A compact Lie algebra can be seen as the smallest real form of a corresponding complex Lie algebra, namely the complexification. Definition. Formally, one may define a compact Lie algebra either as the Lie algebra of a compact Lie group, or as a real Lie algebra whose Killing form is negative definite. These definitions do not quite agree: The Killing form on the Lie algebra of a compact Lie.
Spin(7)-subgroups of SO(8) and Spin(8) V. S. Varadarajan Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA The observations made here are prompted by the paper (1) of De Sapio in which he gives an exposition of the principle of triality and related topics in the context of the Octonion algebra of Cayley. However many of the results discussed there are highly.
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The XYZ antiferromagnetic model in linear spin-wave frame is shown explicitly to have an su(1,2) algebraic structure: the Hamiltonian can be written as a linear function of the su(1,2) algebra generators. Based on it, the energy eigenvalues are obtained by making use of the similar transformations, and the algebraic diagonalization method is investigated.
The large linear superconformal algebra is generated by spin-2 stress tensor, four spin- supersymmetry generators, seven spin-1 currents and and four spin- currents. Six spin-1 currents are the generators of two SU (2) affine algebras where the levels are denoted by and one spin-1 current is the U (1) current.
The algebraic consistency of spin and isospin at the level of an unbroken SU(2) gauge theory suggests the existence of an additional angular momentum besides the spin and isospin and also produces a full quaternionic spinor operator. The latter corresponds to a vector boson in space-time, interpreted as a SU(2) gauge eld. The existence of quaternionic spinor elds implies in a quaternionic.
We obtain the operator product coefficients for primary fields of the extended algebra of the D even type su(2) WZW theories. An important role is played by the Z 2 symmetry which is present in these theories. The primary fields of the extended theory possessing su(2) spin k 4 are identified as linear combinations of the corresponding WZW primaries.
Because SU(2) is connected, the image is in a connected subgroup O(3), so we have a Lie algebra epimorphism The kernel of the Admap is easily seen to be Id, giving a 2-1 covering map; indeed this is a universal covering map of SO(3), as SU(2) is simply-connected. The double cover of a special orthogonal group SO(n) is called its associated.