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Register a new userRelative equivariants under compact Lie groups
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In this work we obtain the general form of polynomial mappings that commute with a linear action of a relative symmetry group. The aim is to give results for relative equivariant polynomials that correspond to the results for relative invariants obtained in a previous paper [P.H. Baptistelli, M. Manoel (2013) Invariants and relative invariants under compact Lie groups, J. Pure Appl. Algebra 217, 2213{2220]. We present an algorithm to compute generators for relative equivariant submodules from the invariant theory applied to the subgroup formed only by the symmetries. The same method provides, as a particular case, generators for equivariants under the whole group from the knowledge of equivariant generators by a smaller subgroup, which is normal of finite index.
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This paper presents algebraic methods for the study of polynomial relative invariants, when the group G formed by the symmetries and relative symmetries is a compact Lie group. We deal with the case when the subgroup H of symmetries is normal in G with index m, m greater or equal to 2. For this, we develop the invariant theory of compact Lie groups acting on complex vector spaces. This is the starting point for the study of relative invariants and the computation of their generators. We first obtain the space of the invariants under the subgroup $H$ of $Gamma$ as a direct sum of $m$ submodules over the ring of invariants under the whole group. Then, based on this decomposition, we construct a Hilbert basis of the ring of G-invariants from a Hilbert basis of the ring of H-invariants. In both results the knowledge of the relative Reynolds operators defined on H-invariants is shown to be an essential tool to obtain the invariants under the whole group. The theory is illustrated with some examples.
In this article, we focus on the left translation actions on noncommutative compact connected Lie groups with topological dimension 3 or 4, consisting of ${rm SU}(2),,{rm U}(2),,{rm SO}(3),,{rm SO}(3) times S^1$ and ${{rm Spin}}^{mathbb{C}}(3)$. We define the rotation vectors (numbers) of the left actions induced by the elements in the maximal tori of these groups, and utilize rotation vectors (numbers) to give the topologically conjugate classification of the left actions. Algebraic conjugacy and smooth conjugacy are also considered. As a by-product, we show that for any homeomorphism $f:L(p, -1)times S^1rightarrow L(p, -1)times S^1$, the induced isomorphism $(picirc fcirc i)_*$ maps each element in the fundamental group of $L(p, -1)$ to itself or its inverse, where $i:L(p,-1)rightarrow L(p, -1)times S^1$ is the natural inclusion and $pi:L(p, -1)times S^1rightarrow L(p, -1)$ is the projection.
Given a Lie group $G$ with finitely many components and a compact Lie group A which acts on $G$ by automorphisms, we prove that there always exists an A-invariant maximal compact subgroup K of G, and that for every such K, the natural map $H^1(A,K)to H^1(A,G)$ is bijective. This generalizes a classical result of Serre [6] and a recent result in [1].
In this paper we consider the re-expansion problems on compact Lie groups. First, we establish weight
Let $G$ be a connected, simply-connected, compact simple Lie group. In this paper, we show that the isometry group of $G$ with a left-invariant pseudo-Riemannan metric is compact. Furthermore, the identity component of the isometry group is compact if $G$ is not simply-connected.
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