Finite-dimensional Group Research Articles

Some groups generated by transvection subgroups

In the 1960’s, J. E. McLaughlin [lo, 111 determined all irreducible finite dimensional linear groups generated by transvection subgroups. (A trans- section of a k-vector space V is linear map t(cp, x): V-t given by vt(cp, x) = 2) + VqJ .Y for all t’ E V, where XE Vand (PE V* are non-zero and xcp = 0; V* is the dual space of V A k-transvection subgroup T(cp, x) consists of the identity and all r(cp’, x’), where q’ and x’ range over the l-dimensional subspaces spanned by cp and x, respectively; it is isomorphic to the additive group of k.) McLaughlin showed that the only such groups are the special linear and symplectic groups and, in the case when k is F, (the field with two elements), the orthogonal and symmetric groups. It is our purpose to give a new proof of a result which includes McLaughlin’s; we discard the assumptions of finiteness of the dimension of V over k and also (to some extent) the irreducibility of the group, while still obtaining a result which is recognizably like McLaughlin’s. Our result for irreducible groups is the following. (Unexplained terminol- ogy will be defined after the statement of the theorem; the more com- plicated result under a weaker hypothesis is stated near the end of the Introduction.)

Duality invariant string algebra and D = 4 effective actions

An infinite-dimensional gauge algebra (DISG) is defined in terms of string vertex operators. The DISG reproduce the gauge algebra of 4D, N = 4 compactifications of the heterotic string, and is invariant under the full internal duality group O(6, 22, Z). The DISG is an indefinite signature lattice algebra which contains affine Lie algebras of any level. It uniquely specifies the N = 4 low-energy effective lagrangian for gravitational and matter N = 4 multiplets. The gauge symmetries are broken on any background to a finite-dimensional gauge group. Orbifold truncations to N = 1, 2 are defined and studied for the case of Z N orbifolds. Inclusion of higher spin fields and the appearance of “duality forms” due to integration of massive fields are discussed.

Friedan-Shenker bundle from Chern-Simons theory

In this letter we present a proof of the invariance of the space of quantum states of the Chern-Simons (CS) theory in the presence of Wilson lines under parallel transport with respect to the Knizhnik-Zamolodchikov (KZ) connection for the case of a simple, simply connected, finite-dimensional group and genus-zero surface. The proof is based on the polynomial realization of the space of tensors in which these quantum states take values.

$T$-equivariant $K$-theory of generalized flag varieties

Let G be a Kac-Moody group with Borel subgroup B and compact maximal torus T. Analogous to Kostant and Kumar [Kostant, B. & Kumar, S. (1986) Proc. Nati. Acad. Sci. USA 83, 1543-1545], we define a certain ring Y, purely in terms of the Weyl group W (associated to G) and its action on T. By dualizing Y we get another ring 1, which, we prove, is canonically isomorphic with the T-equivariant Ktheory KT(G/B) of GIB. Now KT(G/B), apart from being an algebra over KT(pt.) A(T), also has a Weyl group action and, moreover, KT(G/B) admits certain operators {DW},,w similar to the Demazure operators defined on A(T). We prove that these structures on KT(G/B) come naturally from the ring Y. By evaluating the A(T)-module W at 1, we recover K(G/B) together with the above-mentioned structures. We believe that many of the results of this paper are new in the finite case (i.e., G is a finite-dimensional semisimple group over C)

𝐿²-Dolbeault complexes on singular curves and surfaces

Give the smooth part of a singular curve or normal surface the metric induced from the ambient projective space. On this incomplete manifold the minimal L 2 ∂ ¯ {L^2}\bar \partial -complex of ( 0 , q ) (0,q) -forms has finite-dimensional cohomology groups. The Euler characteristic of this cohomology equals the Todd genus of any desingularization of the singular variety.

Infinite alternating groups as finitary linear transformation groups

abstract degree of a classical group as its natural projective representation degree (see (6.2)). Case (1) of (2.6) is dispensed with using Mal’cev’s local characterization of finite dimensional linear groups and their classification. THEOREM (2.7) (Mal’cev, see [ 13, l.L.91). Let G be a Iocally finite group with adeg(G) < 00. Then G is a finite dimensional linear group. THEOREM (2.8) [l, 2,8,21]. Let G be an infinite group. Then is a simple, locally finite, linear group of finite degree if and only if G is isomorphic to an adjoint group of Lie type over an infinite subfield Fp, the algebraic closure of [F,, for some prime p. 3. LOCAL CHARACTERIZATION OF FINITARY LINEAR GROUPS; A THEOREM OF MAL’CEV TYPE Mal’cev’s theorem (2.7) provides a local characterization of locally finite linear groups G of finite degree. The property of having a faithful linear representation of degree n passes from the set of finite subgroups of G to G itself. Under this section we provide a similar result for locally finite simple groups which are linitary linear. Remember that the linear transformation g of the vector space I/ is finitary if g fixes pointwise a subspace of finite codimension in V. That is, the subspace C,(g) = {v E V 1 vg = v} has finite codimension in V, or equivalently the subspace [V, g] = {v(g - 1) 1 v E V} has finite dimension in V. The collection of all linitary members of GL( V) forms a normal sub- group FGL( V), and a group which is (isomorphic to) a subgroup of FGL( V), for some V, is sometimes called a finitary linear group.

Quantization of the Kepler manifold

A representation ofSO(2,n+1), the maximal finite dimensional dynamical group of then-dimensional Kepler problem, is obtained by means of (pseudo) differential operators acting onL 2(S n). This representation is unitary when restricted toSO(2) ⊗SO (n+1), i.e. to the physically relevant subgroup.

On Classical Clifford Theory

Let $k$ be a field, let $N$ be a normal subgroup of a finite group $H$ and let $M$ be a completely reducible $k[N]$-module. We give sufficient conditions for a finite dimensional (finite) group crossed product $k$-algebra to be a Frobenius or symmetric $k$-algebra. These results imply that $k[H]/(J(k[N])k[H])$ and the endomorphism $k$-algebra, ${\operatorname {End} _{k[H]}}({M^H})$, of the induced module ${M^H}$ are symmetric $k$-algebras. We also completely describe the $k[H]$-indecomposable decomposition of ${M^H}$. It follows that the head and socle of an indecomposable component of ${M^H}$ are irreducible isomorphic $k[H]$-modules.

An algorithm for determining the simplicity of a modular group representation

Let G be a finite group an F and arbitrary splitting field for G. Using the Amitsur-Levitzki theorem on the polynomial identities satisfied by a matrix ring over a commutative ring an algorithm is given for finding the minimal dimension t for which there is a simple t-dimensional FG-submodule U in a faithful FG-module M of dimension n <∞. As a corollary a new criterion is obtained for checking the simplicity of a finite-dimensional group representation.

On classical Clifford theory

Let k k be a field, let N N be a normal subgroup of a finite group H H and let M M be a completely reducible k [ N ] k[N] -module. We give sufficient conditions for a finite dimensional (finite) group crossed product k k -algebra to be a Frobenius or symmetric k k -algebra. These results imply that k [ H ] / ( J ( k [ N ] ) k [ H ] ) k[H]/(J(k[N])k[H]) and the endomorphism k k -algebra, End k [ H ] ( M H ) {\operatorname {End} _{k[H]}}({M^H}) , of the induced module M H {M^H} are symmetric k k -algebras. We also completely describe the k [ H ] k[H] -indecomposable decomposition of M H {M^H} . It follows that the head and socle of an indecomposable component of M H {M^H} are irreducible isomorphic k [ H ] k[H] -modules.

T -equivariant K -theory of generalized flag varieties

Let G be a Kac-Moody group with Borel subgroup B and compact maximal torus T. Analogous to Kostant and Kumar [Kostant, B. & Kumar, S. (1986) Proc. Natl. Acad. Sci. USA 83, 1543-1545], we define a certain ring Y, purely in terms of the Weyl group W (associated to G) and its action on T. By dualizing Y we get another ring Psi, which, we prove, is "canonically" isomorphic with the T-equivariant K-theory K(T)(G/B) of G/B. Now K(T)(G/B), apart from being an algebra over K(T)(pt.) approximately A(T), also has a Weyl group action and, moreover, K(T)(G/B) admits certain operators {D(w)}w[unk]W similar to the Demazure operators defined on A(T). We prove that these structures on K(T)(G/B) come naturally from the ring Y. By "evaluating" the A(T)-module Psi at 1, we recover K(G/B) together with the above-mentioned structures. We believe that many of the results of this paper are new in the finite case (i.e., G is a finite-dimensional semisimple group over C) as well.

Loop groups, grassmanians and string theory

The theory of representations of loop groups provides a framework where one can consider Riemann surfaces with arbitrary numbers of handles and nodes on the same footing. Using infinite grassmanians we present a general formulation of some conformal field theories on arbitrary surfaces in terms of an operator formalism. As a by-product, one can obtain some general results for the chiral bosonization of fermions using the vertex operator representation of finite dimensional groups. We believe that this set-up provides the natural arena where the recent proposal of Friedan and Shenker of formulating string theory in the universal moduli space can be discussed.

Finiteness properties of groups

Recall that a group r is said to be of type FP, (resp. FP,) if the ZT-module Z admits a projective resolution which is finitely generated in dimensions <n (resp. in all dimensions), cf. [4] or [S]. For example, r is of type FP, if and only if it is finitely generated, and r is of type FP, if it is finitely presented. (The converse of the last assertion is not known.) In this paper we give a necessary and sufficient condition for r to be of type FP,. The condition involves the homological properties of a suitable topological space X on which r acts. It is quite easy to use in practice, once one has a suitable X. We will illustrate this by giving several non-trivial examples. In Section 1 we recall a well-known sufficient condition for the FP, property. We then give our necessary and sufficient condition in Section 2 (Theorem 2.2). Section 3 contains a similar result about finite presentability. The remainder of the paper deals with examples. In Section 4 we treat several infinite families of groups, which include some finitely presented simple groups first constructed by R.J. Thompson in 1965 and later generalized by Higman. They also include a certain group F, also constructed by Thompson, which later reappeared in homotopy theory and was eventually shown to be of type FP, [lo]. We will give a unified proof that all the groups r in these families are finitely presented and of type FP,. We will also show that these groups all satisfy H*(T, ZT) = 0. This had previously been proven for some of them by Brown and Geoghegan. In Section 5 we consider a sequence of groups H,, (n2 1) introduced by Houghton [18]. Using methods surprisingly similar to those used for the ThompsonHigman groups, we show that H,, is of type FP,_, but not FP,. Finally, in Section 6 we look at a sequence r’ of solvable S-arithmetic groups first studied by Abels [l]. The work of various people (see [2] for references) has led to the result that r, is of type FP, _ 1 but not FP, . We give a new proof of this result by applying the criterion of Section 2 to the action of r, on a certain Bruhat-Tits building.

Holomorphic discrete series and harmonic series unitary irreducible representations of non-compact Lie groups: Sp(2n, R), U(p, q) and SO*(2n)

Generalised characters of the infinite dimensional, holomorphic, discrete series, unitary, irreducible representations of the non-compact groups U(p, q), Sp(2n, R) and SO*(2n) are explicitly expressed in terms of characters of finite dimensional unitary group representations. These formulae are remarkably succinct despite involving certain infinite series of Schur functions. Similar formulae are derived for harmonic series unitary representation of both U(p, q) and Sp(2n, R). Consideration of the branching rules from U(p, q) to U(q)*(p) and from Sp(2n, R) to U(n) enables holomorphic representations to be identified as a subset of the harmonic representations. The branching rules are established in full generality and are then used in the evaluation of tensor products of both holomorphic and harmonic representations. In the case of the former a known result is recast in terms of closed formulae involving Schur functions and for the latter various generalisations of these formulae are given. A conjecture is also made regarding what might be the simplest possible formulae covering all holomorphic and harmonic representations of Sp(2n, R) and U(p, q). Illustrative examples are presented.

Results on BRS cohomologies in gauge theory

We compute the cohomology of the Becchi-Rouet-Stora operator in gauge theory over general space-time M without boundary, with structure group G, for class of polynomial functions of the field. We show that the problem reduces to a standard problem for the finite dimensional group G. As a consequence, we prove that, within this class of polynomials, all anomalies and Schwinger terms are obtained from invariants of G.

Infinite-dimensional metagonal and metaplectic groups. I. General concepts and metagonal groups

We study central S1 -extensions of the groups of orthogonal and symplectic operators on a Hilbert space with Hilbert-Schmidt antilinear part. We investigate in detail the corresponding 2-cocyles on their Lie algebras. The passage to the limit of the corresponding finite-dimensional groups is described and it is shown that both extensions are nontrivial even in the topological sense. Among the applications is a unified construction of the basic modulus for Kac-Moody algebras.

Complex abelian Lie groups with finite-dimensional cohomology groups

Complex abelian Lie groups with finite-dimensional cohomology groups

Absolute convergence of Fourier series on finite-dimensional groups

Absolute convergence of Fourier series on finite-dimensional groups

Group automorphisms with finite entropy

A compact metrie abelian group X with the normalized Haar measure is a Lebesgue probability space. A group automorphism σ ofX is an invertible measure preserving transformation of the probability space. This paper is to show that if the entropy of σ is finite, then there exist totally disconnected subgroupsH andN, a finite-dimensional subgroupS and a subgroupT satisfying the conditions: (i)H, N, S andT are strictly σ-invariant, (ii)N=H∩S∩T, (iii)h(σ| N )=0, (iv) ifS/N is non-trivial then it is a finite-dimensional solenoidal group with condition (**) (see the definition in §1), (v) ifT/N is non-trivial then it is connected and locally connected, such thatX/N splits into a direct sumX/N=H/N⊕S/N⊕T/N. This result characterizes the structure of finite entropy automorphisms.

World-line invariance in predictive mechanics

The Currie–Hill conditions for the relativistic world-line invariance of a Newtonian-like dynamical system of interacting particles are generalized to cover the case of invariance under any given finite-dimensional continuous group of transformations of space–time. Necessary and locally sufficient conditions are obtained both in the general case and in the important particular case when the group of transformations includes time translations as a subgroup.