On linearly ordered H-closed topological semilattices
... τ is Hausdorff, so is τF . Also we
base of a topology τF at the point x and since
observe that the definition of τF implies that ↓e and ↑e are closed subsets of the
topological space (F (E), τF ), and hence the semilattice operation on (F (E), τF ) is
τ ) satisfies the conditions (i) and (iii) o ...
Maximal compact subgroups in the o-minimal setting
... The canonical projection π : G → G/N (G) induces an isomorphism between K and
the maximal definable definably compact subgroup of G/N (G).
Also P = A(G) × K is the smallest definable subgroup of G containing K, and
[P, P ] = [K, K] is a maximal definable semisimple definably compact subgroup of G.
Conjugation spaces - Université de Genève
... additive maps, κ and σ are often easy to construct degree by degree. But we
will show in the “multiplicativity theorem” in Section 3 that in fact σ and κ
are ring homomorphisms. Moreover, given a C-equivariant map f : Y → X
between spaces with involution, along with H ∗ -frames (σX , κX ) and (σY , ...
Positivity for toric vectorbundles
... σ is a maximal cone, then T acts on the fiber of Lw at xσ by χw . It is known that every
equivariant line bundle on Uσ is equivariantly isomorphic to some Lw |Uσ , where the class
of w in M/M ∩ σ ⊥ is uniquely determined.
For every cone σ ∈ ∆, the restriction E|Uσ decomposes as a direct sum of equiv ...
Common fixed point of mappings satisfying implicit contractive
... Huang and Zhang  reintroduced such spaces under the name of cone metric spaces and reintroduced
definition of convergent and Cauchy sequences in the terms of interior points of the underlying cone. They
also proved some fixed point results in framework of cone metric spaces. Subsequently, severa ...
EQUIVARIANT SYMMETRIC MONOIDAL STRUCTURES 1
... 5. Mackey Functors, Tambara Functors, and the transfer
Much of this formalism arose from our attempt to understand the transfer. One
approach in the finite group case is via the Wurtmuller map:
G+ ∧H X −→ FH (G+ , X)
for any H-spectrum X. This is a natural map arising from the the fact that G+ ∧H −
PARTIAL DYNAMICAL SYSTEMS AND C∗
... is faithful if and only if it is faithful on C0 (X). This leads to a sufficient condition for simplicity
of the reduced crossed product in Corollary 2.9.
In Section 3 we consider invariant ideals of a partial action and the ideals they generate
in the crossed product. In Proposition 3.1 we give a ge ...
NSE CHARACTERIZATION OF THE SIMPLE GROUP L2(3n) Hosein
... 2. Notation and preliminaries
For a natural number m, by π(m), we mean the set of all prime divisors of m,
so it is obvious that if G is a finite group, then π(G) = π(|G|). A Sylow p-subgroup
of G is denoted by Gp and by np (G), we mean the number of Sylow p-subgroups
of G. Also, the largest element ...
FROM COMMUTATIVE TO NONCOMMUTATIVE SETTINGS 1
... setting generalize. In [Sme13] this approach is, by means of divisorial one-sided ideal theory, extended
to a class of semigroups that includes commutative and normalizing Krull monoids as special cases. In
particular, this is applied to investigate factorizations in the semigroup of non zero-diviso ...
A Course on Convex Geometry
... V . A is convex if it contains all the segments joining any two points of A, i.e. if
x, y ∈ A and α ∈ [0, 1] implies that αx + (1 − α)y ∈ A. This simple algebraic
property has surprisingly many and far-reaching consequences of geometric nature,
but it also has topological consequences (if V carries ...
An Introduction to Algebra and Geometry via Matrix Groups
... theory of vector spaces over arbitrary fields, and bilinear forms on such vector spaces. We
can then define the orthogonal and symplectic group with respect to the bilinear forms.
The tools we introduce allow us to determine the generators for the general linear group,
the orthogonal group, the symp ...
Invariant convex cone
In mathematics, an invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms. The study of such cones was initiated by Ernest Vinberg and Bertram Kostant.For a simple Lie algebra, the existence of an invariant convex cone forces the Lie algebra to have a Hermitian structure, i.e. the maximal compact subgroup has center isomorphic to the circle group. The invariant convex cone generated by a generator of the Lie algebra of the center is closed and is the minimal invariant convex cone (up to a sign). The dual cone with respect to the Killing form is the maximal invariant convex cone. Any intermediate cone is uniquely determined by its intersection with the Lie algebra of a maximal torus in a maximal compact subgroup. The intersection is invariant under the Weyl group of the maximal torus and the orbit of every point in the interior of the cone intersects the interior of the Weyl group invariant cone.For the real symplectic group, the maximal and minimal cone coincide, so there is only one invariant convex cone. When one is properly contained in the other, there is a continuum of intermediate invariant convex cones.Invariant convex cones arise in the analysis of holomorphic semigroups in the complexification of the Lie group, first studied by Grigori Olshanskii. They are naturally associated with Hermitian symmetric spaces and their associated holomorphic discrete series. The semigroup is made up of those elements in the complexification which, when acting on the Hermitian symmetric space of compact type, leave invariant the bounded domain corresponding to the noncompact dual. The semigroup acts by contraction operators on the holomorphic discrete series; its interior acts by Hilbert–Schmidt operators. The unitary part of their polar decomposition is the operator corresponding to an element in the original real Lie group, while the positive part is the exponential of an imaginary multiple of the infinitesimal operator corresponding to an element in the maximal cone. A similar decomposition already occurs in the semigroup.The oscillator semigroup of Roger Howe concerns the special case of this theory for the real symplectic group. Historically this has been one of the most important applications and has been generalized to infinite dimensions. This article treats in detail the example of the invariant convex cone for the symplectic group and its use in the study of the symplectic Olshanskii semigroup.