On Algebra: On Lie Algebra

"During the last decade the methods of algebraic topology have invaded extensively the domain of pure algebra, and initiated a number of internal revolutions. [...] The invasion of algebra has occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. The three subjects have been given independent but parallel developments." (Henri P Cartan & Samuel Eilenberg, "Homological Algebra", 1956)

"Algebraic deformation represents a method for the quantization of Lie groups and Lie algebras. Quantum groups are not groups, but Hopf algebras. [3]

"Lie groups describe finite symmetries or symmetries which smoothly depend on a finite number of real parameters. Lie algebras are the linearization of Lie groups at the unit element. The passage from Lie groups to Lie algebras simplifies considerably the approach. Lie algebras are frequently called infinitesimal symmetries." (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"Representations of symmetries with the aid of linear operators (e.g., matrices) play a crucial role in modern physics. In particular, this concerns the linear representations of groups, Lie algebras, and quantum groups (Hopf algebras)" (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"Solvable Lie algebras are close to both upper triangular matrices and commutative Lie algebras. In contrast to this, semisimple Lie algebras are as far as possible from being commutative. By Levi’s decomposition theorem, any Lie algebra is built out of a solvable and a semisimple one. The nontrivial prototype of a solvable Lie algebra is the Heisenberg algebra." (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"The fundamental Levi decomposition of a Lie algebra is the prototype of a semidirect product of Lie algebras." (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"There is a fundamental relationship between Lie groupoids and Lie algebroids which is similar to the relationship between Lie groups and Lie algebras. There are, however, significant differences in the conventions involved, and it important to be aware of these." (Mike Crampin & David Saunders, "Cartan Geometries and their Symmetries: A Lie Algebroid Approach", 2016)