Results 191 to 200 of about 107,381 (262)
Some of the next articles are maybe not open access.
Lie Groups, Lie Algebras, and Representations
2003An important concept in physics is that of symmetry, whether it be rotational symmetry for many physical systems or Lorentz symmetry in relativistic systems. In many cases, the group of symmetries of a system is a continuous group, that is, a group that is parameterized by one or more real parameters.
B. Hall
openaire +3 more sources
2021
In the last lecture, we introduced the description of rigid body motion in the three-dimensional world, including the rotation matrix, rotation vector, Euler angle, quaternion, and so on. We focused on the representation of rotation, but in SLAM, we have to estimate and optimize them in addition to the representation.
Tao Zhang, Xiang Gao
openaire +4 more sources
In the last lecture, we introduced the description of rigid body motion in the three-dimensional world, including the rotation matrix, rotation vector, Euler angle, quaternion, and so on. We focused on the representation of rotation, but in SLAM, we have to estimate and optimize them in addition to the representation.
Tao Zhang, Xiang Gao
openaire +4 more sources
Lie groups and Lie algebras [PDF]
Lie Groups, Lie Algebras, and Cohomology. (MN-34), Volume 34, 2008 These are expanded notes of a two-semester course on Lie groups and Lie algebras given by the author at MIT in 2020/2021.
Pavel Etingof
openaire +3 more sources
Lie Groups and Lie Algebras [PDF]
, 1976 As we pointed out in 6.2, there are exactly two simple real Lie algebras of dimension 3. These are: the algebra \( {{\mathfrak{g}}_{1}} = \mathfrak{s}\mathfrak{l}\left( {2,R} \right) \) of real matrices of the second order with zero trace and the algebra \( {{\mathfrak{g}}_{2}} = \mathfrak{s}\mathfrak{o} = \left( {3,R} \right) \) of real skew-symmetric
openaire +1 more source
1988
Whereas discrete groups mainly describe the symmetries of regular geometric structures (crystals), continuous groups are essential in discussing the properties of particles, fields (atoms and all the more elementary particles) and conservation laws. We restrict the investigation here to Lie groups and the Lie algebras connected with them.
W. Ludwig, Claus Falter
openaire +2 more sources
Whereas discrete groups mainly describe the symmetries of regular geometric structures (crystals), continuous groups are essential in discussing the properties of particles, fields (atoms and all the more elementary particles) and conservation laws. We restrict the investigation here to Lie groups and the Lie algebras connected with them.
W. Ludwig, Claus Falter
openaire +2 more sources
Algebra, Lie Group and Lie Algebra
2010Geometry, algebra, and analysis are usually called the three main branches of mathematics. This chapter introduces some fundamental results in algebra that are mostly useful in systems and control. In section 4.1 some basic concepts of group and three homomorphism theorems are discussed. Ring and algebra are introduced briefly in section 4.2. As a tool,
Daizhan Cheng, Tielong Shen, Xiaoming Hu
openaire +2 more sources
2021
The objective of this chapter is to introduce the concepts of Lie groups and their Lie algebras. The Lie algebra \(\mathfrak {g}\) of a Lie group G is defined as the space of invariant vector fields (left or right, depending on choice), with bracket given by the Lie bracket of vector fields.
San Martin, A B Luiz
openaire +2 more sources
The objective of this chapter is to introduce the concepts of Lie groups and their Lie algebras. The Lie algebra \(\mathfrak {g}\) of a Lie group G is defined as the space of invariant vector fields (left or right, depending on choice), with bracket given by the Lie bracket of vector fields.
San Martin, A B Luiz
openaire +2 more sources
2004
In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map of manifolds is used in a fundamental way in §8.1, the notion ...
Joe Harris, William Fulton
openaire +2 more sources
In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map of manifolds is used in a fundamental way in §8.1, the notion ...
Joe Harris, William Fulton
openaire +2 more sources
Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation
Mathematische Annalen, 2023The notions of a post-group and a pre-group are introduced as a unification and enrichment of several group structures appearing in diverse areas from numerical integration to the Yang–Baxter equation.
C. Bai, Li Guo, Y. Sheng, Rong Tang
semanticscholar +1 more source