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Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

Explain why there are four classical Lie groups and algebras.

Classical Lie algebras/groups express the symmetries, notably the dualities, inherent in the possible cognitive interpretations of mathematical expansions such as {$x_1 x_2 \dots x_n$}.

Counting proceeds both forwards and backwards at the same time. We relate these two directions expressly by combining two systems, {$\dots,X,Y,Z$} and {$A,B,C,\dots$}. Apparently, the ways to unite this are by simply connecting them seemlessly, as if folding them, which doubles the ways of counting; or by adding an external zero; or by identifying {$Z=A$} to create an internal zero.

The different dualities are manifest in the four cognitive interpretations of the binomial expansion. This yields three infinite families of polytopes (the simplexes, cross polytopes and hypercubes) and also the coordinate systems (for hypercubes).

A key problem is that the degeneracy in the symmetry group of the coordinate systems matches the degeneracy of the symmetry group of {$D_n$} rather than {$C_n$}. How can I make sense of that?

The different dualities are involved in the Cayley-Hamilton construction.

The classical algebras arise as the composition algebras. Note that they establish the inner norms on the reals (orthogonal algebras), complexes (unitary algebras) and the quaternions (symplectic algebras).

Clarify the four dualities to look for.

On what basis can I relate:

• the root systems and the polytopes
• the root systems and the geometries (real, complex, quaternion) and affine, projective, conformal, symplectic
• {$A_n$} - simplexes
• {$B_n$} - cubes?
• {$C_n$} - cross polytopes?
• {$D_n$} - coordinate systems

Study the constraints on the root systems.

Interpret the Cayley-Hamilton construction.

Study

• Composition algebras, the inner norms, the Cayley-Hamilton construction. See Classical Groups by Kevin McGerty.
• Spinors and how they relate to conjugates. How do they appear in the representation theory of {$D_n$}?

#### FourClassicalAlgebras

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