The aim of this paper is to prove the following analogue of theorem 1. On lorentzian ricci solitons on nilpotent lie groups wears. Let autn denote the group of lie automorphisms of n. Connected lie group an overview sciencedirect topics. In 4we construct parameter rigid actions of nilpotent groups using theorem1. Closed subgroups of a lie group are again lie groups and. Pdf discrete cocompact subgroups of the fivedimensional. Gacting properly on ghwith compact quotient i there is a subalgebra l. An nbyn matrix a with entries from a field is nilpotent if and only if its characteristic polynomial is t n. The orbitmethodreferstoacollection ofrelatedprocedures by which unitary representations of a lie group g can be constructed from certain geometric objects associated to g called coadjoint orbits. It is known that if n is a simply connected nilpotent lie group with lie algebra, then exp. We start with a simply connected nilpotent lie group gof step k, with k. We introduce the notion of pseudodi erential operator in a connected, simply connected nilpotent lie group g using the weyl functional calculus. Given xem, there exist 2dimensional subspaces r, s and t of the tangent.
This generalizes the results of dos santos parameter rigid actions of the heisenberg groups. Then we obtain some conditions for left invariant randers metric to be of douglas type. While any contractible lie group may be made isometric to a solvable group, only those that are solvable and of type r may be made isometric to a nilpotent lie group, in which case the. Henceforth, we will adopt either the point of view that n is an algebraic group or that the group multiplication is polynomial in talking about or working with nilpotent lie groups. Hence, unless explicitly stated to the contrary nonabelian nilpotent lie groups will always be connected and simply connected.
More precisely, we show that for any compact subset k of a connected simply connected nilpotent lie group n, and for any nonzero biinvariant differential operator p on n, there is a compact subset lk with the property that whenever the. Thus for its lie algebra g we have the descending central series g i. Are connected, nilpotent lie groups simply connected. For a connected nilpotent lie group g, its universal cover geis a lie group with the same lie algebra g and hence, by theorem1.
Let us outline the method used to obtained the estimates of x1. We show that a compactly generated locally compact group of polynomial growth having no nontrivial compact normal subgroups can be embedded as a cocompact subgroup into a semidirect product of a connected, simply connected, nilpotent lie group and a compact group. From now we consider that n is a simply connected twostep nilpotent lie group of dimension. Let 3 be the center of lie algebra n, let n be its dual vector space, and let 31 be the annihilator of 3 in n. Let m be a riemannian manifold which admits a transitive connected noncommutative nilpotent lie group of isometries.
Let gbe a connected lie group and autg the group of all continuous automorphisms of g. Pdf coherent state representations of nilpotent lie groups. Let n be a locally compact group, and let g be a discrete subset of n. A compact riemannian nilmanifold is a compact riemannian manifold which is locally isometric to a nilpotent lie group with leftinvariant metric. Parameter rigid actions of simply connected nilpotent lie groups volume 33 issue 6 hirokazu maruhashi. A topological group in particular, a lie group for which the underlying topological space is simply connected. Pdf hardys theorem for simply connected nilpotent lie groups. By eberhard kaniuth fachbereich mathematikinformatik. Some examples of quasiisometries of nilpotent lie groups xiangdong xie abstract we construct quasiisometries of nilpotent lie groups. Let g be a simply connected nilpotent lie group, its lie algebra. Poguntke studied the action of an abelian compact group k on a nilpotent lie group and characterised the kprime ideals as kernels of korbits. In particular, for any simply connected nilpotent lie group n, we construct quasiisometries from n to itself that is not at nite distance from any map that is a composition of left translations and automorphisms. The significance of simply connected groups in the theory of lie groups is explained by the following theorems.
Let g be a simply connected nilpotent lie group, g the lie algebra of g and z the centre of g. Our grading of the lie algebra relates to the hausdorff dimensions of the group. Make that a connected, simply connected nilpotent lie group. Proposition the category of super lie groups is equivalent to a category of harishchandra pairs, i. Using this, we construct parameter rigid actions of simply connected nilpotent lie groups whose lie algebras admit rational structures with graduations. Hardys theorem for simply connected nilpotent lie groups. Probabilities on simply connected nilpotent lie groups. Let n be a connected and simply connected nilpotent lie group endowed with a leftinvariant riemannian metric determined by an inner product h,i on its lie algebra n. The main goal of this note is to prove the following characterization of nilpotent groups. On nonformality of a simplyconnected symplectic 8manifold. If g is a nilpotent group then the exponential map g g is onto, and has no singular points. The orbit method unitary representations of g quantization of symplectic g. Precisely, we prove that there do not exist quasiisometric embeddings of such a.
There is no classification of nilpotent lie groups, or even of nilpotent lie algebras. The converse is true if the action has a dense orbit. This generalizes the results of dos santos concerning the heisenberg groups. Since g is a separable, type i, unimodular group 1, there exists by the general plancherel theorem 3 a unique borel measure p. N is a diffeomorphism, where exp denotes the lie group exponential map. Discrete cocompact subgroups of the fivedimensional connected and simply connected nilpotent lie groups.
Levy processes in a step 3 nilpotent lie group 3 some of these properties have analogs in more general spaces, for example, in some classes of lie groups. See hilgert and neeb, structure and geometry of lie groups, p. The subbundle is called the horizontal bundle and the. This means that g is the complex heisenberg group hc, that is, the complex nilpotent lie group of complex matrices of the form 1 u2 u3 0 1 u1 0 0 1 in fact, in terms of the natural complex coordinate functions. The paper 1938a contains an interesting theorem on the approximation of lie groups by finite groups. Let g be a connected simply connected nilpotent lie group. Unitary representations of nilpotent super lie groups. It follows from lie s third theorem and the preceding result that every finitedimensional real lie algebra is the lie algebra of a unique simply connected lie group. Pdf hardys theorem for simply connected nilpotent lie. Finally, the third section is devoted to determining the simply connected lie subgroups of the lie group g4, showing each of them together their associated lie algebra which is a subalgebra of g4. The main result shows that the irreducible unitary representations of a simply connected nilpotent lie group are in onetoone correspondence with the orbits of g in the dual of the lie.
Pauls in this paper, we prove results concerning the large scale geometry of connected, simply connected nonabelian nilpotent lie groups equipped with left invariant riemannian metrics. Let g be a simply connected nilpotent lie group with cocompact subgroup then any abelian invariant complex structure on m. Title parameter rigid actions of simply connected nilpotent. This means that g is the complex heisenberg group hc, that is, the complex nilpotent lie group of complex matrices of the form 1 u2 u3 0 1 u1 0 0 1 in fact, in terms of. Nilpotent lie groups are solvable, so the properties of solvable lie groups carry over them, and often in a strengthened from, since every nilpotent lie group is triangular. If g is simply connected, the remark above asserts that m g exp gm is both left invariant and right invariant.
Some examples of quasiisometries of nilpotent lie groups. No nilpotent element can be a unit except in the trivial ring 0, which has only a single element 0 1. On the structure of groups with polynomial growth iii. Maximal operators associated to discrete subgroups. Let be a lattice in a simply connected nilpotent lie group n, as above. The method can be made to work perfectly for the class ofsimply connected.
It turns out that if this holds, then gis a nilpotent lie group see 4. Description of the representations of nilpotent lie groups. Let l be a simply connected nilpotent lie group with a left invariant riemannian metric and let f c l be a discrete subgroup with compact quotient xlp. Parameter rigid actions of simply connected nilpotent lie. An example of a simply connected group is the special unitary group su2, which as a manifold is the 3sphere. The rst step is to translate the problem, which concerns lattices in a simply connected nilpotent lie group, into a problem about discrete cocompact subrings in its lie algebra.
As above, this group is compact because it is closed and bounded with respect to the hilbertschmidt norm. The main result shows that the irreducible unitary representations of a simply connected nilpotent lie group are in onetoone correspondence with the orbits of g in the dual of the lie algebra of g i. For connected, simply connected, nilpotent lie groups, j. Following some work of siebert, we show that the group is simply connected and nilpotent with positively graduable lie algebra. Group theory nilpotent subalgebras of semisimple lie algebras paul levya. On lorentzian ricci solitons on nilpotent lie groups. The riemannian manifold n,h,i is said to be a homogeneous nilmanifold cf. The large scale geometry of nilpotent lie groups scott d. Let n be a simply connected nilpotent lie group and n the lie algebra of n.
Note that the weyl group is, on the one hand, in very close proximity to the combinatorial data that determines a simply connected, simple, complex lie group and, on the other hand, also arises waydowntheroad as a nite group associated with the representation theory of the group. There is also a uniqueness statement for this extension. According to the kirillov theory 6, 12 the irreducible representations are parameterized by the. In fact, connected and simply connected nilpotent lie groups are the noncommutative groups \closest in appearance to rn. Let gbe a simply connected nilpotent lie group, h a closed connected therefore simply connected subgroup. Although we shall make some progress in studying fourier analysis on them, we are still very far from a complete understanding. This chapter is devoted to investigations into semi stability phenomena on simply connected nilpotent lie groups, showing that probabilities on this considerably small class of groups have similar behavior as on vector spaces. We prove an analogue of hardys theorem for fourier transform pairs in open face r for arbitrary simply connected nilpotent lie groups, thus extending earlier work on open face rn and the. Then there is a nontrivial closed simply connected subgroup l. The analysis is carried out by classifying all left invariant lorentzian metrics on the connected, simply. In particular, since nilpotent lie groups are solvable, simply connected nilpotent lie groups are matrix groups. We also want to point out that even though there are choices to be made for an appropriate measure on a general subriemannian manifold, the use of the gft forces one to use a haar measure.
Let n be a connected and simply connected nilpotent lie group and 0, 3be lattices in n. The coherent state representations of a connected and simply connected nilpotent lie group are characterized in terms of the kirillov correspondence, as being those irreducible unitary. Consider the action of 3on 0nn by right multiplication and let. On the irreducibility of some restrictions in nilpotent. Introduction a carnot group g is a connected, simply connected, strati. In this article we study the rational homology of the space of group homomorphisms homg gn, endowed with the subspace topology from gn.
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