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Let $M^K_n$ be the moduli space of framed $K$-instantons over $S^4$ with instanton number $n$ when $K$ is a compact simple Lie group of classical type. Let $U^{K}_{n}$ be the Uhlenbeck partial compactification of $M^{K}_{n}$. A scheme structure on $U^{K}_{n}$ is endowed by Donaldson as an algebro-geometric Hamiltonian reduction of ADHM data. In this paper, for $K=SO(N,R)$, $N\ge5$, we prove that $U^{K}_{n}$ is an irreducible normal variety with smooth locus $M^{K}_{n}$. Hence, together with the author's previous result, the K-theoretic Nekrasov partition function for any simple classical group other than $SO(3,R)$, is interpreted as a generating function of Hilbert series of the instanton moduli spaces. Using this approach we also study the case $K=SO(4,R)$ which is the unique semisimple but non-simple classical group., 40 pages, v.2: typos corrected, v.3: dedicatory and acknowledgement messages added, references updated, to appear in Adv. Math

A group G is saturated with groups from a set ℜ of groups if every finite subgroup of G is contained in a subgroup of G that is isomorphic to some group in ℜ. Previously [Kourovka Notebook, Quest. 14.101], the question was posed whether a periodic group saturated with finite simple groups of Lie type whose ranks are bounded in totality is itself a simple group of Lie type. A partial answer to this question is given for groups of Lie type of rank 1. We prove the following: Let a periodic group G be saturated with finite simple groups of Lie type of rank 1. Then G is isomorphic to a simple group of Lie type of rank 1 over a suitable locally finite field.

We give a sufficient condition for complex manifolds for automorphism groups to become Lie groups. As an application, we see that the automorphism group of any strictly pseudoconvex domain or finite-type pseudoconvex domain has a Lie group structure.

We expose explicit calculations of the spectra of the Laplace operators for smooth real or complex functions on all connected compact simple Lie groups of rank 3 with bi-invariant Riemannian metric and establish the relationship of the obtained formulas with number theory and integer-valued ternary and binary quadratic forms.

We classify Alt n and its perfect covers for n ≥ 6 up to conjugacy as subgroups of the exceptional complex Lie groups (with some exceptions where we give lower bounds and discuss the possibilities of Lie primitive subgroups).

International audience; A simple Almost-Riemannian Structure on a Lie group G is defined by a linear vector field (that is an infinitesimal automorphism) and dim(G) − 1 left-invariant ones. It is first proven that two different ARSs are isometric if and only if there exists an isometry between them that fixes the identity. Such an isometry preserves the left-invariant distribution and the linear field. If the Lie group is nilpotent it is an automorphism. These results are used to state a complete classification of the ARSs on the 2D affine and the Heisenberg groups.

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally compact group which has a compact quotient group modulo its identity component and thus, in particular, each compact and each connected locally compact group; it also includes all locally compact abelian groups. This paper provides an overview of the structure theory and Lie theory of pro-Lie groups including results more recent than those in the authors' reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function which links the two. (A topological vector space is weakly complete if it is isomorphic to a power $\R^X$ of an arbitrary set of copies of $\R$. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected with pro-Lie groups., Comment: 19 pages

It is well known that on any Lie group, a left-invariant Riemannian structure can be defined. For other left-invariant geometric structures, for example, complex, symplectic, or contact structures, there are difficult obstructions for their existence, which have still not been overcome, although a lot of works were devoted to them. In recent years, substantial progress in this direction has been made; in particular, classification theorems for low-dimensional groups have been obtained. This paper is a brief review of left-invariant complex, symplectic, pseudo-Kahlerian, and contact structures on low-dimensional Lie groups and classification results for Lie groups of dimension 4, 5, and 6.

Let G be a group. An element x ∈ G is called real if x is conjugate to x−1 in G. In this paper we study the structure of real elements in the compact connected Lie group of type F4 and algebraic groups of type F4 defined over an arbitrary field.

We show that a surface group of high genus contained in a classical simple Lie group can be deformed to become Zariski dense, unless the Lie group is $SU(p,q)$ (resp. $SO^* (2n)$, $n$ odd) and the surface group is maximal in some $S(U(p,p)\times U(q-p))\subset SU(p,q)$ (resp. $SO^* (2n-2)\times SO(2)\subset SO^* (2n)$). This is a converse, for classical groups, to a rigidity result of S. Bradlow, O. Garc\'{\i}a-Prada and P. Gothen.

This is a survey of our research (conducted over the last few years) on invariant control systems, the associated optimal control problems, and the associated Hamilton–Poisson systems. The focus is on equivalence and classification. State space and detached feedback equivalence of control systems are characterized in simple algebraic terms; several classes of systems (in three dimensions, on the Heisenberg groups, and on the six-dimensional orthogonal group) are classified. Equivalence of cost-extended systems is shown to imply equivalence of the associated Hamilton–Poisson systems. Cost-extended systems of a certain kind are reinterpreted as invariant sub-Riemannian structures. A classification of quadratic Hamilton–Poisson systems in three dimensions is presented. As an illustrative example, the stability and integration of a typical system is investigated.

Let R be a commutative ring of characteristic zero and G be an arbitrary group. In the present paper we classify the groups G for which the set of symmetric elements with respect to the classical involution of the group ring RG is Lie metabelian.

In this paper we give a bound to the number of conjugacy classes of maximal subgroups of any almost simple group whose socle is a classical group of Lie type. The bound is 2 n 5.2 + n log 2 log 2 q , where n is the dimension of the classical socle and q is the size of the defining field. To obtain the bound, we first bound the number of projective cross-characteristic representations of simple groups of Lie type as a function of the representation degree. These bounds are computed for different families of groups separately. In the computation, we use information on conjugacy class numbers, minimal character degrees and gaps between character degrees.

We consider the degeneration of a simple Lie group which is a semidirect product of its Borel subgroup and a normal Abelian unipotent subgroup. We introduce a class of highest weight representations of the degenerate group of type A, generalizing the construction of PBW-graded representations of the classical group (PBW is an abbreviation for “Poincare-Birkhoff-Witt”). Following the classical construction of flag varieties, we consider the closures of orbits of the Abelian unipotent subgroup in projectivizations of the representations. We show that the degenerate flag varieties Fna and their desingularizations Rn can be obtained via this construction. We prove that the coordinate ring of Rn is isomorphic as a vector space to the direct sum of the duals of the highest weight representations of the degenerate group. At the end we state several conjectures on the structure of the highest weight representations of the degenerate group of type A.

By definition a homogeneous Lie group is a Lie group equipped with a family of dilations compatible with the group law. The abelian group \( \left( {{\mathbb{R}}^n , + } \right) \) is the very first example of homogeneous Lie group. Homogeneous Lie groups have proved to be a natural setting to generalise many questions of Euclidean harmonic analysis. Indeed, having both the group and dilation structures allows one to introduce many notions coming from the Euclidean harmonic analysis. There are several important differences between the Euclidean setting and the one of homogeneous Lie groups. For instance the operators appearing in the latter setting are usually more singular than their Euclidean counterparts. However it is possible to adapt the technique in harmonic analysis to still treat many questions in this more abstract setting

Let $M^K_n$ be the moduli space of framed $K$-instantons with instanton number $n$ when $K$ is a compact simple Lie group of classical type. Due to Donaldson's theorem, its scheme structure is given by the regular locus of a GIT quotient of $\mu^{-1}(0)$ where $\mu$ is the moment map on the associated symplectic vector space of ADHM data. A main theorem of this paper asserts that $\mu$ is flat for $K=SO(3,\mathbb{R})$ and any $n\ge0$. Hence we complete the interpretation of the K-theoretic Nekrasov partition function for the classical groups in Nekrasov-Shadchin's work in term of Hilbert series of the instanton moduli spaces together with the author's previous results. We also write ADHM data for the second symmetric and exterior products of the associated vector bundle of an instanton. This gives an explicit quiver-theoretic description of the isomorphism $M^{K}_{n}\cong M^{K'}_{n}$ for all the pairs $K,K'$ with isomorphic Lie algebras., Comment: v.1: 25 pages; v.2: 29 pages, Introduction expanded, to appear in J. Algebra

For an element $g$ in a group $X$, we say that $g$ has 2-part order $2^{a}$ if $2^{a}$ is the largest power of 2 dividing the order of $g$. We prove lower bounds on the proportion of elements in finite classical groups in odd characteristic that have certain 2-part orders. In particular, we show that the proportion of odd order elements in the symplectic and orthogonal groups is at least $C/\ell^{3/4}$, where $\ell$ is the Lie rank, and $C$ is an explicit constant. We also prove positive constant lower bounds for the proportion of elements of certain 2-part orders independent of the Lie rank. Furthermore, we describe how these results can be used to analyze part of Yal\c{c}inkaya's Black Box recognition algorithm for finite classical groups in odd characteristic., Comment: 28 pages

Given a group G which is not necessarily finite, and a Noetherian commutative ring R , an important question is when the cohomology ring H ∗ ( G , R ) is a Noetherian ring. For finite groups, this is the Venkov–Evens theorem. This theorem is extended to finite dimensional restricted Lie algebras when R is the ground field and to more general finite groups schemes over a field R by Friedlander and Suslin. Recently, van der Kallen and Touze have shown this for reductive algebraic groups over a field of positive characteristics with rational group cohomology. In this note, we prove a finite generation theorem for a large class of infinite groups which include all arithmetic subgroups of linear algebraic groups, lattices of semisimple Lie groups, mapping class groups of surfaces, outer automorphism groups of free groups, Gromov hyperbolic groups, outer automorphism groups of free groups, and many other natural infinite groups.

In this paper, we will show that nonabelian simple classical groups of Lie type are uniquely determined by the structure of their complex group algebras., Comment: 10 pages, to appear in Journal of Algebra

We study the determinants on the sum of orbits of two elements in the Lie algebra of a compact connected subgroup in the unitary group. As an application, the extremal determinant expressions are obtained for the symplectic group.

For each simply connected 3-dimensional unimodular Lie group the left invariant Riemannian metrics up to automorphism are classified in Ha and Lee (2009) [2] . Using this classification, we determine on each such a Lie group the full group of isometries which depends on left invariant Riemannian metrics.

UDC 512.542 Abstract: Let L be a simple linear or unitary group of dimension larger than 3 over a finite field of characteristic p. We deal with the class of finite groups isospectral to L. It is known that a group of this class has a unique nonabelian composition factor. We prove that if L = U4(2) ,U 5(2) then this factor is isomorphic to either L or a group of Lie type over a field of characteristic different from p.

We prove that all finite simple groups of Lie type, with the exception of the Suzuki groups, can be made into a family of expanders in a uniform way. This confirms a conjecture of Babai, Kantor and Lubotzky from 1989, which has already been proved by Kassabov for sufficiently large rank. The bounded rank case is deduced here from a uniform result for SL2 which is obtained by combining results of Selberg and Drinfeld via an explicit construction of Ramanujan graphs by Lubotzky, Samuels and Vishne.

We show that certain closure properties on subsets Q of a finite group G of Lie type enable the ratio | Q | / | G | to be determined by finding the proportions of elements of Q in the maximal tori of G and the proportions of certain related subsets of the Weyl group. We prove fundamental results about these subsets Q, including those necessary for moving between these groups and their automorphism groups, normal subgroups and central quotients. As a sample application of these new techniques, we derive upper and lower bounds for the proportion of elements with order divisible by certain natural sets of primes, for the finite classical groups, thereby improving existing results.

We give a review of infinite-dimensional Lie groups and algebras and show some applications and examples in mathematical physics. This includes diffeomorphism groups and their natural subgroups like volume-preserving and symplectic transformations, as well as gauge groups and loop groups. Applications include fluid dynamics, Maxwell's equations, and plasma physics. We discuss applications in quantum field theory and relativity (gravity) including BRST and supersymmetries.

In quantum physical theories, interactions in a system of particles are commonly understood as perturbations to certain observables, including the Hamiltonian, of the corresponding interaction-free system. The manner in which observables undergo perturbations is subject to constraints imposed by the overall symmetries that the interacting system is expected to obey. Primary among these are the spacetime symmetries encoded by the unitary representations of the Galilei group and Poincare group for the non-relativistic and relativistic systems, respectively. In this light, interactions can be more generally viewed as perturbations to unitary representations of connected Lie groups, including the non-compact groups of spacetime symmetry transformations. In this paper, we present a simple systematic procedure for introducing perturbations to (infinite dimensional) unitary representations of finite dimensional connected Lie groups. We discuss applications to relativistic and non-relativistic particle systems.

The spectrum of a finite group is the set of its element orders. Two groups are said to be isospectral if their spectra coincide. We deal with the class of finite groups isospectral to simple and orthogonal groups over a field of an arbitrary positive characteristic p. It is known that a group of this class has a unique nonabelian composition factor. We prove that this factor cannot be isomorphic to an alternating or sporadic group. We also consider the case where this factor is isomorphic to a group of Lie type over a field of the same characteristic p.

We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic stationary measures always exist, and when such a measure has full support, we show the following. 1) Either the manifold admits a smooth equivariant map onto a homogeneous projective variety, defined on an open dense conull invariant set, or the Lie algebra of the Zariski closure of the Gromov representation of the fundamental group contains a Lie subalgebra isomorphic to the Lie algebra of the acting group. As a corollary, a smooth non-trivial homogeneous projective factor does exist whenever the fundamental group of $M$ admits only virtually solvable linear representations, and thus in particular when $M$ is simply connected, regardless of the real rank. 2) There exist explicit examples showing that analytic rigid actions of certain simple groups (of real rank one) may indeed fail to have a smooth projective factor. 3) It is possible to generalize Gromov's theorem on the algebraic hull of the representation of the fundamental group of the manifold to the case of analytic rigid non-unimodular structures, for actions of simple groups of any real rank. An important ingredient in the proofs is a generalization of Gromov's centralizer theorem beyond the case of invariant measures., Comment: The existence of a smooth projective factor is now established for actions of groups of arbitrary real rank, provided that the fundamental group is amenable. Geometric and Functional Analysis, to appear

We establish analogues of Hardy’s uncertainty principle for the Fourier transform on ℝ for locally compact Abelian groups and for some classes of solvable Lie groups, such as diamond groups and exponential Lie groups with non-trivial centre.

A 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.

In this work we introduce Kobayashi hyperbolic manifolds with their automorphism groups which are Lie groups. Bounded Stein manifolds are Kobayashi hyperbolic with Lie groups as automorphism groups. One and two-Stein spaces are in close connection with dimensions of El Naschie’s exceptional Lie groups hierarchies. The sum of seventeen one and two-Stein spaces can be expressed in terms of the dimensions of certain Lie groups. The number of high energy elementary particles in an extended standard model and their fuzzy values are found.

The notions of the order of a symmetry group may be extended to that of an average, non-integer order. Building on this extension, it can be shown that the five classical exceptional Lie symmetry groups could be extended to a hierarchy, the total sum of which is four times α ¯ 0 = 137 + k 0 of the electromagnetic field. Subsequently it can be shown that all known and conjectured physical fields may be derived by E-infinity transfinite scaling transformation. Consequently E8E8 exceptional Lie symmetry groups manifold, the SL(2,7)c holographic modular curve boundary Γ(7), Einstein–Kaluza gravity R(n=4) and R(n=5) as well as the electromagnetic field are all topological transformations of each other. It is largely a matter of mathematical taste to choose E8 or the electromagnetic field associated with α ¯ 0 as derived or as fundamental. However since E8 has been extensively studied by the founding father of group theory and has recently been mapped completely, it seems beneficial to discuss at least high energy physics starting from the largest of the exceptional groups.

In this work and following some major developments in our understanding of the connection between the exceptional Lie symmetry groups hierarchy and physics, we present finite and infinite dimensional Lie groups and their connections to the orthogonal and unitary groups. The dimension of the Lie group E 7 1 2 is included in the derivation of the number of elementary particle-like super states 576 and 672. Connections to the dimensions of Lie groups and the 72 and 84 elementary particles recently considered by the Author are discussed.

In the present work, different derivations of the 548 isometries of E-infinity symmetry group are presented. The connection between the dimensions of exceptional Lie groups, orthogonal, unitary groups and the 548 is found. The work gives some arguments for deriving the inverse electromagnetic fine structure constant from 1152 bosons and an equal number of fermions following the light cone quantization of the GS action of a super Maxwell theory.

The Guillemin–Sternberg conjecture states that “quantisation commutes with reduction” in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved forcompactLie groupsGacting oncompactsymplectic manifolds, and, largely due to the use of SpincDirac operator techniques, has reached a high degree of perfection under these compactness assumptions. In this paper we formulate an appropriate Guillemin–Sternberg conjecture in the general case, under the main assumptions that the Lie group action is proper and cocompact. This formulation is motivated by our interpretation of the “quantisation commuates with reduction” phenomenon as a special case of the functoriality of quantisation, and uses equivariantK-homology and theK-theory of the groupC*-algebraC*(G) in a crucial way. For example, the equivariant index – which in the compact case takes values in the representation ringR(G) – is replaced by the analytic assembly map – which takes values inK0(C*(G)) – familiar from the Baum–Connes conjecture in noncommutative geometry. Under the usual freeness assumption on the action, we prove our conjecture for all Lie groupsGhaving a discrete normal subgroup Γ with compact quotientG/Γ, but we believe it is valid for all unimodular Lie groups.

We show that the alternating groups, the sporadic groups and the exceptional covering groups of groups of Lie type satisfy the inductive condition required to verify the McKay conjecture for finite...

The structure of Lie groups acting transitively on the direct product of a circle and an even-dimensional sphere is described. For products of two spheres of dimension >1 a similar problem has already been solved by other authors. The minimal transitive Lie groups on and are also indicated. As an application of these results, the structure of the automorphism group of one class of geometric structures, generalized quadrangles (a special case of Tits buildings) is considered. A conjecture put forward by Kramer is proved: the automorphism group of a connected generalized quadrangle of type (1,2m) always contains a transitive subgroup that is the direct product of a compact simple Lie group and a one-dimensional Lie group. Bibliography: 16 titles.

A pro-Lie group is a projective limit of finite dimensional Lie groups. It is proved that a surjective continuous group homomorphism between connected pro-Lie groups is open. In fact this remains true for almost connected pro-Lie groups where a topological group is called almost connected if the factor group modulo the identity component is compact. As consequences we get a Closed Graph Theorem and the validity of the Second Isomorphism Theorem for pro-Lie groups in the almost connected context.

The general solution of the 2-cocycle functional equation on abelian groups is well known. Methods are also known for finding regular solutions on Lie groups. Here we find general solutions of the 2-cocycle equation on certain non-abelian groups. We consider groups that are semidirect products, and in particular solvable groups that can be built up by splitting extensions.

This paper is devoted to the investigation of semidirect products of loop groups and homeomorphism or diffeomorphism groups of finite-and infinite-dimensional real, complex, and quaternion manifolds. Necessary statements about quaternion manifolds with quaternion holomorphic transition mappings between charts of atlases are proved. It is shown that these groups exist and have the structure of infinite-dimensional Lie groups, i.e., they are continuous or differentiable manifolds and the composition (f, g) ↦ f −1 g is continuous or differentiable depending on the smoothness class of groups. Moreover, it is proved that in the cases of complex and quaternion manifolds, these groups have the structures of complex and quaternion manifolds, respectively. Nevertheless, it is proved that these groups do not necessarily satisfy the Campbell-Hausdorff formula even locally outside of the exceptional case of a group of holomorphic diffeomorphisms of a compact complex manifold. Unitary representations of these groups G′, including irreducible ones, are constructed by using quasi-invariant measures on groups G relative to dense subgroups G′. It is proved that this procedure provides a family of cardinality card(ℝ) of pairwise nonequivalent, irreducible, unitary representations. The differentiabilty of such representations is studied.

We determine the 2-adic $K$-localizations for a large class of $H$-spaces and related spaces. As in the odd primary case, these localizations are expressed as fibers of maps between specified infinite loop spaces, allowing us to approach the 2-primary $v_1$-periodic homotopy groups of our spaces. The present $v_1$-periodic results have been applied very successfully to simply-connected compact Lie groups by Davis, using knowledge of the complex, real, and quaternionic representations of the groups. We also functorially determine the united 2-adic $K$-cohomology algebras (including the 2-adic $KO$-cohomology algebras) for all simply-connected compact Lie groups in terms of their representation theories, and we show the existence of spaces realizing a wide class of united 2-adic $K$-cohomology algebras with specified operations.

We complete the classification of irreducible 2F-modules for groups of Lie type acting in the natural characteristic by dealing with the three open cases from [R.M. Guralnick, G. Malle, Classification of 2F-modules, II, in: C. Ho, P. Sin, P. Tiep, A. Turull (Eds.), Finite Groups 2003, Proceedings of the Gainesville Conference on Finite Groups, March 6–12, 2003, de Gruyter, Berlin, 2004, pp. 117–183]. We also finish the classification of such modules for almost quasi-simple groups, and show that in all cases there is an offender with cubic action.

In a recent paper we found conditions for a nilpotent Lie group $N$ to have a filtration by normal subgroups whose successive quotients have square integrable representations, and such that these square integrable representations fit together nicely to give an explicit construction of Plancherel almost all representations of $N$. That resulted in explicit character formulae, Plancherel formulae and multiplicity formulae. We also showed that nilradicals $N$ of minimal parabolic subgroups $P = MAN$ enjoy that "stepwise square integrable" property. Here we extend those results to direct limits of stepwise square integrable nilpotent Lie groups. This involves some development of the corresponding Schwartz spaces. The main result is an explicit Fourier inversion formula for that class of infinite dimensional Lie groups. One important consequence is the Fourier inversion formula for nilradicals of classical minimal parabolic subgroups of finitary real reductive Lie groups such as $GL(\infty;R)$, $Sp(\infty;C)$ and $SO(\infty,\infty)$., Comment: 12 pages. arXiv admin note: text overlap with arXiv:1306.6392

We begin with a very important class of groups, the general linear groups. The groups we will study in this book will all be subgroups (of a certain sort) of one of the general linear groups. This chapter makes use of various standard results from linear algebra that are summarized in Appendix B. This chapter also assumes basic facts and definitions from the theory of abstract groups; the necessary information is provided in Appendix A.