scholarly journals Differentiable semigroups are Lie groups

1995 ◽  
Vol 18 (3) ◽  
pp. 509-530
Author(s):  
John P. Holmes ◽  
Mitch Anderson
Keyword(s):  
Banach Space ◽  
Lie Groups ◽  
Lie Group ◽  
Infinite Dimensional ◽  
Detailed Proof ◽  
The One ◽  
Hilbert’S Fifth Problem ◽  
Local Semigroup

We present here a modern, detailed proof to the following theorem which was introduced by Garrett Birkhoff [1] in 1938. IfSis a local semigroup with neighborhood of1homeomorphic to a Banach space and with multiplication strongly differentiable at1, thenSis a local Lie Group. Although this theorem is more than 50 years old and remains the strongest result relating to Hilbert's fifth problem in the infinite dimensional setting, it is frequently overlooked in favor of weaker results. Therefore, it is the goal of the authors here to clarify its importance and to demonstrate a proofwhich is more accessible to contemporary readers than the one offered by Birkhoff.

scholarly journals Homological Stability for Spaces of Commuting Elements in Lie Groups

2020 ◽  
Author(s):  
Daniel A Ramras ◽  
Mentor Stafa
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Stable Range ◽  
Rational Homology ◽  
Infinite Dimensional ◽  
Fixed Group ◽  
Character Varieties ◽  
Representation Stability ◽  
Equivariant Homology ◽  
Homological Stability

Abstract In this paper, we study homological stability for spaces $\textrm{Hom}({{\mathbb{Z}}}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $n\geqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, $\textrm{Comm}(G)$ and $B_{\textrm{com}} G$, introduced by Cohen–Stafa and Adem–Cohen–Torres-Giese, respectively. In addition, we show that the rational homology of the space of unordered commuting $n$-tuples in a fixed group $G$ stabilizes as $n$ increases. Our proofs use the theory of representation stability—in particular, the theory of $\textrm{FI}_W$-modules developed by Church–Ellenberg–Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.

A New Form of the Segal-Bargmann Transform for Lie Groups of Compact Type

1999 ◽  
Vol 51 (4) ◽  
pp. 816-834 ◽  
Author(s):  
Brian C. Hall
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Joint Work ◽  
Compact Type ◽  
Bargmann Transform ◽  
Infinite Dimensional ◽  
Infinite Dimensional Analysis ◽  
Finite Dimensional ◽  
New Form ◽  
Two Parameter

AbstractI consider a two-parameter family Bs,t of unitary transforms mapping an L2-space over a Lie group of compact type onto a holomorphic L2-space over the complexified group. These were studied using infinite-dimensional analysis in joint work with B. Driver, but are treated here by finite-dimensional means. These transforms interpolate between two previously known transforms, and all should be thought of as generalizations of the classical Segal-Bargmann transform. I consider also the limiting cases s → ∞ and s → t/2.

scholarly journals Cylindrical representations of some infinite dimensional nuclear Lie groups

1992 ◽  
Vol 46 (2) ◽  
pp. 295-310 ◽  
Author(s):  
Jean Marion
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Additive Group ◽  
Test Functions ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations ◽  
Direct Product Group ◽  
Necessary And Sufficient

Let Γ.𝒜 be the semi-direct product group of a nuclear Lie group Γ with the additive group 𝒜 of a real nuclear vector space. We give an explicit description of all the continuous representations of Γ.𝒜 the restriction of which to 𝒜 is a cyclic unitary representation, and a necessary and sufficient condition for the unitarity of such cylindrical representations is stated. This general result is successfully used to obtain irreducible unitary representations of the nuclear Lie groups of Riemannian motions, and, in the setting of the theory of multiplicative distributions initiated by I.M. Gelfand, it is proved that for any connected real finite dimensional Lie groupGand for any strictly positive integerkthere exist non located and non trivially decomposable representations of orderkof the nuclear Lie group(M;G) of all theG-valued test-functions on a given paracompact manifoldM.

scholarly journals Pro-Lie groups which are infinite-dimensional Lie groups

2009 ◽  
Vol 146 (2) ◽  
pp. 351-378 ◽  
Author(s):  
K. H. HOFMANN ◽  
K.-H. NEEB
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Exponential Function ◽  
Lie Group ◽  
Smooth Manifold ◽  
Projective Limit ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Infinite Dimensional Lie Groups

AbstractA pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.

Lie groups of 𝐶𝑘-maps on non-compact manifolds and the fundamental theorem for Lie group-valued mappings

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hamza Alzaareer
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Fundamental Theorem ◽  
Smooth Manifold ◽  
Rough Boundary ◽  
Infinite Dimensional ◽  
New Class ◽  
Infinite Dimensional Lie Group ◽  
Group Structures ◽  
Infinite Dimensional Lie Groups

Abstract We study the existence of Lie group structures on groups of the form C k ⁢ ( M , K ) C^{k}(M,K) , where 𝑀 is a non-compact smooth manifold with rough boundary and 𝐾 is a, possibly infinite-dimensional, Lie group. Motivated by introducing this new class of infinite-dimensional Lie groups, we obtain a new version of the fundamental theorem for Lie algebra-valued functions.

Infinite Dimensional DeWitt Supergroups and their Bodies

2014 ◽  
Vol 57 (2) ◽  
pp. 283-288 ◽  
Author(s):  
Ronald Fulp
Keyword(s):  
Banach Space ◽  
Lie Algebra ◽  
Lie Group ◽  
Grassmann Algebra ◽  
The Body ◽  
Exponential Mapping ◽  
Infinite Dimensional ◽  
Group Operation ◽  
Banach Lie Group ◽  
Quotient Manifold

AbstractFor Dewitt super groups G modeled via an underlying finitely generated Grassmann algebra it is well known that when there exists a body group BG compatible with the group operation on G, then, generically, the kernel K of the body homomorphism is nilpotent. This is not true when the underlying Grassmann algebra is infinitely generated. We show that it is quasi-nilpotent in the sense that as a Banach Lie group its Lie algebra κ has the property that for each a ∊ κ ada has a zero spectrum. We also show that the exponential mapping from κ to K is surjective and that K is a quotient manifold of the Banach space κ via a lattice in κ.

scholarly journals Smoothing operators and C∗-algebras for infinite dimensional Lie groups

2017 ◽  
Vol 28 (05) ◽  
pp. 1750042 ◽  
Author(s):  
Karl-Hermann Neeb ◽  
Hadi Salmasian ◽  
Christoph Zellner
Keyword(s):  
Lie Groups ◽  
Bounded Operator ◽  
Representation Formula ◽  
Unitary Representations ◽  
Direct Integral ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Lie Group ◽  
The One ◽  
Infinite Dimensional Lie Groups

A smoothing operator for a unitary representation [Formula: see text] of a (possibly infinite dimensional) Lie group [Formula: see text] is a bounded operator [Formula: see text] whose range is contained in the space [Formula: see text] of smooth vectors of [Formula: see text]. Our first main result characterizes smoothing operators for Fréchet–Lie groups as those for which the orbit map [Formula: see text] is smooth. For unitary representations [Formula: see text] which are semibounded, i.e. there exists an element [Formula: see text] such that all operators [Formula: see text] from the derived representation, for [Formula: see text] in a neighborhood of [Formula: see text], are uniformly bounded from above, we show that [Formula: see text] coincides with the space of smooth vectors for the one-parameter group [Formula: see text]. As the main application of our results on smoothing operators, we present a new approach to host [Formula: see text]-algebras for infinite dimensional Lie groups, i.e. [Formula: see text]-algebras whose representations are in one-to-one correspondence with certain continuous unitary representations of [Formula: see text]. We show that smoothing operators can be used to obtain host algebras and that the class of semibounded representations can be covered completely by host algebras. In particular, the latter class permits direct integral decompositions.

scholarly journals FUNCTORIAL ASPECTS OF THE RECONSTRUCTION OF LIE GROUPOIDS FROM THEIR BISECTIONS

2016 ◽  
Vol 101 (2) ◽  
pp. 253-276 ◽  
Author(s):  
ALEXANDER SCHMEDING ◽  
CHRISTOPH WOCKEL
Keyword(s):  
Lie Groups ◽  
Present Article ◽  
Lie Group ◽  
Lie Groupoids ◽  
Infinite Dimensional ◽  
Lie Groupoid ◽  
Equivalence Of Categories ◽  
Infinite Dimensional Lie Group

To a Lie groupoid over a compact base $M$, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present article we consider functorial aspects of these construction principles. The first observation is that this procedure is functorial (for morphisms fixing $M$). Moreover, it gives rise to an adjunction between the category of Lie groupoids over $M$ and the category of Lie groups acting on $M$. In the last section we then show how to promote this adjunction to almost an equivalence of categories.

scholarly journals Covariant Homogeneous Nets of Standard Subspaces

2021 ◽  
Author(s):  
Vincenzo Morinelli ◽  
Karl-Hermann Neeb
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Free Field ◽  
Large Family ◽  
Parameter Group ◽  
Semisimple Lie Algebras ◽  
New Models ◽  
The One

AbstractRindler wedges are fundamental localization regions in AQFT. They are determined by the one-parameter group of boost symmetries fixing the wedge. The algebraic canonical construction of the free field provided by Brunetti–Guido–Longo (BGL) arises from the wedge-boost identification, the BW property and the PCT Theorem. In this paper we generalize this picture in the following way. Firstly, given a $$\mathbb Z_2$$ Z 2 -graded Lie group we define a (twisted-)local poset of abstract wedge regions. We classify (semisimple) Lie algebras supporting abstract wedges and study special wedge configurations. This allows us to exhibit an analog of the Haag–Kastler one-particle net axioms for such general Lie groups without referring to any specific spacetime. This set of axioms supports a first quantization net obtained by generalizing the BGL construction. The construction is possible for a large family of Lie groups and provides several new models. We further comment on orthogonal wedges and extension of symmetries.

Feynman approximation to integrals with respect to Brownian sheet on Lie groups

2015 ◽  
Vol 18 (01) ◽  
pp. 1550008 ◽  
Author(s):  
A. A. Kalinichenko
Keyword(s):  
Brownian Motion ◽  
Lie Groups ◽  
Lie Group ◽  
Functional Space ◽  
Brownian Sheet ◽  
Functional Integrals ◽  
Finite Dimensional ◽  
Approximation Formulas ◽  
The One ◽  
Connected Lie Group

We consider the Feynman-type approximations to functional integrals over the distribution of the Brownian sheet on a compact connected Lie group M, which give a representation of the integrals over the functional space C([0, 1] × [0, 1], M) as the limit of integrals over the finite-dimensional manifolds M × ⋯ × M. The known approximation formulas for the one-parameter Brownian motion are generalized to the case of the Brownian sheet.

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