Automorphisms Of Products Research Articles

IF-AUTOMORPHISMS OF A FREE SOLVABLE Sp-PERMUTABLE ALGEBRA OF A FINITE RANK OVER A LEFT CANCELLATIVE MONOID ARE PSEUDO-TAME

Consider a variety V of acts over a left cancellative monoid S that have a ternary Maltsev operation p(〈p, S〉- algebras ). Using the Magnus–Artamonov representation, the construction of the free Abelian extension of an arbitrary V-algebra was obtained and explored by the author. In the present article we prove that each IF-automorphism of a free k-step solvable V-algebra of a finite rank (k > 1) is pseudo-tame, i.e. it is presented as a product of elementary automorphisms of a special module over a ring with several objects.

Automorphic products, generalized Kac-Moody algebras and string amplitudes

We review automorphic products and generalized Kac-Moody algebras from a physics point of view. We discuss the appearance of automorphic products in BPS-saturated one-loop quantities in heterotic string theory. At particular points in moduli space, with enhanced gauge symmetry \U0001d525, these products can be used to define a generalized Kac-Moody algebra \U0001d50a(\U0001d524++) as an automorphic correction of the Lorentzian Kac-Moody algebra \U0001d524++, which is obtained through double extension of the complement \U0001d524 = (\U0001d5228 ⊕ \U0001d5228)/\U0001d525. The root multiplicities of \U0001d50a(\U0001d524++) are then encoded in the Fourier coefficients of certain modular forms, which appear directly in the integrand of the one-loop quantities. We review particular examples of this extension for compactifications with \U0001d4a9 = 2 and \U0001d4a9 = 4 space-time supersymmetry.

Regularized theta lifts for orthogonal groups over totally real fields

We define a regularized theta lift from SL2 to orthogonal groups over totally real fields. It takes harmonic ‘Whittaker forms’ to automorphic Green functions and weakly holomorphic Whittaker forms to meromorphic modular forms on orthogonal groups with zeros and poles supported on special divisors, generalizing Borcherds' work on automorphic products. To prove our results, we use the spectral expansion of the lift and study its relationship with the cohomological theta lift of Kudla and Millson.

Products of loxodromic automorphisms of pretrees

We extend some basic properties of hyperbolic isometries of \({\mathbb{R}}\)-trees to the case of automorphisms of median pretrees.

AUTOMORPHISMS OF THE UHF ALGEBRA THAT DO NOT EXTEND TO THE CUNTZ ALGEBRA

AbstractThe automorphisms of the canonical core UHF subalgebra ℱn of the Cuntz algebra 𝒪n do not necessarily extend to automorphisms of 𝒪n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras Mn. In that case, necessary and sufficient conditions for the extension property are presented. Also addressed is the problem of extending to 𝒪n the automorphisms of the diagonal 𝒟n, which is a regular maximal abelian subalgebra with Cantor spectrum. In particular, it is shown that there exist product-type automorphisms of 𝒟n that do not extend to (possibly proper) endomorphisms of 𝒪n.

Random products of automorphisms of Heisenberg nilmanifolds and Weil’s representation

AbstractForn≥1, letHbe the (2n+1)-dimensional real Heisenberg group, and let Λ be a lattice inH. Let Γ be the group of automorphisms of the corresponding nilmanifold Λ∖HandUthe associated unitary representation of Γ onL2(Λ∖H) . Denote byTthe maximal torus factor associated to Λ∖H. Using Weil’s representation (also known as the metaplectic representation), we show that a dense set of matrix coefficients of the restriction ofUto the orthogonal complement ofL2(T) inL2(Λ∖H) belong toℓ4n+2+ε(Γ) for every ε>0 . We give the following application to random walks on Λ∖Hdefined by a probability measureμon Aut (Λ∖H) . Denoting by Γ(μ) the subgroup of Aut (Λ∖H) generated by the support ofμand byU0andV0the restrictions ofUto, respectively, the subspaces ofL2(Λ∖H) andL2(T) with zero mean, we prove the following inequality:whereλis the left regular representation of Γ(μ) onℓ2(Γ(μ)) . In particular, the action of Γ(μ) on Λ∖Hhas a spectral gap if and only if the corresponding action of Γ(μ) onThas a spectral gap.

The Weil Representation of and Some Applications

The theta function of a positive definite even lattice of even rank generates a representation of on the group algebra of the discriminant form of the lattice. This representation goes back to Jacobi and is called Weil representation. We derive an explicit formula for the action in terms of the genus of the lattice. This generalizes classical results of Schoeneberg and Weil. We use the formula to calculate the lift from scalar-valued modular forms on Γ 0 (N) to modular forms for the Weil representation. We also show that the elements of the Mathieu group M 23 correspond naturally to reflective automorphic products of singular weight, and we construct three generalized Kac-Moody superalgebras representing supersymmetric superstrings in dimensions 10, 6, and 4.

Automorphisms of Sylow p-subgroups of Chevalley groups over p-primary residue rings of integers

In this paper, we prove that any automorphism of a Sylow p-subgroup of the Chevalley group over the ring \( \mathbb{Z}_{p^m } \) (where p is prime and m ≥ 1) is a product of graph, inner, diagonal, and hypercentral automorphisms.

Natural constructions of some generalized Kac–Moody algebras as bosonic strings

There are 10 generalized Kac-Moody algebras whose denominator identities are completely reflective automorphic products of singular weight on lattices of squarefree level. Under the assumption that the meromorphic vertex operator algebra of central charge 24 and spin-1 algebra $\hat{A}_{p-1,p}^r$ exists we show that four of them can be constructed in a uniform way from bosonic strings moving on suitable target spaces.

Homomorphisms of two-dimensional linear groups over a ring of stable range one

Let R be a commutative domain of stable range 1 with 2 a unit. In this paper we describe the homomorphisms between SL 2 ( R ) and GL 2 ( K ) where K is an algebraically closed field. We show that every non-trivial homomorphism can be decomposed uniquely as a product of an inner automorphism and a homomorphism induced by a morphism between R and K. We also describe the homomorphisms between GL 2 ( R ) and GL 2 ( K ) . Those homomorphisms are found of either extensions of homomorphisms from SL 2 ( R ) to GL 2 ( K ) or the products of inner automorphisms with certain group homomorphisms from GL 2 ( R ) to K.

Decomposition of Lie automorphisms of upper triangular matrix algebra over commutative rings

Let T n+1 ( R) be the algebra of all upper triangular square matrices of order n + 1 over a commutative ring R with the identity 1 and unit 2. For n ⩾ 2, we prove that any Lie automorphism of T n+1 ( R) can be uniquely written as a product of graph, central, inner and diagonal automorphisms.

On the classification of automorphic products and generalized Kac–Moody algebras

On the classification of automorphic products and generalized Kac–Moody algebras

Birational morphisms of the plane

Let A 2 A^2 be the affine plane over a field K K of characteristic 0 0 . Birational morphisms of A 2 A^2 are mappings A 2 → A 2 A^2 \to A^2 given by polynomial mappings φ \varphi of the polynomial algebra K [ x , y ] K[x,y] such that for the quotient fields, one has K ( φ ( x ) , φ ( y ) ) = K ( x , y ) K(\varphi (x), \varphi (y)) = K(x,y) . Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping τ x \tau _x given by x → x , y → x y x \to x, ~y \to xy . For a while, it was an open question whether every birational morphism is a product of polynomial automorphisms and copies of τ x \tau _x . This question was answered in the negative by P. Russell (in an informal communication). In this paper, we give a simple combinatorial solution of the same problem. More importantly, our method yields an algorithm for deciding whether a given birational morphism can be factored that way.

The Nagata automorphism is wild.

It is proved that the well known Nagata automorphism of the polynomial ring in three variables over a field of characteristic zero is wild, that is, it can not be decomposed into a product of elementary automorphisms.

A natural construction of Borcherds' fake baby monster Lie algebra

We use a Z 2 -orbifold of the vertex operator algebra associated to the Niemeier lattice with root lattice A 8 3 and the no-ghost theorem of string theory to construct a generalized Kac-Moody algebra. Borcherds' theory of automorphic products allows us to determine the simple roots and identify the algebra with the fake baby monster Lie algebra.

Eisenstein series attached to lattices¶and modular forms on orthogonal groups

We study certain vector valued Eisenstein series on the metaplectic cover of SL2(ℝ), which transform with the Weil representation associated with the discriminant group of an even lattice L. We find a closed formula for the Fourier coefficients in terms of Dirichlet L-series and representation numbers of L modulo “bad” primes. Such Eisenstein series naturally occur in the context of Borcherds' theory of automorphic products. We indicate some applications to modular forms on the orthogonal group of L with zeros on Heegner divisors.

Automorphisms of the Lie algebra of strictly upper triangular matrices over certain commutative rings

Let n be the nilpotent Lie algebra consisting of all strictly upper triangular (n+1)×(n+1) matrices over a commutative ring R. In this paper, we discuss the automorphism group of n. We prove that any automorphism ϕ of n can be uniquely expressed as ϕ=ω·η·ξ·μ·σ, where ω, η, ξ, μ and σ are graph, diagonal, external, central and inner automorphisms, respectively, of n when n⩾3 and R is a local ring that contains 2 as a unit or an integral domain of characteristic other than two. In the case n=2 we also prove that any automorphism of n can be expressed as a product of graph, diagonal, extremal and inner automorphisms for an arbitrary local ring R.

Local Borcherds products

The local Picard group at a generic point of the one-dimensional Baily-Borel boundary of a Hermitean symmetric quotient of type O(2,n) is computed. The main ingredient is a local version of Borcherds’ automorphic products. The local obstructions for a Heegner divisor to be principal are given by certain theta series with harmonic coefficients. Sometimes they generate Borcherds’ space of global obstructions. In these particular cases we obtain a simple proof of a result due to the first author: Suppose that Γ⊂O(2,n) is the orthogonal group attached to an even unimodular lattice. Then every meromorphic modular form for Γ, whose zeros and poles lie on Heegner divisors, is given by a Borcherds product.

A public key system with signature and master key functions

Let K be a finite field of 2m elements. Let Ø4Ø3Ø2Ø1be tame automorphisms of the n + r-dimensional affine space Kn+r . Let the composition Ø4Ø3Ø2Ø1 be π. The automorphism π and some of the Ø i ’S will be hidden. Let the component expression of π be . Let the restriction of π to a subspace be as . The field K and the polynomial map will be announced as the public key. Given a plaintext , then the ciphertext will be . Given Ø i and , it is easy to find . Therefore the plaintext can be recovered by . The private key will be the set of maps Ø4Ø3Ø2Ø1. The security of the system rests in part on the difficulty of finding the map π from the partial informations provided by the map and the factorization of the map π into a product (i.e., composition) of tame automorphisms Ø i 's.

A structural result of irreducible inclusions of type 𝐼𝐼𝐼_{𝜆} factors, 𝜆∈(0,1)

Given an irreducible inclusion of factors with finite index N ⊂ M N\subset M , where M M is of type I I I λ 1 / m {III}_{\lambda ^{1/m}} , N N of type I I I λ 1 / n {III}_{\lambda ^{1/n}} , 0 > λ > 1 0>\lambda >1 , and m , n m,n are relatively prime positive integers, we will prove that if N ⊂ M N\subset M satisfies a commuting square condition, then its structure can be characterized by using fixed point algebras and crossed products of automorphisms acting on the middle inclusion of factors associated with N ⊂ M N\subset M . Relations between N ⊂ M N\subset M and a certain G G -kernel on subfactors are also discussed.