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3 days ago · Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.)
Jun 25, 2024 · A Lie group is a smooth manifold whose underlying set of elements is equipped with the structure of a group such that the group multiplication and inverse -assigning functions are smooth functions. In other words, a Lie group is a group object internal to the category SmthMfd of smooth manifolds.
Jun 27, 2024 · This course provides an introduction to Lie groups (the general object responsible for smooth symmetries) and Lie algebras (their infinitesimal counterpart). A particular focus will be on compact Lie groups, including a discussion of their structure theory and classification.
Jun 19, 2024 · The group of all proper and improper rotations in n dimensions is called the orthogonal group O(n), and the subgroup of proper rotations is called the special orthogonal group SO(n), which is a Lie group of dimension n(n − 1)/2.
Jun 21, 2024 · The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representations.
Jun 25, 2024 · Key Takeaways. Lie algebras are the mathematical language of symmetry, with deep connections to geometry, topology, and theoretical physics. Lie algebras are defined as vector spaces equipped with a special Lie bracket operation, which captures the essence of infinitesimal transformations.
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Jun 17, 2024 · Let $G$ be any Lie group with a Lie algebra $\mathfrak{g}(n, \mathbb{R})$; but assume that I only deal with a locally compact, connected (topologically) component around its $e \in G$ element, say $K \leqslant G$.
