Lie Superalgebras Based on 𝔤𝔩n Associated to the Adjoint Representation, and Invariant Geometric Structures Defined on Them

Finite-dimensional real and complex Lie superalgebras whose underlying Lie algebra is ????_n and whose odd module is ????_n itself under the adjoint representation are classified up to isomorphism. It is shown that for n≥3 there are one-parameter families of nonisomorphic such Lie superalgebras, plus another set of finitely many different isomorphism classes. For n=2 there are 10 different isomorphism classes over the real field, and 8 different over the complex numbers. For n=1 there are 2 different isomorphism classes over either ground field. Representatives on each isomorphism class are given, and their automorphism groups are determined. The question as to which representatives admit ?₂-graded, ad-invariant geometric structures (of orthogonal or symplectic type) is also addressed, and a precise list of which of such geometric structures can be defined on each isomorphism class is given. In particular, it is shown that ?₂-homogeneous, orthogonal, ad-invariant geometric structures must be odd. The case of ????₂ over the real field is further analyzed in order to determine for which of the equivalence classes that admit such a structure, that structure can be induced by an underlying Minkowski metric on the 4-dimensional (nongraded) ????₂.     

Entropy Treatment of Evolution Algebras

In this paper, by introducing an entropy of Markov evolution algebras, we treat the isomorphism of S-evolution algebras. A family of Markov evolution algebras is defined through the Hadamard product of structural matrices of non-negative real S-evolution algebras, and their isomorphism is studied by means of their entropy. Furthermore, the isomorphism of S-evolution algebras is treated using the concept of relative entropy.     

Isomorphisme de Duflo et la cohomologie tangentielle

In the present note we show that the Duflo isomorphism can be extended to an isomorphism of associative algebras of tangential cohomologies. This result confirms the Shoikhet’s conjecture.     

On the Connections of W Algebras from Different Reductions of WZNW Theory

The possible isomorphic relations between W algebras from various conformal reductions of WZNW model are investigated and the criterion for the isomorphism of W algebras is given.In the special case of sl₃ WZNW theory,we show explicitly the isomorphism between W[(111)₂] and W[(12)₁] and discuss the constraint-gauge condition transmutation between the O'Raifeartaigh gauges and the transmutation of conformal weights under the isomorphism transformation.     

Automorphisms of Order Structures of Abelian Parts of Operator Algebras and Their Role in Quantum Theory

It is shown that any order isomorphism between the structures of unital associative JB subalgebras of JB algebras is given naturally by a partially linear Jordan isomorphism. The same holds for nonunital subalgebras and order isomorphisms preserving the unital subalgebra. Finally, we recover usual action of time evolution group on a von Neumann factor from group of automorphisms of the structure of Abelian subalgebras.     

SAYD Modules over Lie-Hopf Algebras

In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.     

Effect test spaces and effect algebras

The concept of an effect test space, which is equivalent to a D-test space of Dvurečenskij and Pulmannová, is introduced. Connections between effect test space. (E-test space, for short) morphisms, and event-morphisms as well as between algebraic E-test spaces and effect algebras, are studied. Bimorphisms and E-test space tensor products are considered. It is shown that any E-test space admits a unique (up to an isomorphism) universal group and that this group, considered as a test group, determines the E-test space uniquely (up to an isomorphism).     

Line bundles and divisors on a super riemann surface

Line bundles and divisors are defined on a super Riemann surface. The isomorphism between them is shown.     

On the local central limit theorem for Gibbs processes

We derive a sufficient condition for the validity of the local central limit theorem for Gibbs processes and their isomorphism with a Bernoulli shift.     

关于规范群中含有Abel子群和有Higgs场的情况下重正化前后规范群的同构

在规范群含有Abel子群和有Higgs场的情况下,对重正化前后的规范群的同构性质作了一个严格的证明。 关键词：     

La méthode de la fonction génératrice et les développements en quasibosons

An isomorphism between the oH' subspace of the physical bound states and a normalized oF functional space is established. This is done by introducing a generating function for these oH' states. By choosing for the oH-space the Bargmann's space, an isomorphism is obtained between an oH_F space of the fermion states and an oH_B space of the boson states. An expansion of any quasiboson operator of oH_B is thus feasible. This is particularly interesting for the Hamiltonian operator. In this case the expansion obtained, similar to the Beliaev-ZeIevinski and Marumori expansions, appear to be simple and more convergent.     

Path integral formulation for quantized fermion and boson fluids

An isomorphism between fully quantized fermion and boson fluids and classical polymer mixtures is used as a point of departure to initiate an analytic treatment of quantum fluids.     

The mass gap for Higgs models on a unit lattice

An isomorphism is established between eertain compact and noncompact formulations of Abelian gauge theory on a lattice. For weak coupling, the mass gap predicted by the Higgs mechanism is then established.     

Differential operators on a Riemann surface with projective structure

Let X be a Riemann surface equipped with a projective structure and a theta characteristic on X, or in other words, is a holomorphic line bundle equipped with a holomorphic isomorphism with the holomorphic cotangent bundle Ω_X. The complement of the zero section in the total space of the line bundle has a natural holomorphic symplectic structure, and using , this symplectic structure has a canonical quantization. Using this quantization, holomorphic differential operators on X are constructed. The main result is the construction of a canonical isomorphism

, n≥0, provided i[−2(k−1),0].     

Harish-Chandra isomorphisms for quantum algebras

The center of the quantum algebra is studied. Especially an analogue of the Harish-Chandra isomorphism is established.     

GAUGE CONDITIONS AND THE RENORMALIZABILITY OF A GENERALIZED GAUGE TIiEORY CONSISTING OF SEVERAL ABELIAN AND NON-ABELIAN SUB-GROUPS

We discuss generalized gauge conditions and the renormalizability of a generalized gauge theory, which contains finite numbers of Abe-Zian and non-Abelian sub-groups. The isomorphism between gauge groups before and after renormalization is demonstrated.     

Canonical involutions in three-dimensional generalised Lie algebras

All involutions in complex three-dimensional generalised Lie algebras with one-dimensional or zero-dimensional homogeneous components are described up to isomorphism.     

Constructing Hadamard States via an Extended Møller Operator

We consider real scalar field theories, whose dynamics is ruled by normally hyperbolic operators differing only by a smooth potential V. By means of an extension of the standard definition of Møller operator, we construct an isomorphism between the associated spaces of smooth solutions and between the associated algebras of observables. On the one hand, such isomorphism is non-canonical, since it depends on the choice of a smooth time-dependant cut-off function \({\chi}\). On the other hand, given any quasi-free Hadamard state for a theory with a given V, such isomorphism allows for the construction of another quasi-free Hadamard state for a different potential. The resulting state preserves also the invariance under the action of any isometry, whose associated Killing field \({\xi}\) is complete and fulfilling both \({\mathcal{L}_\xi V=0 \,\, {\rm and} \,\, \mathcal{L}_\xi\chi=0}\). Eventually, we discuss a sufficient condition to remove on static spacetimes, the dependence on the cutoff via a suitable adiabatic limit.     

Space-time isomorphism problem is intractable (NP-hard)

The problem of whether or not two different mathematical models of space-time describe the same space-time is not trivial. For example, the first spherically symmetric solution—the Schwarzschild metric—describes only part of the process of falling into a black hole, and other metrics were discovered for the same situation that also describe the subsequent events. These metrics turned out to be isomorphic in the sense that some 1-1 correspondence (coordinates transformations) transform one into another. But in the general case to find whether such an isomorphism exists is a difficult computational problem. There are some algorithms for smooth metrics, but the problem is also important for the nonsmooth metrics involving singularities. We prove that in the general case this isomorphism problem is intractable.