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Full-Text Articles in Applied Mathematics
Symmetric Criticality In General Relativity, Charles G. Torre
Symmetric Criticality In General Relativity, Charles G. Torre
Research Vignettes
In this worksheet I explore the local Lagrangian version of the Principle of Symmetric Criticality (PSC) due to Anderson, Fels, and Torre], which asserts the commutativity of the processes (i) of symmetry reduction (for finding group-invariant fields) and (ii) forming Euler-Lagrange equations. There are two obstructions to PSC, which I will call the Lie algebra obstruction and the isotropy obstruction. In this worksheet I will illustrate these obstructions in the General Theory of Relativity.
The Octonions And The Exceptional Lie Algebra G2, Ian M. Anderson
The Octonions And The Exceptional Lie Algebra G2, Ian M. Anderson
Research Vignettes
The octonions O are an 8-dimensional non-commutative, non-associative normed real algebra. The set of all derivations of O form a real Lie algebra. It is remarkable fact, first proved by E. Cartan in 1908, that the the derivation algebra of O is the compact form of the exceptional Lie algebra G2. In this worksheet we shall verify this result of Cartan and also show that the derivation algebra of the split octonions is the split real form of G2.
PDF and Maple worksheets can be downloaded from the links below.