Invariants of the adjoint action in the nilradical of a parabolic subalgebra of types Bn, Cn, and Dn

We study invariants of the adjoint action of the unipotent group in the nilradical of a parabolic subalgebra of types B _n , C _n , D _n . We introduce the notion of expanded base in the set of positive roots and construct an invariant for every root of the expanded base. We prove that these invariants are algebraically independent. We also give an estimate of the transcendence degree of the invariant field. Bibliography: 6 titles.     

Representation fields for quaternionic skew-hermitian forms

Classes of indefinite quadratic forms in a genus are in correspondence with the Galois group of an abelian extension called the spinor class field (Estes and Hsia, Japanese J. Math. 16, 341–350 (1990)). Hsia has proved (Hsia et al., J. Reine Angew. Math. 494, 129–140 (1998)) the existence of a representation field F with the property that a lattice in the genus represents a fixed given lattice if and only if the corresponding element of the Galois group is trivial on F. This far, the corresponding result for skew-hermitian forms was known only in some special cases, e.g., when the ideal (2) is square free over the base field. In this work we prove the existence of representation fields for quaternionic skew-hermitian forms in complete generality.     

ADDITIVE FAMILIES OF INVARIANTS OF FORMS OF DEGREE d

Abstract

Let d be a positive integer, and F be a field of characteristic zero. Suppose that for each positive integer n, I _n, is a GL _n,(F)- invariant of forms of degree d in x₁, …, x _n, over F. We call {I _n} an additive family of invariants if I _p+q (f ⊥ g) = I _p(f).I _q(g) whenever f; g are forms of degree d over F in x _l, …, x _p; …, x _q respectively, and where (f ⊥ g)(x _l, …, x _p+q) = f(x ₁, …, x _p,) + g (x _p+1, …, x _p+q). It is well-known that the family of discriminants of the quadratic forms is additive. We prove that in odd degree d each invariant in an additive family must be a constant. We also give an example in each even degree d of a nontrivial family of invariants of the forms of degree d. The proofs depend on the symbolic method for representing invariants of a form, which we review.     

Quaternionic and para-quaternionic <Emphasis Type="Italic">CR</Emphasis> structure on (4n+3)-dimensional manifolds

We define notion of a quaternionic and para-quaternionic CR structure on a (4n+3)-dimensional manifold M as a triple (ω₁,ω₂,ω₃) of 1-forms such that the corresponding 2-forms satisfy some algebraic relations. We associate with such a structure an Einstein metric on M and establish relations between quaternionic CR structures, contact pseudo-metric 3-structures and pseudo-Sasakian 3-structures. Homogeneous examples of (para)-quaternionic CR manifolds are given and a reduction construction of non homogeneous (para)-quaternionic CR manifolds is described.     

Nonperturbative approach to finite-dimensional non-Gaussian integrals

We study the homogeneous non-Gaussian integral $ J_{n|r} (S) = \int {e^{ - S(x_1 , \ldots ,x_n )} } d^n x $ J_{n|r} (S) = \int {e^{ - S(x_1 , \ldots ,x_n )} } d^n x , where S(x₁,…,x_n) is a symmetric form of degree r in n variables. This integral is naturally invariant under SL(n) transformations and therefore depends only on the invariants of the form. For example, in the case of quadratic forms, it is equal to the (−1/2)th power of the determinant of the form. For higher-degree forms, the integral can be calculated in some cases using the so-called Ward identities, which are second-order linear differential equations. We describe the method for calculating the integral and present detailed calculations in the case where n = 2 and r = 5. It is interesting that the answer is a hypergeometric function of the invariants of the form.     

For a fixed Turaev shadow Jones-Vassiliev invariants depend polynomially on the gleams

We use Turaev's technique of shadows and gleams to parametrize the set of all knots in S ³ with the same Hopf projection. We show that the Vassiliev invariants arising from the Jones polynomial J _t (K) are polynomials in the gleams, i.e., for , the n-th order Vassiliev invariant u _n , defined by , is a polynomial of degree 2n in the gleams. Received: April 30, 1996     

Laplace Invariants of Two-Dimensional Open Toda Lattices

We show that Toda lattices with the Cartan matrices A _n , B _n , C _n , and D _n are Liouville-type systems. For these systems of equations, we obtain explicit formulas for the invariants and generalized Laplace invariants. We show how they can be used to construct conservation laws (x and y integrals) and higher symmetries.     

Scalar differential invariants of symplectic Monge-Ampère equations

All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce a series of invariant differential forms and vector fields which allow us to construct numerous scalar differential invariants of higher order. The introduced invariants give a solution of the symplectic equivalence of Monge-Ampère equations. As an example we study equations of the form u _xy + f(x, y, u _x , u _y ) = 0 and in particular find a simple linearization criterion.     

Lipschitzian retracts and curves as invariant sets

By an invariant set in a metric space we mean a non-empty compact set K such that K = ⋃ _i=1 ⁿ T _i (K) for some contractions T ₁, …, T _n of the space. We prove that, under not too restrictive conditions, the union of finitely many invariant sets is an invariant set. Hence we establish collage theorems for non-affine invariant sets in terms of Lipschitzian retracts. We show that any rectifiable curve is an invariant set though there is a simple arc which is not an invariant set.      

ADDITIVE FAMILIES OF INVARIANTS OF SKEWSYMMETRIC TENSORS

Abstract

Let d be a positive integer and F be a field of characteristic 0. Suppose that for each positive integer n, I _n is a polynomial invariant of the usual action of GL_n (F) on Λ^d(Fⁿ), such that for t ? Λ^d(F ^k) and s ? Λ^d(F ^l), I _{k + l} (t l s) = I _k(t)I _t (s), where t ⊥ s is defined in §1.4. Then we say that {I_n} is an additive family of invariants of the skewsymmetric tensors of degree d, or, briefly, an additive family of invariants. If not all the I_n are constant we say that the family is non-trivial. We show that in each even degree d there is a non-trivial additive family of invariants, but that this is not so for any odd d. These results are analogous to those in our paper [3] for symmetric tensors. Our proofs rely on the symbolic method for representing invariants of skewsymmetric tensors. To keep this paper self-contained we expound some of that theory, but for the proofs we refer to the book [2] of Grosshans, Rota and Stein.     

Pseudo-conformal quaternionic CR structure on (4n + 3)-dimensional manifolds

We study an integrable, nondegenerate codimension 3-subbundle ${\mathcal{D}}We study an integrable, nondegenerate codimension 3-subbundle on a (4n + 3)-manifold M whose fiber supports the structure of 4n-dimensional quaternionic vector space. It is thought of as a generalization of quaternionic CR structure. We single out an -valued 1-form ω locally on a neighborhood U such that and construct the curvature invariant on (M, ω) whose vanishing gives a uniformization to flat quaternionic CR geometry. The invariant obtained on M has the same formula as that of pseudo-quaternionic K?hler 4n-manifolds. From this viewpoint, we exhibit a quaternionic analogue of Chern-Moser’s CR structure. The authors are grateful to ESI for financial support and hospitality during the preparation of this work. The first author acknowledge the support by Grant FWF Project P17108-N04 (Vienna) and Grant N MSM 0021622409 of the Ministry of Education, Youth and Sports (Brno).     

The Yamabe problem on quaternionic contact manifolds

By constructing normal coordinates on a quaternionic contact manifold M, we can osculate the quaternionic contact structure at each point by the standard quaternionic contact structure on the quaternionic Heisenberg group. By using this property, we can do harmonic analysis on general quaternionic contact manifolds, and solve the quaternionic contact Yamabe problem on M if its Yamabe invariant satisfies λ(M) < λ(ℍ ⁿ ). Mathematics Subject Classification (2000) 53C17, 53D10, 35J70     

Quaternionic matrices: Unitary similarity, simultaneous triangularization and some trace identities

We construct six unitary trace invariants for 2×2 quaternionic matrices which separate the unitary similarity classes of such matrices, and show that this set is minimal. We have discovered a curious trace identity for two unit-speed one-parameter subgroups of Sp(1). A modification gives an infinite family of trace identities for quaternions as well as for 2×2 complex matrices. We were not able to locate these identities in the literature. We prove two quaternionic versions of a well known characterization of triangularizable subalgebras of matrix algebras over an algebraically closed field. Finally we consider the problem of describing the semi-algebraic set of pairs (X,Y) of quaternionic n×n matrices which are simultaneously triangularizable. Even the case n=2, which we analyze in more detail, remains unsolved.     

The curvature invariant of a non-commutingn-tuple

Non-commutative versions of Arveson's curvature invariant and Euler characteristic for a commutingn-tuple of operators are introduced. The non-commutative curvature invariant is sensitive enough to determine if ann-tuple is free. In general both invariants can be thought of as measuring the freeness or curvature of ann-tuple. The connection with dilation theory provides motivation and exhibits relationships between the invariants. A new class of examples is used to illustrate the differences encountered in the non-commutative setting and obtain information on the ranges of the invariants. The curvature invariant is also shown to be upper semi-continuous.     

The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order

We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group \mathbbZ_p{\mathbb{Z}_p}, where p is an odd prime. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p = n = 3. We use the expressions obtained to show that any such manifold is trivial in the appropriate reduced equivariant spin bordism group.     

Dynamical generalizations of the lagrange spectrum

We compute two invariants of topological conjugacy, the upper and lower limits of the inverse of Boshernitzan??s ne _n , where e _n is the smallest measure of a cylinder of length n, for three families of symbolic systems: the natural codings of rotations, three-interval exchanges, and Arnoux-Rauzy systems. The sets of values of these invariants for a given family of systems generalize the Lagrange spectrum, which is what we obtain for the family of rotations with the upper limit of 1/ne _n.     

Lower degree bounds for modular invariants and a question of I. Hughes

We prove two statements. The first one is a conjecture of Ian Hughes which states that iff ₁, ..., f_n are primary invariants of a finite linear groupG, then the least common multiple of the degrees of thef _i is a multiple of the exponent ofG.The second statement is about vector invariants: IfG is a permutation group andK a field of positive characteristicp such thatp divides |G|, then the invariant ringK[V ^m]^G ofm copies of the permutation moduleV overK requires a generator of degreem(p–1). This improves a bound given by Richman [6], and implies that there exists no degree bound for the invariants ofG that is independent of the representation.     

The quaternionic evolution operator

In the recent years, the notion of slice regular functions has allowed the introduction of a quaternionic functional calculus. In this paper, motivated also by the applications in quaternionic quantum mechanics, see Adler (1995) [1], we study the quaternionic semigroups and groups generated by a quaternionic (bounded or unbounded) linear operator T=T₀+iT₁+jT₂+kT₃. It is crucial to note that we consider operators with components T?(?=0,1,2,3) that do not necessarily commute. Among other results, we prove the quaternionic version of the classical Hille–Phillips–Yosida theorem. This result is based on the fact that the Laplace transform of the quaternionic semigroup etT is the S-resolvent operator , the quaternionic analogue of the classical resolvent operator. The noncommutative setting entails that the results we obtain are somewhat different from their analogues in the complex setting. In particular, we have four possible formulations according to the use of left or right slice regular functions for left or right linear operators.     

Characterization and Transformation of Invariants

We consider difference equations of order k _n+k ≥ 2 of the form: y_n+k = f(yn,…,yn+k-1), n= 0,1,2,… where f: D ^k → D is a continuous function, and D?R. We develop a necessary and sufficient condition for the existence of a symmetric invariant I(x ₁,…,x_k ) ∈C^∞[D^k,D]. This condition will be used to construct invariants for linear and rational difference equations. Also, we investigate the transformation of invariants under invertible maps. We generalize and extend several results that have been obtained recently.