# Book: LieGroupInverses

Investigation: Understand Lie groups in terms of the inverses of elements.

• Relate the inverse elements to the orthogonality of rows and columns (as given by the transpose matrix).
• Consider the case of the transpose of odd and even orthogonal Lie groups. It should make for a distinction.
• What are the inverses of the exceptional Lie groups?
• Consider the role of the symmetric group in the determinant in the denominator of Cramer's rule. How are the various "short-cut" inverses related to this symmetric group?
• Study the fundamental representations of the Lie groups and their relationships to the root systems, and how inverses are expressed in them.
• Study and interpret Cartan's procedure by which any representation

Thoughts

• My thinking about inverses may be more relevant and central than the notion of continuity. Continuity might be a concept that arises as a consequence of deeper constraints, namely, of restrictions on inverses.
• The distinction between the two cases i=j and i!=j among matrix indices is extremely important. Certainly for the odd and even dimensional orthogonal matrices, but perhaps also for the notion of whether two indices (such as conjugates) can be the same or not.
• Two conjugates are not the same but in every other sense are equivalent.
• Lie groups link dimensions in a way needed for a list, not just a set.
• Key idea: Inversion of composition. The basis also for Yoneda's lemma.'''

Inverses

The complex special orthogonal group is defined as {$\textrm{SO}(n,\mathbb{C})=\{Q\in \textrm{SL}(n,\mathbb{C})|Q^TQ=QQ^T=I\}$}. The inverse {$Q^{-1}=Q^T$}.

In the case of {$G_2$}, we should be dealing with the octonion transpose, as this is the automorphism group of the octonions.