Yang-Mills Algebra Research Articles

Hochschild homology and cohomology of down–up algebras

We present a detailed computation of the cyclic and the Hochschild homology and cohomology of generic and 3-Calabi–Yau homogeneous down–up algebras. This family was defined by Benkart and Roby in [3] in their study of differential posets. Our calculations are completely explicit, by making use of the Koszul bimodule resolution and some arguments similar to those used in [13] to compute the Hochschild cohomology of Yang–Mills algebras.

Non-abelian whole gauge symmetry

The wholeness principle is analysed for non-abelian gauge symmetry. This principle states that nature acts through grouping. It says that physical laws should be derived from elds associations. At this work, we consider on the possibility of introducting a non-abelian elds set fAaIg under a common gauge parameter.A Yang-Mills extension is studied. Taking the SU(N) symmetry group with different potential elds rotating under a same group, new elds strengths are developed. They express covariant entities which are granular, collective, correlated, and not necessarily Lie algebra valued. They yield new scalars and a Lagrangian beyond Yang-Mills is obtained. Classical equations are derived and (2N + 7) equations are developed. A further step is on how such non-abelian whole symmetry is implemented at SU(N) gauge group. For this, it is studied on the algebra closure and Jacobi identities, Bianchi identities, Noether theorem, gauge xing, BRST symmetry, conservation laws, covariance, charges algebra. As result, one notices that it is installed at SU(N) symmetry independentlyon the number of involved elds. Given this consistency, Yang-Mills should not more be considered as the unique Lagrangian performed from SU(N).Introducting the BRST symmetry an invariant Leff is stablished. The BRST charge associated to the N-potential elds system is calculated and its nilpotency property obtained.Others conservations laws involving ghost scale, global charges are evalued showing that this whole symmetry extension preserve the original Yang-Mills algebra. Also the ghost number is conserved. These results imply that Yang-Mills should be understood as a pattern and not as a specic Lagrangian.Concluding, an extended Lagrangian can be constructed. It is possible to implement a non-abelian whole gauge symmetry based on a elds set fAaIg. Its physical feature is a systemic interpretation for the physical processes. Understand complexity from whole gauge symmetry.

Some remarks on representations of Yang-Mills algebras

In this article, we present some new properties of representations of Yang-Mills algebras. We first show that any free Lie algebra with m generators is a quotient of the Yang-Mills algebra 𝔶𝔪(n) on n generators, for n ≥ 2m. We derive from this that any semisimple Lie algebra and even any affine Kac-Moody algebra is a quotient of 𝔶𝔪(n) for n ≥ 4. Combining this with previous results on representations of Yang-Mills algebras given in [Herscovich and Solotar, Ann. Math. 173(2), 1043–1080 (2011)], one may obtain solutions to the Yang-Mills equations by differential operators acting on sections of twisted vector bundles on the affine space of dimension n ≥ 4 associated to representations of any semisimple Lie algebra. We also show that this quotient property does not hold for n = 3, since any morphism of Lie algebras from 𝔶𝔪(3) to 𝔰𝔩(2, k) has in fact solvable image.

GRAVIWEAK UNIFICATION, INVISIBLE UNIVERSE AND DARK ENERGY

We consider a graviweak unification model with the assumption of the existence of a hidden (invisible) sector of our Universe, parallel to the visible world. This Hidden World (HW) is assumed to be a Mirror World (MW) with broken mirror parity. We start with a diffeomorphism invariant theory of a gauge field valued in a Lie algebra [Formula: see text], which is broken spontaneously to the direct sum of the space–time Lorentz algebra and the Yang–Mills algebra: [Formula: see text] — in the ordinary world, and [Formula: see text] — in the hidden world. Using an extension of the Plebanski action for general relativity, we recover the actions for gravity, SU(2) Yang–Mills and Higgs fields in both (visible and invisible) sectors of the Universe, and also the total action. After symmetry breaking, all physical constants, including the Newton's constants, cosmological constants, Yang–Mills couplings, and other parameters, are determined by a single parameter g present in the initial action, and by the Higgs VEVs. The dark energy problem of this model predicts a too large supersymmetric breaking scale (M SUSY ~1010 GeV ), which is not within the reach of the LHC experiments.

On the multi-Koszul property for connected algebras.

\small In this article we introduce the notion of emphmulti-Koszul algebra for the case of a locally finite dimensional nonnegatively graded connected algebra, as a generalization of the notion of (generalized) Koszul algebras defined by R. Berger for homogeneous algebras. This notion also extends and generalizes the one recently introduced by the author and A. Rey, which was for the particular case of algebras further assumed to be finitely generated in degree 1 and with a finite dimensional space of relations. The idea of this new notion for this generality, which should be perhaps considered as a probably interesting common property for several of these algebras, was to find a grading independent description of some of the more appealing features shared by all generalized Koszul algebras. It includes several new interesting examples, textite.g. the super Yang-Mills algebras introduced by M. Movshev and A. Schwarz, which are not generalized Koszul or even multi-Koszul for the previous definition given by the author and Rey in any natural manner. On the other hand, we provide an equivalent description of the new definition in terms of the textrmTor (or textrmExt) groups, similar to the existing one for homogeneous algebras, and we show that several of the typical homological computations performed for the generalized Koszul algebras are also possible in this more general setting. In particular, we give a very explicit description of the A_{\infty} -algebra structure of the Yoneda algebra of a multi-Koszul algebra, which has a similar pattern as for the case of generalized Koszul algebras. We also show that a finitely generated multi-Koszul algebra with a finite dimensional space of relations is a K_{2} algebra in the sense of T. Cassidy and B. Shelton.

Representations of Super Yang-Mills Algebras

We study in this article the representation theory of a family of super algebras, called the \emph{super Yang-Mills algebras}, by exploiting the Kirillov orbit method \textit{\`a la Dixmier} for nilpotent super Lie algebras. These super algebras are a generalization of the so-called \emph{Yang-Mills algebras}, introduced by A. Connes and M. Dubois-Violette in \cite{CD02}, but in fact they appear as a "background independent" formulation of supersymmetric gauge theory considered in physics, in a similar way as Yang-Mills algebras do the same for the usual gauge theory. Our main result states that, under certain hypotheses, all Clifford-Weyl super algebras $\Cliff_{q}(k) \otimes A_{p}(k)$, for $p \geq 3$, or $p = 2$ and $q \geq 2$, appear as a quotient of all super Yang-Mills algebras, for $n \geq 3$ and $s \geq 1$. This provides thus a family of representations of the super Yang-Mills algebras.

Hochschild and cyclic homology of Yang–Mills algebras

The aim of this article is to present a detailed algebraic computation of the Hochschild and cyclic homology groups of the Yang–Mills algebras YM( n ) ( n ∈ ℕ ≧2 ) defined by A. Connes and M. Dubois-Violette in [8], continuing thus the study of these algebras that we have initiated in [17]. The computation involves the use of a spectral sequence associated to the natural filtration on the universal enveloping algebra YM( n ) provided by a Lie ideal 𝔱𝔶𝔪( n ) in 𝔶𝔪( n ) which is free as Lie algebra. As a corollary, we describe the Lie structure of the first Hochschild cohomology group.

Representations of Yang-Mills algebras

The aim of this article is to describe families of representations of the Yang-Mills algebras YM(n) (n 2 N 2) dened by A. Connes and M. DuboisViolette. We rst describe some irreducible nite dimensional representations. Next, we provide families of innite dimensional representations of YM, big enough to separate points of the algebra. In order to prove this result, we prove and use that all Weyl algebras Ar(k) are epimorphic images of YM(n).

Unification of gravity, gauge fields and Higgs bosons

We consider a diffeomorphism invariant theory of a gauge field valued in a Lie algebra that breaks spontaneously to the direct sum of the spacetime Lorentz algebra, a Yang–Mills algebra and their complement. Beginning with a fully gauge invariant action—an extension of the Plebanski action for general relativity—we recover the action for gravity, Yang–Mills and Higgs fields. The low-energy coupling constants, obtained after symmetry breaking, are all functions of the single parameter present in the initial action and the vacuum expectation value of the Higgs.

Discrete Minimal Surface Algebras

We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sl(n) (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d<=4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.

Covariant field equations, gauge fields and conservation laws from Yang-Mills matrix models

The effective geometry and the gravitational coupling of nonabelian gauge and scalar fields on generic NC branes in Yang-Mills matrix models is determined. Covariant field equations are derived from the basic matrix equations of motions, known as Yang-Mills algebra. Remarkably, the equations of motion for the Poisson structure and for the nonabelian gauge fields follow from a matrix Noether theorem, and are therefore protected from quantum corrections. This provides a transparent derivation and generalization of the effective action governing the SU(n) gauge fields obtained in [1], including the would-be topological term. In particular, the IKKT matrix model is capable of describing 4-dimensional NC space-times with a general effective metric. Metric deformations of flat Moyal-Weyl space are briefly discussed.

Cohomology of Yang–Mills algebras

In this paper we compute cyclic and Hochschild homology of the universal envelope U(YM) of the Yang–Mills Lie algebra YM . We also compute Hochschild cohomology with coefficients in U(YM) , considered as a bimodule over itself. The result of the calculations depends on the number of generators n of YM . The semidirect product \mathfrak{so}(n) ⋉ ℂ^n acts by derivations upon U(YM) . One of the important consequences of our results is that if n ≥ 3 then the Lie algebra of outer derivations of U(YM) coincides with \mathfrak{so}(n) ⋉ ℂ^n .

Multilinear forms and graded algebras

In this paper we investigate the class of the connected graded algebras which are finitely generated in degree 1, which are finitely presented with relations of degrees greater or equal to 2 and which are of finite global dimension D and Gorenstein. For D greater or equal to 4 we add the condition that these algebras are homogeneous and Koszul. It is shown that each such algebra is completely characterized by a multilinear form satisfying a twisted cyclicity condition and some other nondegeneracy conditions depending on the global dimension D. This multilinear form plays the role of a volume form and canonically identifies in the quadratic case to a nontrivial Hochschild cycle of maximal degree. Several examples including the Yang–Mills algebra and the extended 4-dimensional Sklyanin algebra are analyzed in this context. Actions of quantum groups are also investigated.

Inhomogeneous Yang-Mills Algebras

We determine all inhomogeneous Yang-Mills algebras and super Yang-Mills algebras which are Koszul. Following a recent proposal, a non-homogeneous algebra is said to be Koszul if the homogeneous part is Koszul and if the PBW property holds. In this paper, the homogeneous parts are the Yang-Mills algebra and the super Yang-Mills algebra.

Yang–Mills Algebra

Some unexpected properties of the cubic algebra generated by the covariant derivatives of a generic Yang–Mills connection over the (s+1)-dimensional pseudo Euclidean space are pointed out. This algebra is Koszul of global dimension 3 and Gorenstein but except for s=1 (i.e. in the two-dimensional case) where it is the universal enveloping algebra of the Heisenberg Lie algebra and is a cubic Artin–Schelter regular algebra, it fails to be regular in that it has exponential growth. We give an explicit formula for the Poincare series of this algebra $$\mathcal{A}$$ and for the dimension in degree n of the graded Lie algebra of which $$\mathcal{A}$$ is the universal enveloping algebra. In the four-dimensional (i.e. s=3) Euclidean case, a quotient of this algebra is the quadratic algebra generated by the covariant derivatives of a generic (anti) self-dual connection. This latter algebra is Koszul of global dimension 2 but is not Gorenstein and has exponential growth. It is the universal enveloping algebra of the graded Lie algebra which is the semi-direct product of the free Lie algebra with three generators of degree one by a derivation of degree one.

Super-particles, superfields and Yang-Mills Lagrangians

The N = 1 scalar, and spinor, and the N = 2 scalar, chiral, causal superfields are derived directly from the known irreducible particle representations, using standard methods for conventional theories. The conventional methods for developing the algebra of non-Abelian Yang-Mills theories are used to derive the generalized supersymmetric Yang-Mills algebras for N = 1, N = 2 and N = 4 ‘ special ’ theories with central charge, from which the chiral superfields and Lagrangians are constructed.