A topological conjugacy of invariant flows on some class of Lie groups (original) (raw)
On topological conjugacy of left invariant flows on semisimple and affine Lie groups
Proyecciones (Antofagasta), 2011
In this paper, we study the flows of nonzero left invariant vector fields on Lie groups with respect to topological conjugacy. Using the fundamental domain method, we are able to show that on a simply connected nilpotent Lie group any such flows are topologically conjugate. Combining this result with the Iwasawa decomposition, we find that on a noncompact semisimple Lie group the flows of two nilpotent or abelian fields are topologically conjugate. Finally, for affine groups G = HV , V ∼ = n , we show that the conjugacy class of a left invariant vector field does not depend on its Euclidean component.
Transitivity of Families of Invariant Vector Fields on The Semidirect Products of Lie Groups
Transactions of the American Mathematical Society, 1982
In this paper we give necessary and sufficient conditions for a family of right (or left) invariant vector fields on a Lie group G to be transitive. The concept of transitivity is essentially that of controllability in the literature on control systems. We consider families of right (resp. left) invariant vector fields on a Lie group G which is a semidirect product of a compact group K and a vector space V on which K acts linearly. If 5F is a family of right-invariant vector fields, then the values of the elements of if at the identity define a subset T of 7.(0) the Lie algebra of G. We say that if is transitive on G if the semigroup generated by U XE¡,{exp(tX): t » 0} is equal to G. Our main result is that if is transitive if and only if Lie(F), the Lie algebra generated by T, is equal to L(G).
Algorithmic Lie Theory for Solving Ordinary Differential Equations, 2007
Suppose G is a Lie group and M is a manifold (G and M are not necessarily finite dimensional). Let D(M) denote the group of diffeomorphisms on M and V(M) denote the Lie algebra of vector fields on M. If X: isv a. complete-vector: field-then; Exp tX will denote the one-parameter group of X. A local action <£ of G oh M gives rise to a Lie algebra homomorphism + from L(G) into V(M). In particular if G is a subgroup-of D(M> and <|> : G x M-> M is the natural global action (g»p)-> g(p). then G is called a Lie transformation group of M. If M is a Hausdorff manifold and G is a Lie transformation group of M we show that is an isomorphism of L(G) onto (L(G)) and L = + (L(G)) satisfies the following conditions : (A) L consists of complete vector fields. (B) L has a Banach Lie algebra structure satisfying the following two conditions : (BI) the evaluation map ev : (X,p)-> X(p) is a vector bundle morphism from the trivial bundle L x M into T(M), (B2) there exists an open ball B r (0) of radius r at 0 such that Exp : L-> D(M) is infective on B r (0). Conversely, if L is a suba-lgebra of V(M) (M Hausdorff) satisfying conditions (A) and (B) we show there exists a unique connected Lie transfor-+ mation group with natural action : G x M-> M such that tj) is a Banach Lie algebra isomorphism of L(G) onto L .
Left invariant geometry of Lie groups
Cubo, 2004
Section 3 Poisson manifolds (3.1) Definition of a Poisson manifold (3.2) Reformulation of the Jacobi identity (3.3) Examples of Poisson manifolds (brief summary) Example 1 Symplectic manifolds Example 2 Dual space H* of a Lie algebra H Example 3 Lie algebra H with an inner ...
Translations in simply transitive affine actions of Heisenberg type Lie groups
Linear Algebra and its Applications, 2003
Let G be a 2-step nilpotent Lie group with a 1-dimensional commutator subgroup. We prove that for any simply transitive and affine action of G, there exists a non-trivial subgroup of G acting as pure translations. This result no longer holds in case the commutator subgroup is higher dimensional.
Notes on group actions, manifolds, lie groups, and lie algebras
2005
CHAPTER 2. REVIEW OF GROUPS AND GROUP ACTIONS 3. Similarly, the sets R of real numbers and C of complex numbers are groups under addition (with identity element 0), and R * = R − {0} and C * = C − {0} are groups under multiplication (with identity element 1). 4. The sets R n and C n of n-tuples of real or complex numbers are groups under componentwise addition: (x 1 ,. .. , x n) + (y 1 , • • • , y n) = (x 1 + y n ,. .. , x n + y n), with identity element (0,. .. , 0). All these groups are abelian. 5. Given any nonempty set S, the set of bijections f : S → S, also called permutations of S, is a group under function composition (i.e., the multiplication of f and g is the composition g • f), with identity element the identity function id S. This group is not abelian as soon as S has more than two elements. 6. The set of n × n matrices with real (or complex) coefficients is a group under addition of matrices, with identity element the null matrix. It is denoted by M n (R) (or M n (C)). 7. The set R[X] of polynomials in one variable with real coefficients is a group under addition of polynomials. 8. The set of n × n invertible matrices with real (or complex) coefficients is a group under matrix multiplication, with identity element the identity matrix I n. This group is called the general linear group and is usually denoted by GL(n, R) (or GL(n, C)). 9. The set of n × n invertible matrices with real (or complex) coefficients and determinant +1 is a group under matrix multiplication, with identity element the identity matrix I n. This group is called the special linear group and is usually denoted by SL(n, R) (or SL(n, C)). 10. The set of n × n invertible matrices with real coefficients such that RR = I n and of determinant +1 is a group called the orthogonal group and is usually denoted by SO(n) (where R is the transpose of the matrix R, i.e., the rows of R are the columns of R). It corresponds to the rotations in R n. 11. Given an open interval ]a, b[, the set C(]a, b[) of continuous functions f : ]a, b[ → R is a group under the operation f + g defined such that (f + g)(x) = f (x) + g(x) for all x ∈]a, b[.
2 Invariant generalized complex structures on Lie groups
2012
We describe a class (called regular) of invariant generalized complex structures on a real semisimple Lie group G. The problem reduces to the description of admissible pairs (k, ω), where k ⊂ g C is an appropriate regular subalgebra of the complex Lie algebra g C associated to G and ω is a closed 2-form on k, such that Im ω| k∩g is non-degenerate. In the case when G is a semisimple Lie group of inner type (in particular, when G is compact) a classification of regular generalized complex structures on G is given. We show that any invariant generalized complex structure on a compact semisimple Lie group is regular, provided that an additional natural condition is satisfied. In the case when G is a semisimple Lie group of outer type, we describe the subalgebras k in terms of appropriate root subsystems of a root system of g C and we construct a large class of admissible pairs (k, ω) (hence, regular generalized complex structures of G). 2 Preliminary material 2.1 Invariant complex structures on Lie groups and homogeneous manifolds 2.1.1 Invariant complex structures on homogeneous manifolds The Lie algebra of a Lie group will be identified as usual with the tangent space at the identity element or with the space of left-invariant vector fields. Let G be a real Lie group, with Lie algebra g, and L a closed connected subgroup of G, with Lie algebra l. Suppose that the space M = G/L of left cosets is reductive, i.e. g has an Ad L-invariant decomposition g = l ⊕ m.
Invariant generalized complex structures on Lie groups
Proceedings of the London Mathematical Society, 2012
We describe a class (called regular) of invariant generalized complex structures on a real semisimple Lie group G. The problem reduces to the description of admissible pairs (k, ω), where k ⊂ g C is an appropriate regular subalgebra of the complex Lie algebra g C associated to G and ω is a closed 2-form on k, such that Im ω| k∩g is non-degenerate. In the case when G is a semisimple Lie group of inner type (in particular, when G is compact) a classification of regular generalized complex structures on G is given. We show that any invariant generalized complex structure on a compact semisimple Lie group is regular, provided that an additional natural condition is satisfied. In the case when G is a semisimple Lie group of outer type, we describe the subalgebras k in terms of appropriate root subsystems of a root system of g C and we construct a large class of admissible pairs (k, ω) (hence, regular generalized complex structures of G). 2 Preliminary material 2.1 Invariant complex structures on Lie groups and homogeneous manifolds 2.1.1 Invariant complex structures on homogeneous manifolds The Lie algebra of a Lie group will be identified as usual with the tangent space at the identity element or with the space of left-invariant vector fields. Let G be a real Lie group, with Lie algebra g, and L a closed connected subgroup of G, with Lie algebra l. Suppose that the space M = G/L of left cosets is reductive, i.e. g has an Ad L-invariant decomposition g = l ⊕ m.
Invariant Poisson-Nijenhuis structures on Lie groups and classification
2017
We study right-invariant (resp., left-invariant) Poisson-Nijenhuis structures on a Lie group G and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra g. We show that r-n structures can be used to find compatible solutions of the classical Yang-Baxter equation. Conversely, two compatible r-matrices from which one is invertible determine an r-n structure. We classify, up to a natural equivalence, all r-matrices and all r-n structures with invertible r on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by r-matrices on a phase space whose symmetry group is Lie group G, can be specifically determined.