On the existence of an invariant non-degenerate bilinear form under a linear map

Let V be a vector space over a field F. Assume that the characteristic of F is large, i.e. char(F)>dimV. Let T:V→V be an invertible linear map. We answer the following question in this paper. When doesVadmit a T-invariant non-degenerate symmetric (resp. skew-symmetric) bilinear form? We also answer the infinitesimal version of this question.Following Feit and Zuckerman 2, an element g in a group G is called real if it is conjugate in G to its own inverse. So it is important to characterize real elements in GL(V,F). As a consequence of the answers to the above question, we offer a characterization of the real elements in GL(V,F).Suppose V is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form B. Let S be an element in the isometry group I(V,B). A non-degenerate S-invariant subspace W of (V,B) is called orthogonally indecomposable with respect to S if it is not an orthogonal sum of proper S-invariant subspaces. We classify the orthogonally indecomposable subspaces. This problem is non-trivial for the unipotent elements in I(V,B). The level of a unipotent T is the least integer k such that (T-I)^k=0. We also classify the levels of unipotents in I(V,B).     

Weighted Berezin transform in the polydisc

For c>−1, let νc denote a weighted radial measure on C normalized so that νc(D)=1. If f is harmonic and integrable with respect to νc over the open unit disc D, then for every ψ∈Aut(D). Equivalently f is invariant under the weighted Berezin transform; Bcf=f. Conversely, does the invariance under the weighted Berezin transform imply the harmonicity of a function? In this paper, we prove that for any 1?p<∞ and c₁,c₂>−1, a function f∈Lp(D²,νc₁×νc₂) which is invariant under the weighted Berezin transform; Bc₁,c₂f=f needs not be 2-harmonic.     

The Berezin transform and Laplace-Beltrami operator

We study the Berezin transform of bounded operators on the Bergman space on a bounded symmetric domain Ω in Cn. The invariance of range of the Berezin transform with respect to G=Aut(Ω), the automorphism group of biholomorphic maps on Ω, is derived based on the general framework on invariant symbolic calculi on symmetric domains established by Arazy and Upmeier. Moreover we show that as a smooth bounded function, the Berezin transform of any bounded operator is also bounded under the action of the algebra of invariant differential operators generated by the Laplace-Beltrami operator on the unit disk and even on the unit ball of higher dimensions.     

Wave and Dirac operators,and representations of the conformal group

Let M be the flat Minkowski space. The solutions of the wave equation, the Dirac equations, the Maxwell equations, or more generally the mass 0, spin s equations are invariant under a multiplier representation U_s, of the conformal group. We provide the space of distributions solutions of the mass 0, spin s equations with a Hilbert space structure H_s, such that the representation U_s, will act unitarily on H_s. We prove that the mass 0 equations give intertwining operators between representations of principal series. We relate these representations to the Segal-Shale-Weil (or “ladder”) representation of U(2, 2).     

Plancherel Formula for Berezin Deformation of L on Riemannian Symmetric Space

Consider natural representations of the pseudounitary group U(p, q) in the space of holomorphic functions on the Cartan domain (Hermitian symmetric space) U(p, q)/(U(p)×U(q)). Berezin representations of O(p, q) are the restrictions of such representations to the subgroup O(p, q). We obtain the explicit Plancherel formula for the Berezin representations. The support of the Plancherel measure is a union of many series of representations. The density of the Plancherel measure on each piece of the support is an explicit product of Γ-functions. We also show that the Berezin representations give an interpolation between L² on noncompact symmetric space O(p, q)/O(p)×O(q) and L² on compact symmetric space O(p+q)/O(p)×O(q).     

Volume comparison and the σk-Yamabe problem

In this paper we study the problem of finding a conformal metric with the property that the kth elementary symmetric polynomial of the eigenvalues of its Weyl-Schouten tensor is constant. A new conformal invariant involving maximal volumes is defined, and this invariant is then used in several cases to prove existence of a solution, and compactness of the space of solutions (provided the conformal class admits an admissible metric). In particular, the problem is completely solved in dimension four, and in dimension three if the manifold is not simply connected.     

Full field algebras, operads and tensor categories

We study the operadic and categorical formulations of (conformal) full field algebras. In particular, we show that a grading-restricted R×R-graded full field algebra is equivalent to an algebra over a partial operad constructed from spheres with punctures and local coordinates. This result is generalized to conformal full field algebras over VL⊗VR, where VL and VR are two vertex operator algebras satisfying certain finiteness and reductivity conditions. We also study the geometry interpretation of conformal full field algebras over VL⊗VR equipped with a nondegenerate invariant bilinear form. By assuming slightly stronger conditions on VL and VR, we show that a conformal full field algebra over VL⊗VR equipped with a nondegenerate invariant bilinear form exactly corresponds to a commutative Frobenius algebra with a trivial twist in the category of VL⊗VR-modules. The so-called diagonal constructions [Y.-Z. Huang, L. Kong, Full field algebras, arXiv: math.QA/0511328] of conformal full field algebras are given in tensor-categorical language.     

Space of invariant bilinear forms

Let \({\mathbb {F}}\) be a field, V a vector space of dimension n over \({\mathbb {F}}\). Then the set of bilinear forms on V forms a vector space of dimension \(n^2\) over \({\mathbb {F}}\). For char \({\mathbb {F}}\ne 2\), if T is an invertible linear map from V onto V then the set of T-invariant bilinear forms, forms a subspace of this space of forms. In this paper, we compute the dimension of T-invariant bilinear forms over \({\mathbb {F}}\). Also we investigate similar type of questions for the infinitesimally T-invariant bilinear forms (T-skew symmetric forms). Moreover, we discuss the existence of nondegenerate invariant (resp. infinitesimally invariant) bilinear forms.     

On Conformal Invariants Built from Spinor Fields

By means of a conformal covariant differentiation process we construct generating systems for conformally invariant symmetric (r, s)–spinors in an arbitrary curved space–time. Extending this method to conformally invariant linear differential operators acting on symmetric spinor fields some classes of such operators are derived.     

Semi-Invariants of Symmetric Quivers of Tame Type

A symmetric quiver (Q, σ) is a finite quiver without oriented cycles Q?=?(Q ₀, Q ₁) equipped with a contravariant involution σ on $Q_0\sqcup Q_1$ . The involution allows us to define a nondegenerate bilinear form $\langle -,-\rangle_V$ on a representation V of Q. We shall say that V is orthogonal if $\langle -,-\rangle_V$ is symmetric and symplectic if $\langle -,-\rangle_V$ is skew-symmetric. Moreover, we define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. So we prove that if (Q, σ) is a symmetric quiver of tame type then the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type c ^V and, when the matrix defining c ^V is skew-symmetric, by the Pfaffians pf ^V . To prove it, moreover, we describe the symplectic and orthogonal generic decomposition of a symmetric dimension vector.     

Finite dimensional imbeddings of harmonic spaces

For a noncompact harmonic manifoldM we establish finite dimensionality of the eigensubspacesV _γ generated by radial eigenfunctions of the form coshr+c. As a consequence, for such harmonic manifolds, we give an isometric imbedding ofM into (V _γ,B), whereB is a nondegenerate symmetric bilinear indefinite form onV _γ (analogous to the imbedding of the real hyperbolic spaceH ⁿ into ? ⁿ⁺¹ with the indefinite formQ(x,x)=?x ₀ ² +Σx _i ² ). This imbedding is minimal in a ‘sphere’ in (V _γ,B). Finally we give certain conditions under whichM is symmetric.     

Invariant bilinear forms on a vertex algebra

In this paper we construct a linear space that parameterizes all invariant bilinear forms on a given vertex algebra with values in a arbitrary vector space. Also we prove that every invariant bilinear form on a vertex algebra is symmetric. This is a generalization of the result of Li (J. Pure Appl. Algebra 96(3) (1994) 279), who proved this for the case when the vertex algebra is non-negatively graded and has finite dimensional homogeneous components.As an application, we introduce a notion of a radical of a vertex algebra. We prove that a radical-free vertex algebra A is non-negatively graded, and its component A₀ of degree 0 is a commutative associative algebra, so that all structural maps and operations on A are A₀-linear. We also show that in this case A is simple if and only if A₀ is a field.     

On products of two involutions in the orthogonal group of a vector space

Let V be a finite-dimentional vector space over a commutative field of characteristic distinct from 2. Let V carry a symmetric nondegenerate bilinear form. Results: (A) Let π = ρσ, where π, ρ, σ ∈ O(V) and ρ, σ are involutions. There exists an orthogonal decomposition of V into orthogonally indecomposable π-modules which are simultaneously invariant under ρ and σ. (B) Let π ∈ O(V).One can find involutions ρ, σ ∈ O(V) such that π = ρσ and B(π) = B(ρ) + B(σ) holds if and only if an orthogonal decomposition of V into orthogonally indecomposable π-modules does not contain a term whose minimum polynomial is (x−1)^α where α is even.     

Some ergodic theorems for a parabolic Anderson model

In this paper,we study some ergodic theorems of a class of linear systems of interacting diffusions,which is a parabolic Anderson model.First,under the assumption that the transition kernel a=(a(i,j)) i,j∈s is doubly stochastic,we obtain the long-time convergence to an invariant probability measure νh starting from a bounded a-harmonic function h based on self-duality property,and then we show the convergence to the invariant probability measure νh holds for a broad class of initial distributions.Second,if(a(i,j)) i,j∈S is transient and symmetric,and the diffusion parameter c remains below a threshold,we are able to determine the set of extremal invariant probability measures with finite second moment.Finally,in the case that the transition kernel(a(i,j)) i,j∈S is doubly stochastic and satisfies Case I(see Case I in [Shiga,T.:An interacting system in population genetics.J.Math.Kyoto Univ.,20,213-242(1980)]),we show that this parabolic Anderson model locally dies out independent of the diffusion parameter c.     

Intersection Theorems for t-Valued Functions

This paper investigates the maximum possible size of families ℱ of t-valued functions on an n-element set S = {1, 2, . . . , n}, assuming any two functions of ℱ agree in sufficiently many places. More precisely, given a family ℬ of k-element subsets of S, it is assumed for each pair h, g ∈ ℱ that there exists a B in ℬ such that h = g on B. If ℬ is ‘not too large’ it is shown that the maximal families have t^n−k members.     

New results on q-positivity

In this paper we discuss symmetrically self-dual spaces, which are simply real vector spaces with a symmetric bilinear form. Certain subsets of the space will be called q-positive, where q is the quadratic form induced by the original bilinear form. The notion of q-positivity generalizes the classical notion of the monotonicity of a subset of a product of a Banach space and its dual. Maximal q-positivity then generalizes maximal monotonicity. We discuss concepts generalizing the representations of monotone sets by convex functions, as well as the number of maximally q -positive extensions of a q-positive set. We also discuss symmetrically self-dual Banach spaces, in which we add a Banach space structure, giving new characterizations of maximal q-positivity. The paper finishes with two new examples.     

Monodromy Groups and Bilinear Monodromy Invariant Combinations of Generalized Hypergeometric Type Integrals

The problem of determining bilinear combinations of holomorphic and antiholomorphic generalized hypergeometric type integrals left invariant under the action of the monodromy groups of the integrals is studied. In the special cases of simple Pochhammer type integrals and of twofold hypergeometric type integrals the existence and uniqueness of the bilinear invariants are proved, and the bilinear invariants are explicitly computed. Preparing the tools it is shown how to linearize and iterate representations of the braid group B_n as automorphism groups of certain free subgroups of the braid group B_n+1, and how the resulting iterated linear representations of the braid group in a natural way provide an algorithm to compute the monodromy group of generalized hypergeometric type integrals. Explicit formulae for different types of integration contours are given in the case of simple and twofold integrals.     

The Berezin Transform and Radial Operators on the Bergman Space of the Unit Ball

In this paper, we investigate the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a class of operators that we call radial operators, an oscillation criterion and diagonal are sufficient conditions under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit sphere. We further study a special class of radial operators, i.e., Toeplitz operators with a radial L ¹(B _n ) symbol.     

Specht property for the 2-graded identities of the Jordan algebra of a bilinear form

Let K be a field of characteristic zero and V a vector space of dimension m>1 with a nondegenerate symmetric bilinear form f:V×V→K. The Jordan algebra B_m = K⊕V of the form f is a ?₂-graded algebra with this decomposition. We prove that the ideal of all the ?₂-graded identities of B_m satisfies the Specht property and we compute the ?₂-graded cocharacter sequence of B_m.     

Tensor products of holomorphic representations and bilinear differential operators

Let be the weighted Bergman space on a bounded symmetric domain D=G/K. It has analytic continuation in the weight ν and for ν in the so-called Wallach set still forms unitary irreducible (projective) representations of G. We give the irreducible decomposition of the tensor product of the representations for any two unitary weights ν and we find the highest weight vectors of the irreducible components. We find also certain bilinear differential intertwining operators realizing the decomposition, and they generalize the classical transvectants in invariant theory of . As applications, we find a generalization of the Bol's lemma and we characterize the multiplication operators by the coordinate functions on the quotient space of the tensor product modulo the subspace of functions vanishing of certain degree on the diagonal.