomorphic embeddings in strictly almost Kähler ambient manifolds, and the corresponding BRST algebra, preprint (1999), arXiv: math-ph/9904026].