External Algebra Research Articles

О векторных блочных элементах в задачах механики

A plane dynamic problem of the second kind for the Lame equation is considered in the first quadrant. For the first time the exact solution of this boundary value problem in the form of a Packed vector block element is constructed by the block element method. A system of Two lame differential equations is considered in the boundary value problem. To solve it by the block element method, operations of external algebra, external analysis are performed and a vector block element consisting of two components is constructed. For its construction there is a problem of differential factorization of a matrix-function of the second order-coefficient of the functional equation necessary for the correct construction of pseudo-differential equations. Their solution allows you to build components of the external form and the packaged block element itself. The solution of the considered boundary value problem is of interest because it serves the purpose of substantiating the existence and investigation of the properties of cracks of a new type, where previously the antiplane problem was used for these purposes. Also, the solution of the problem is important in the development of methods for designing materials based on block elements, in the analysis of landslides, in the problems of seismology in the analysis preparation of crustal earthquakes.

Classification of External Zonotopal Algebras

In this paper we work with power algebras associated to hyperplane arrangements. There are three main types of these algebras, namely, external, central, and internal zonotopal algebras. We classify all external algebras up to isomorphism in terms of zonotopes. Also, we prove that unimodular external zonotopal algebras are in one to one correspondence with regular matroids. For the case of central algebras we formulate a conjecture.

Algebraic description of external and internal attributes of fundamental fermions

To describe external and internal attributes of fundamental fermions, a theory of multi-spinor fields is developed on an algebra, a triplet algebra, which consists of all the triple-direct-products of Dirac γ-matrices. The triplet algebra is decomposed into the product of two subalgebras, an external algebra and an internal algebra, which are exclusively related with external and internal characteristic of the multi-spinor field named triplet fields. All elements of the external algebra which is isomorphic to the original Dirac algebra Aγ are invariant under the action of permutation group S3 which works to exchange the order of the Aγ elements in the triple-direct-product. The internal algebra is decomposed into the product of two 42 dimensional algebras, called the family and color algebras, which describe the family and color degrees of freedom. The family and color algebras have fine substructures with "trio plus solo" (3 + 1) conformations which are irreducible under the action of S3. The triplet field has trio plus solo family modes with ordinary tricolor quark and colorless solo lepton components. To incorporate the Weinberg-Salam mechanism, it is required to introduce two types of triplet fields, a left-handed doublet and right-handed singlets of electroweak iso-spin. It is possible to qualify the Yukawa interaction and to make a new interpretation of its coupling constants naturally in an intrinsic mechanism of the triplet field formalism. The ordinary Higgs mechanism leads to the Dirac mass matrices which can explain all data of quark sector within experimental accuracy.

Noncommutative differential calculus for Moyal subalgebras

We build a differential calculus for subalgebras of the Moyal algebra on R 4 starting from a redundant differential calculus on the Moyal algebra, which is suitable for reduction. In some cases we find a frame of one-forms which allows to realize the complex of forms as a tensor product of the noncommutative subalgebras with the external algebra Λ ∗ .

Spin geometry on quantum groups via covariant differential calculi

Let A be a cosemisimple Hopf ∗-algebra with antipode S and let Γ be a left-covariant first-order differential ∗-calculus over A such that Γ is self-dual (see Section 2) and invariant under the Hopf algebra automorphism S 2. A quantum Clifford algebra Cl( Γ, σ, g) is introduced which acts on Woronowicz’ external algebra Γ ∧. A minimal left ideal of Cl( Γ, σ, g) which is an A -bimodule is called a spinor module. Metrics on spinor modules are investigated. The usual notion of a linear left connection on Γ is extended to quantum Clifford algebras and also to spinor modules. The corresponding Dirac operator and connection Laplacian are defined. For the quantum group SL q (2) and its bicovariant 4 D ±-calculi these concepts are studied in detail. A generalization of Bochner's theorem is given. All invariant differential operators over a given spinor module are determined. The eigenvalues of the Dirac operator are computed.

Notes on q-Electroweak

A gauged SU q (2) theory is characterized by two dual algebras, the first lying close to the Lie algebra of SU(2) while the second introduces new degrees of freedom that may be associated with nonlocality or solitonic structure. The first and second algebras, here called the external and internal algebras respectively, define two sets of fields, also called external and internal. The gauged external fields agree with the Weinberg–Salam model at the level of the doublet representation but differ at the level of the adjoint representation. For example, the g-factor of the charged W-boson differs in the two models. The gauged internal fields remain speculative but are analogous to color fields.

On the Noncommutative Geometry of Twisted Spheres

We describe noncommutative geometric aspects of twisted deformations, in particular of the spheres in Connes and Landi [8] and in Connes and Dubois Violette [7], by using the differential and integral calculus on these spaces that is covariant under the action of their corresponding quantum symmetry groups. We start from multiparametric deformations of the orthogonal groups and related planes and spheres. We show that only in the twisted limit of these multiparametric deformations the covariant calculus on the plane gives by a quotient procedure a meaningful calculus on the sphere. In this calculus the external algebra has the same dimension of the classical one. We develop the Haar functional on spheres and use it to define an integral on forms. In the twisted limit (differently from the general multiparametric case) the Haar functional is a trace and we thus obtain a cycle on the algebra. Moreover we explicitely construct the *-Hodge operator on the space of forms on the plane and then by quotient on the sphere. We apply our results to even spheres and we compute the Chern-Connes pairing between the character of this cycle, i.e. a cyclic 2n-cocycle, and the instanton projector defined in [7].

De Rham Cohomology and Hodge Decomposition For Quantum Groups

ISTVAN HECKENBERGER and AXEL SCHULER Abstra t Let Γ = Γτ,z be one of the N -dimensional bicovariant first order differential calculi for the quantum groups GLq(N), SLq(N), SOq(N), or Spq(N), where q is a transcendental complex number and z is a regular parameter. It is shown that the de Rham cohomology of Woronowicz’ external algebra Γ coincides with the de Rham cohomologies of its left-coinvariant, its right-coinvariant and its (twosided) coinvariant subcomplexes. In the cases GLq(N) and SLq(N) the cohomology ring is isomorphic to the coinvariant external algebra Γ Inv and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator.

Kähler geometry and SUSY mechanics

We present two examples of SUSY mechanics related with Kähler geometry. The first system is the N = 4 supersymmetric one-dimensional sigma-model proposed in hep-th/0101065. Another system is the N = 2 SUSY mechanics whose phase space is the external algebra of an arbitrary Kähler manifold. The relation of these models with antisymplectic geometry is discussed.

Structural aspects of functional integral bosonization

We discuss structural aspects of functional integral bosonization of two-dimensional models. We show that the use of auxiliary vector fields enlarges the Hilbert space by the introduction of an external field algebra that should not be considered as an element of the intrinsic algebraic structure defining the model. These aspects are discussed in a model with well known and established results in the literature, by considering the Abelian reduction of the Wess-Zumino-Witten theory to reconstruct in the Hilbert space of states Coleman's proof of the fermion-boson mapping between the massive Thirring and sine-Gordon theories. We show that the factorization of the partition function will generally lead to incorrect conclusions concerning the physical content of the model, such as the existence of infinitely delocalized states and the violation of the asymptotic factorization property of the Wightman functions. In order to exert control on the effect of the redundant Bose fields and obtain the fermion-boson mapping in the Hilbert space of states, the functional integral bosonization must be performed on the generating functional.

Differential Hopf Algebras on Quantum Groups of Type A

Let A be a Hopf algebra and Γ be a bicovariant first order differential calculus over A. It is known that there are three possibilities to construct a differential Hopf algebra Γ∧=Γ⊗/Jthat contains Γ as its first order part. Corresponding to the three choices of the idealJ, we distinguish the “universal” exterior algebra, the “second antisymmetrizer” exterior algebra, and Woronowicz' external algebra, respectively. Let Γ be one of theN2-dimensional bicovariant first order differential calculi on the quantum groupGLq(N) orSLq(N), and letqbe a transcendental complex number. For Woronowicz' external algebra we determine the dimension of the space of left-invariant and of bi-invariantk-forms, respectively. Bi-invariant forms are closed and represent different de Rham cohomology classes. The algebra of bi-invariant forms is graded anti-commutative. ForN≥3 the three differential Hopf algebras coincide. However, in case of the 4D±-calculi onSLq(2) the universal differential Hopf algebra is strictly larger than Woronowicz' external algebra. The bi-invariant 1-form is not closed.

Hodge and Laplace-Beltrami Operators for Bicovariant Differential Calculi on Quantum Groups

For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace–Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinite-dimensional. Using Jucys–Murphy elements of the Hecke algebra, the eigenvalues of the Laplace–Beltrami operator for the Hopf algebra (SLq(N)) are computed.

Graded poisson-lie structures on general linear groups

The external algebra on GL(N) is equipped with graded Poisson brackets compatible with the group action. We prove that there exist two and only two corresponding Poisson-Hopf superalgebras on M and obtain their explicit descriptions. We show that precisely these superalgebras arise in the semiclassical limit of bicovariant differential calculi on the quantum linear group GLq(N).

Graded Poisson Lie structures on classical complex Lie groups

The external algebra over holomorphic first order differential forms on a complex Lie groupG is endowed with the structure of a graded Poisson Lie algebra. This structure is introduced via graded bicovariant brackets that are shown to be in one to one correspondence withG-invariant tensors of special symmetry. Complete classification of graded Poisson Lie structures defined by homogeneous brackets is obtained for the case of classical complex Lie groups.

On Poisson–Lie structure on the external algebra of the classical Lie groups

The general expression for the bicovariant bracket for odd generators of the external algebra on a Poisson–Lie group is given. The graded Poisson–Lie structures derived before for GL(N) and SL(N) are shown to be the special cases of this bracket. The formula is universal and can be applied to any matrix Lie group.

Covariant differential complexes on quantum linear groups

We consider the possible covariant external algebra structures for Cartan's 1-forms ( Omega ) on GLq(N) and SLq(N). Our starting point is that the Omega realize an adjoint representation of quantum group and all monomials of the Omega possess the unique ordering. For the external algebras obtained define the differential mapping d possessing the usual nilpotence condition, and the generally deformed version of the Leibnitz rules. The status of the known examples of GLq(N)-differential calculi in the proposed classification scheme and the problems of SLq(N)-reduction are discussed.

Bicovariant 3D calculus for SLq(2) from Poisson–Lie structures

All possible graded Poisson–Lie structures on the external algebra of SL(2) are described. We prove that differential Poisson–Lie structures prolonging the Sklyanin brackets do not exist on SL(2). There are two and only two graded Poisson–Lie structures on SL(2) and neither of them can be obtained by a reduction of graded Poisson–Lie structures on the external algebra of GL(2). Both of them can be quantized and as a result we get new graded algebras of quantum right-invariant forms on SLq(2) with three generators.

Gauge group and gauge transformation in the continual theory of defects

A representation of the algebra (3)=t(3) ⊕ S0(3, ℝ) by differential Schaefer's operators is proposed, and an external algebra of (3)-valued differential forms is constructed. The requirement of local gauge invariance is formulated in the model of the (3)-valued field, which enables a group of gauge transformations of the continual theory of defects to be obtained.

Differential calculus on compact matrix pseudogroups (quantum groups)

The paper deals with non-commutative differential geometry. The general theory of differential calculus on quantum groups is developed. Bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied. Tensor algebra and external algebra constructions are described. It is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra, so the external derivative of higher order differential forms is well defined and obeys the usual properties. The proper form of the Cartan Maurer formula is found. The vector space dual to the space of left-invariant differential forms is endowed with a bilinear operation playing the role of the Lie bracket (commutator). Generalized antisymmetry relation and Jacobi identity are proved.

Spinor space and mass formula for hadrons

In this note a representation of space-time as a bilinear product of spinors and a suggestion concerning the nature of space-time in the region occupied by an elementary particle are examined. These ideas were put forward by S~Rz (1), who defined the internalsymmetry algebra Ua in terms of differential operators in the spinor variables and then derived the Gell-Nfann-Okubo (2) mass formula for hadrons from the tensor properties of the mass operator of the external (de Sitter) space-time symmetry algebra. The novelty of this approach prompts us to consider its relationship to the way in which this problem has been attempted previously, i.e: by embedding the external and internal symmetry algebras in a larger higher-symmetry or noninvariance algebra Let W~ be a four-component complex spinor, providing a basis for the fourdimensional representation of SUe.e, the group of 4 x 4 complex unimodular matrices satisfying