Lie group - Wikipedia
WEBIn mathematics, a Lie group (pronounced / liː / LEE) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
WEBHis big discovery was that the behaviour of a Lie group near the group unit is completely determined by an algebraic structure on the tangent space at the group unit, its Lie algebra.
WEBPart I: Lie Groups Richard Borcherds, Mark Haiman, Nicolai Reshetikhin, Vera Serganova, and Theo Johnson-Freyd October 5, 2016
WEBdetailed study of Lie groups and Lie algebras and interactions between them, with numerous examples. These notes are based on a year-long introductory course on Lie
Lie Group -- from Wolfram MathWorld
WEBAug 22, 2024 · A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable.
WEBThe first chapter introduces the notion of a Lie group and provides a number of classical examples. These examples are matrix groups, such as groups of invertible matrices, orthogonal matrices, unitary matrices, and others. We introduce the algebra H of quaternions and matrices of quater-nions, and certain compact Lie groups of such matrices.
Lie group - Encyclopedia of Mathematics
WEBJun 13, 2020 · In other words, a Lie group is a set endowed with compatible structures of a group and an analytic manifold. A Lie group is said to be real, complex or $ p $ -adic, depending on the field over which its analytic manifold is considered.
What is a Lie Group in layman's terms?
WEBMar 24, 2017 · The collection of all rotations forms a Lie group. The group property basically means that if we rotate the sphere over any angle α α, after this over an angle β β, it is the same if we would have rotated it in one go (over some different angle).
WEBDefinition 1 (Lie Group). A Lie group is a group object in the category of smooth (C∞) manifolds. In other words, a Lie group is a smooth manifold, with product and inverse maps that satisfy the group axioms and are smooth. We’ll restrict attention to finite dimensional Lie groups, but infinite-dimensional Lie groups are quite interesting.
Representation of a Lie group - Wikipedia
WEBIn mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry.
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