On the Structure Groups of Decomposable Algebraic Curvature Tensors
نویسنده
چکیده
This paper examines the action of GLN (R) on decomposable algebraic curvature tensors. The main result is that the structure group of a decomposable algebraic curvature tensor can only permute the subspaces into which the tensor decomposes: if the tensor decomposes into tensors Ri on Vi where V = ⊕k i=1 Vi, then for any matrix A in the structure group there exists a permutation σ such that A : Vi → Vσ(i). Restrictions are found for when σ(i) = j; for one, dim(Vi) must equal dim(Vj). We find A restricted to Vi, namely if we set W = Vi then the set of possible maps A|W : W → Vj is isomorphic to a particular left coset of the structure group of Ri.
منابع مشابه
On the decomposable numerical range of operators
Let $V$ be an $n$-dimensional complex inner product space. Suppose $H$ is a subgroup of the symmetric group of degree $m$, and $chi :Hrightarrow mathbb{C} $ is an irreducible character (not necessarily linear). Denote by $V_{chi}(H)$ the symmetry class of tensors associated with $H$ and $chi$. Let $K(T)in (V_{chi}(H))$ be the operator induced by $Tin text{End}(V)$. Th...
متن کاملOn the Structure Group of a Decomposable Model Space
We study the structure group of a canonical algebraic curvature tensor built from a symmetric bilinear form, and show that in most cases it coincides with the isometry group of the symmetric form from which it is built. Our main result is that the structure group of the direct sum of such canonical algebraic curvature tensors on a decomposable model space must permute the subspaces Vi on which ...
متن کاملCurvature collineations on Lie algebroid structure
Considering prolongation of a Lie algebroid equipped with a spray, defining some classical tensors, we show that a Lie symmetry of a spray is a curvature collineation for these tensors.
متن کاملComplex Osserman Algebraic Curvature Tensors and Clifford Families
We use methods of algebraic topology to study the eigenvalue structure of a complex Osserman algebraic curvature tensor. We classify the algebraic curvature tensors which are both Osserman and complex Osserman in all but a finite number of exceptional dimensions.
متن کاملExamples of non-quasicommutative semigroups decomposed into unions of groups
Decomposability of an algebraic structure into the union of its sub-structures goes back to G. Scorza's Theorem of 1926 for groups. An analogue of this theorem for rings has been recently studied by A. Lucchini in 2012. On the study of this problem for non-group semigroups, the first attempt is due to Clifford's work of 1961 for the regular semigroups. Since then, N.P. Mukherjee in 1972 studied...
متن کامل