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In this paper, we give an explicit realization of all irreducible modules in Chari's category \widetilde {\mathcal O} for the affine Kac-Moody algebra A_{1}^{(1)} by using the idea of free fields. We work on a much more general setting which also gives us explicit realizations of all simple weight modules for certain current algebra of \mathfrak {sl}_2(\mathbb {C}) with finite weight multiplicities, including the polynomial current algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t], the loop algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1}] and the three-point Lie algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1},(t-1)^{-1}] arisen in the work by Kazhdan-Lusztig. [ABSTRACT FROM AUTHOR]

We review some recent progress in the theory of electroweak radiative corrections in semileptonic decay processes. The resurrection of the so-called Sirlin’s representation based on current algebra relations permits a clear separation between the perturbatively-calculable and incalculable pieces in the O(GFα) radiative corrections. The latter are expressed as compact hadronic matrix elements that allow systematic non-perturbative analysis such as dispersion relation and lattice QCD. This brings substantial improvements to the precision of the electroweak radiative corrections in semileptonic decays of pion, kaon, free neutron and JP=0+ nuclei that are important theory inputs in precision tests of the Standard Model. Unresolved issues and future prospects are discussed.

We review some recent progress in the theory of electroweak radiative corrections in semileptonic decay processes. The resurrection of the so-called Sirlin's representation based on current algebra relations permits a clear separation between the perturbatively-calculable and incalculable pieces in the O (G F α) radiative corrections. The latter are expressed as compact hadronic matrix elements that allow systematic non-perturbative analysis such as dispersion relation and lattice QCD. This brings substantial improvements to the precision of the electroweak radiative corrections in semileptonic decays of pion, kaon, free neutron and J P = 0 + nuclei that are important theory inputs in precision tests of the Standard Model. Unresolved issues and future prospects are discussed. [ABSTRACT FROM AUTHOR]

In this note we realise current algebra with anomalous terms in terms of a |$U(1)$| gauge theory, in the space of maps |$M$| from |$S^1$| into a compact Lie group corresponding to the current algebra. The Wilson loop around a closed curve in |$M$| is shown to be the Wess–Zumino–Witten term. This discussion enables a simple understanding of the non-Abelian anomaly in the Schrödinger picture. [ABSTRACT FROM AUTHOR]

We review a non-relativistic current algebra symmetry approach to constructing the Bogolubov generating functional of many-particle distribution functions and apply it to description of invariantly reduced Hamiltonian systems of the Boltzmann type kinetic equations, related to naturally imposed constraints on many-particle correlation functions. As an interesting example of deriving Vlasov type kinetic equations, we considered a quantum-mechanical model of spinless particles with delta-type interaction, having applications for describing so called Benney-type hydrodynamical praticle flows. We also review new results on a special class of dynamical systems of Boltzmann–Bogolubov and Boltzmann–Vlasov type on infinite dimensional functional manifolds modeling kinetic processes in many-particle media. Based on algebraic properties of the canonical quantum symmetry current algebra and its functional representations, we succeeded in dual analysis of the infinite Bogolubov hierarchy of many-particle distribution functions and their Hamiltonian structure. Moreover, we proposed a new approach to invariant reduction of the Bogolubov hierarchy on a suitably chosen correlation function constraint and deduction of the related modified Boltzmann–Bogolubov kinetic equations on a finite set of multi-particle distribution functions. There are also presented results of application of devised methods to describing kinetic properties of a many-particle system with an adsorbent surface, in particular, the corresponding kinetic equation for the occupation density distribution function is derived.

Abstract Twisted current algebras are fixed point subalgebras of current algebras under a finite group action. Special cases include equivariant map algebras and twisted forms of current algebras. Their finite-dimensional simple modules fall into two categories, those which factor through an evaluation map and those which do not. We show that there are no nontrivial extensions between finite-dimensional simple evaluation and non-evaluation modules. We then compute extensions between any pair of finite-dimensional simple modules for twisted current algebras, and use this information to study the block decomposition for the category. In the special case of twisted forms, this decomposition can be described in terms of maps to the fundamental group of the underlying root system. [ABSTRACT FROM AUTHOR]

We study structural properties of truncated Weyl modules. A truncated Weyl module is a local Weyl module for , where is a finite-dimensional simple Lie algebra. It has been conjectured that, if N is sufficiently small with respect to λ, the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when λ is a multiple of certain fundamental weights, including all minuscule ones for simply laced. We also take a further step towards proving the conjecture for all multiples of fundamental weights by proving that the corresponding truncated Weyl module is isomorphic to a natural quotient of a fusion product of Kirillov–Reshetikhin modules. One important part of the proof of the main result shows that any truncated Weyl module is isomorphic to a Chari–Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that related to Demazure flags and chains of inclusions of truncated Weyl modules. [ABSTRACT FROM AUTHOR]

We construct invariant polynomials on truncated multicurrent algebras, which are Lie algebras of the form g ⊗ F F [ t 1 , … , t ℓ ] / I , where g is a finite-dimensional Lie algebra over a field F of characteristic zero, and I is a finite-codimensional ideal of F [ t 1 , … , t ℓ ] generated by monomials. In particular, when g is semisimple and F is algebraically closed, we construct a set of algebraically independent generators for the algebra of invariant polynomials. In addition, we describe a transversal slice to the space of regular orbits in g ⊗ F F [ t 1 , … , t ℓ ] / I . As an application of our main result, we show that the center of the universal enveloping algebra of g ⊗ F F [ t 1 , … , t ℓ ] / I acts trivially on all irreducible finite-dimensional representations provided I has codimension at least two. [ABSTRACT FROM AUTHOR]

Abstract We construct the all loop effective action representing, for small couplings, simultaneously self- and mutually interacting current algebra CFTs realized by WZW models. This non-trivially generalizes our previous works where such interactions were, at the linear level, not simultaneously present. For the two coupling case we prove integrability and calculate the coupled RG flow equations. We also consider non-Abelian T-duality type limits. Our models provide concrete realizations of integrable flows between exact CFTs and exhibit several new features which we discuss in detail. [ABSTRACT FROM AUTHOR]

A presentation à la Faddeev-Reshetikhin-Takhtajan (FRT) of the Onsager, augmented Onsager and sl2-invariant Onsager algebras is given, using the framework of the nonstandard classical Yang-Baxter algebras. Associated current algebras are identified, and generating functions of mutually commuting quantities are obtained. [ABSTRACT FROM AUTHOR]

Let \(\mathbf{H}\) be the quaternion algebra. Let \(\mathfrak{g}\) be a complex Lie algebra and let \(U(\mathfrak{g})\) be the enveloping algebra of \(\mathfrak{g}\). The quaternification \(\mathfrak{g}^{\mathbf{H}}=\)\(\,(\,\mathbf{H}\otimes U(\mathfrak{g}),\,[\quad,\quad]_{\mathfrak{g}^{\mathbf{H}}}\,)\) of \(\mathfrak{g}\) is defined by the bracket \( \big[\,\mathbf{z}\otimes X\,,\,\mathbf{w}\otimes Y\,\big]_{\mathfrak{g}^{\mathbf{H}}}\,=\)\(\,(\mathbf{z}\cdot \mathbf{w})\otimes\,(XY)\,- \)\(\, (\mathbf{w}\cdot\mathbf{z})\otimes (YX)\,,\nonumber \) for \(\mathbf{z},\,\mathbf{w}\in \mathbf{H}\) and {the basis vectors \(X\) and \(Y\) of \(U(\mathfrak{g})\).} Let \(S^3\mathbf{H}\) be the ( non-commutative) algebra of \(\mathbf{H}\)-valued smooth mappings over \(S^3\) and let \(S^3\mathfrak{g}^{\mathbf{H}}=S^3\mathbf{H}\otimes U(\mathfrak{g})\). The Lie algebra structure on \(S^3\mathfrak{g}^{\mathbf{H}}\) is induced naturally from that of \(\mathfrak{g}^{\mathbf{H}}\). We introduce a 2-cocycle on \(S^3\mathfrak{g}^{\mathbf{H}}\) by the aid of a tangential vector field on \(S^3\subset \mathbf{C}^2\) and have the corresponding central extension \(S^3\mathfrak{g}^{\mathbf{H}} \oplus(\mathbf{C}a)\). As a subalgebra of \(S^3\mathbf{H}\) we have the algebra of Laurent polynomial spinors \(\mathbf{C}[\phi^{\pm}]\) spanned by a complete orthogonal system of eigen spinors \(\{\phi^{\pm(m,l,k)}\}_{m,l,k}\) of the tangential Dirac operator on \(S^3\). Then \(\mathbf{C}[\phi^{\pm}]\otimes U(\mathfrak{g})\) is a Lie subalgebra of \(S^3\mathfrak{g}^{\mathbf{H}}\). We have the central extension \(\widehat{\mathfrak{g}}(a)= (\,\mathbf{C}[\phi^{\pm}] \otimes U(\mathfrak{g}) \,) \oplus(\mathbf{C}a)\) as a Lie-subalgebra of \(S^3\mathfrak{g}^{\mathbf{H}} \oplus(\mathbf{C}a)\). Finally we have a Lie algebra \(\widehat{\mathfrak{g}}\) which is obtained by adding to \(\widehat{\mathfrak{g}}(a)\) a derivation \(d\) which acts on \(\widehat{\mathfrak{g}}(a)\) by the Euler vector field \(d_0\). That is the \(\mathbf{C}\)-vector space \(\widehat{\mathfrak{g}}=\left(\mathbf{C}[\phi^{\pm}]\otimes U(\mathfrak{g})\right)\oplus(\mathbf{C}a)\oplus (\mathbf{C}d)\) endowed with the bracket \( \bigl[\,\phi_1\otimes X_1+ \lambda_1 a + \mu_1d\,,\phi_2\otimes X_2 + \lambda_2 a + \mu_2d\,\,\bigr]_{\widehat{\mathfrak{g}}} \, =\)\( (\phi_1\phi_2)\otimes (X_1\,X_2) \, -\,(\phi_2\phi_1)\otimes (X_2X_1)+\mu_1d_0\phi_2\otimes X_2- \) \(\mu_2d_0\phi_1\otimes X_1 + \) \( (X_1\vert X_2)c(\phi_1,\phi_2)a\,. \) When \(\mathfrak{g}\) is a simple Lie algebra with its Cartan subalgebra \(\mathfrak{h}\) we shall investigate the weight space decomposition of \(\widehat{\mathfrak{g}}\) with respect to the subalgebra \(\widehat{\mathfrak{h}}= (\phi^{+(0,0,1)}\otimes \mathfrak{h} )\oplus(\mathbf{C}a) \oplus(\mathbf{C}d)\).

I examine the Generalized Kähler (GK) geometry of classical superconformal WZW model on a compact group and relate the right-moving and left-moving Kac-Moody superalgebra currents to the GK geometry data using biholomorphic gerbe formulation and Hamiltonian formalism. It is shown that the canonical Poisson homogeneous space structure induced by the GK geometry of the group manifold is crucial to provide superconformal -model with the Kac-Moody superalgebra symmetries. Then, the biholomorphic gerbe geometry is used to prove that Kac-Moody superalgebra currents are globally defined. [ABSTRACT FROM AUTHOR]

We study the projective spectrum of the Rees algebra of a module and characterize it by a universal property. As applications, we give descriptions of universal flatifications of modules and of birational projective morphisms. [ABSTRACT FROM AUTHOR]

We introduce the notion of generalized Weyl modules for twisted current algebras. We study their representation-theoretic and combinatorial properties and also their connection with nonsymmetric Macdonald polynomials. As an application, we compute the dimension of the classical Weyl modules in the remaining unknown case. [ABSTRACT FROM AUTHOR]

We explicitly show how the field dependent 2-cocycle that arises in the current algebra of 4 dimensional asymptotically flat spacetimes can be used as a central extension to turn the BMS4 Lie algebra, or more precisely, the BMS4 action Lie algebroid, into a genuine Lie algebroid with field dependent structure functions. Both a BRST formulation, where the extension appears as a ghost number 2 cocyle, and a formulation in terms of vertex operator algebras are introduced. The mapping of the celestial sphere to the cylinder then implies zero mode shifts of the asymptotic part of the shear and of the news tensor. [ABSTRACT FROM AUTHOR]

We present and discuss near horizon boundary conditions for flat space higher-spin gravity in three dimensions. As in related work our boundary conditions ensure regularity of the solutions independently of the charges. The asymptotic symmetry algebra is given by a set of $$ \widehat{\mathfrak{u}}(1) $$ current algebras. The associated charges generate higher-spin soft hair. We derive the entropy for solutions that are continuously connected to flat space cosmologies and find the same result as in the spin-2 case: the entropy is linear in the spin-2 zero-mode charges and independent from the spin-3 charges. Using twisted Sugawara-like constructions of higher-spin currents we show that our simple result for entropy of higherspin flat space cosmologies coincides precisely with the complicated earlier results expressed in terms of higher-spin zero mode charges. [ABSTRACT FROM AUTHOR]

Let g be a finite-dimensional complex simple Lie algebra with highest root θ and let g [ t ] be the corresponding current algebra. In this paper, we consider the g [ t ] -stable Demazure modules associated to integrable highest weight representations of the affine Lie algebra g ˆ . We prove that the fusion product of Demazure modules of a given level with a single Demazure module of a different level and with highest weight a multiple of θ is a generalized Demazure module, and also give defining relations. This also shows that the fusion product of such Demazure modules is independent of the chosen parameters. As a consequence we obtain generators and relations for certain types of generalized Demazure modules. We also establish a connection with the modules defined by Chari and Venkatesh. [ABSTRACT FROM AUTHOR]

In this article, we formalize in the Mizar system [3] the notion of the derivative of polynomials over the field of real numbers [4]. To define it, we use the derivative of functions between reals and reals [9]. [ABSTRACT FROM AUTHOR]

Quantum M-theory is formulated using the current algebra technique. The current algebra is based on a Kac-Moody algebra rather than usual finite dimensional Lie algebra. Specifically, I study the Kac-Moody algebra that was shown recently to contain all the ingredients of M-theory. Both the internal symmetry and the external Lorentz symmetry can be realized inside , so that, by constructing the current algebra of , I obtain both internal gauge theory and gravity theory. The energy-momentum tensor is constructed as the bilinear form of the currents, yielding a system of quantum equations of motion of the currents/fields. Supersymmetry is incorporated in a natural way. The so-called 'field-current identity' is built in and, for example, the gravitino field is itself a conserved supercurrent. One unanticipated outcome is that the quantum gravity equation is not identical to the one obtained from the Einstein-Hilbert action. [ABSTRACT FROM AUTHOR]

We identify the trace, or zeroth Hochschild homology, of categorified quantum groups of ADE type with the corresponding current algebra of the same type. To prove this, we show that 2-representations defined using categories of modules over cyclotomic (or deformed cyclotomic) quotients of Khovanov-Lauda-Rouquier algebras correspond to local (or global) Weyl modules. We also investigate the implications for centers of categories in 2-representations of categorified quantum groups. [ABSTRACT FROM AUTHOR]

Given a finite-dimensional module V for a finite-dimensional, complex semi-simple Lie algebra $\mathcal {g}$ , and a positive integer m, we construct a family of graded modules for the current algebra $\mathcal {g}[t]$ indexed by simple C $\mathcal {S}_{m}$ -modules. These modules are free of finite rank for the ring of symmetric polynomials and so can be localized to give finite-dimensional graded $\mathcal {g}[t]$ -modules. We determine the graded characters of these modules and show that these graded characters admit a curious duality. [ABSTRACT FROM AUTHOR]

We compute the all-loop anomalous dimensions of current and primary field operators in deformed current algebra theories based on a general semi-simple group, but with different (large) levels for the left and right sectors. These theories, unlike their equal level counterparts, possess a new non-trivial fixed point in the IR. By computing the exact in λ two- and three-point functions for these operators we deduce their OPEs and their equal-time commutators. Using these we argue on the nature of the CFT at the IR fixed point. The associated to the currents Poisson brackets are a two-parameter deformation of the canonical structure of the isotropic PCM. [ABSTRACT FROM AUTHOR]

Consequences of QCD current algebra formulated on a light-like hyperplane are derived for the forward scattering of vector and axial–vector currents on an arbitrary hadronic target. It is shown that current algebra gives rise to a special class of sum rules that are direct consequences of the independent chiral symmetry that exists at every point on the two-dimensional transverse plane orthogonal to the lightlike direction. These sum rules are obtained by exploiting the closed, infinite-dimensional algebra satisfied by the transverse moments of null-plane axial–vector and vector charge distributions. In the special case of a nucleon target, this procedure leads to the Adler–Weisberger, Gerasimov–Drell–Hearn, Cabibbo–Radicati and Fubini–Furlan–Rossetti sum rules. Matching to the dispersion-theoretic language which is usually invoked in deriving these sum rules, the moment sum rules are shown to be equivalent to algebraic constraints on forward S -matrix elements in the Regge limit. [ABSTRACT FROM AUTHOR]

The transconductance of current differencing transconductance amplifier (CDTA) has been enhanced by combining the two methods. The first method is reported recently in Rai and Gupta (2015) while the second method is a novel one. These methods can also be merged together to enhance the transconductance of other analog building blocks (ABBs). In the first method, MOSFETs used in the differential pair of transconductance amplifier (TA) has been replaced by the networks N1 and N2 which contains the ‘n’ symmetrical MOSFETs in parallel; where n is the number of MOSFETs. Thus, the voltage which was applied earlier to the differential pairs has now been applied to these networks (N1 and N2) which inject equal currents in all the MOSFETs used. The total current (I T ) coming out of the networks (N1 and N2) will be the sum of currents drawn by all the MOSFETs used in these networks. In the second method, the total current (I T ) copied by various current mirrors are added at the output node (X+ and X−) of CDTA in order to enhance the transconductance effectively. The advantage of this method is that the transconductance has been boosted while the power dissipation of CDTA remains unaffected for a particular biasing current. [ABSTRACT FROM AUTHOR]

Let p(t) ∈ ℂ[t] be a polynomial with distinct roots and nonzero constant term. We describe, using Faà de Bruno's formula and Bell polynomials, the universal central extension in terms of generators and relations for the hyperelliptic current Lie algebras g ⊗ R whose coordinate ring is of the form R = ℂ[t, t-1, u ∣ u² = p(t)]. [ABSTRACT FROM AUTHOR]

We construct a family of vertex algebras associated to the affine Lie algebra of polynomial current algebras of finite-dimensional abelian Lie algebras, along with their modules and logarithmic modules. These vertex algebras and their (logarithmic) modules are strongly -graded and quasi-conformal. We then show that matrix elements of products and iterates of logarithmic intertwining operators among these logarithmic modules satisfy certain systems of differential equations. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory developed by Huang, Lepowsky and Zhang. [ABSTRACT FROM AUTHOR]

Let H be the quaternion algebra. Let g be a complex Lie algebra and let U(g) be the enveloping algebra of g. The quaternification gH = (H⊗U(g); [ , ]gH ) of g is defined by the bracket [ z⊗X ; w⊗Y ] gH = (z . w)⊗(XY ) - (w . z)⊗(Y X) ; for z; w 2 H and the basis vectors X and Y of U(g). Let S3H be the ( non-commutative) algebra of H-valued smooth mappings over S3 and let S3gH = S3H⊗U(g). The Lie algebra structure on S3gH is induced naturally from that of gH. We introduce a 2-cocycle on S3gH by the aid of a tangential vector field on S3 ⊂ C2 and have the corresponding central extension S3gH⊕(Ca). As a subalgebra of S3H we have the algebra of Laurent polynomial spinors C[ϕ±] spanned by a complete orthogonal system of eigen spinors fϕ±(m;l;k)gm;l;k of the tangential Dirac operator on S3. ThenC[ϕ±] U(g) is a Lie subalgebra of S3gH. We have the central extensionbg(a) = (C[ϕ±]⊗U(g) ) ⊕ (Ca) as a Lie-subalgebra of S3gH ⊕ (Ca). Finally we have a Lie algebrabg which is obtained by adding tobg(a) a derivation d which acts onbg(a) by the Euler vector field d0. That is the C-vector spacebg = (C[ϕ±]⊗U(g)) ⊕ (Ca)⊕(Cd) endowed with the bracket [ ϕ1 X1 +λ1a+μ1d ; ϕ2 X2 +λ2a+μ2d ] bg = (ϕ1ϕ2) (X1 X2) - (ϕ2ϕ1) (X2X1)+μ1d0ϕ2 X2-μ2d0ϕ1 X1+(X1∣X2)c(ϕ1; ϕ2)a : When g is a simple Lie algebra with its Cartan subalgebra h we shall investigate the weight space decomposition ofbg with respect to the subalgebra h = (ϕ+(0;0;1) ⊗h)⊕(Ca)⊕(Cd). [ABSTRACT FROM AUTHOR]

Let H be the quaternion algebra. Let g be a complex Lie algebra and let U(g) be the enveloping algebra of g. The quaternification gH = (H⊗U(g); [ , ]gH ) of g is defined by the bracket [ z⊗X ; w⊗Y ] gH = (z . w)⊗(XY ) - (w . z)⊗(Y X) ; for z; w 2 H and the basis vectors X and Y of U(g). Let S3H be the ( non-commutative) algebra of H-valued smooth mappings over S3 and let S3gH = S3H⊗U(g). The Lie algebra structure on S3gH is induced naturally from that of gH. We introduce a 2-cocycle on S3gH by the aid of a tangential vector field on S3 ⊂ C2 and have the corresponding central extension S3gH⊕(Ca). As a subalgebra of S3H we have the algebra of Laurent polynomial spinors C[ϕ±] spanned by a complete orthogonal system of eigen spinors fϕ±(m;l;k)gm;l;k of the tangential Dirac operator on S3. ThenC[ϕ±] U(g) is a Lie subalgebra of S3gH. We have the central extensionbg(a) = (C[ϕ±]⊗U(g) ) ⊕ (Ca) as a Lie-subalgebra of S3gH ⊕ (Ca). Finally we have a Lie algebrabg which is obtained by adding tobg(a) a derivation d which acts onbg(a) by the Euler vector field d0. That is the C-vector spacebg = (C[ϕ±]⊗U(g)) ⊕ (Ca)⊕(Cd) endowed with the bracket [ ϕ1 X1 +λ1a+μ1d ; ϕ2 X2 +λ2a+μ2d ] bg = (ϕ1ϕ2) (X1 X2) - (ϕ2ϕ1) (X2X1)+μ1d0ϕ2 X2-μ2d0ϕ1 X1+(X1∣X2)c(ϕ1; ϕ2)a : When g is a simple Lie algebra with its Cartan subalgebra h we shall investigate the weight space decomposition ofbg with respect to the subalgebra h = (ϕ+(0;0;1) ⊗h)⊕(Ca)⊕(Cd). [ABSTRACT FROM AUTHOR]

In this paper we study the category of graded modules for the current algebra associated to sl 2 . The category enjoys many nice properties, including a tilting theory which was established in [1] . We show that the indecomposable tilting modules for sl 2 [ t ] are the exterior powers of the fundamental global Weyl module and give the filtration multiplicities in the standard and costandard filtration. An interesting consequence of our result (which is far from obvious from the abstract definition) is that an indecomposable tilting module admits a free right action of the ring of symmetric polynomials in finitely many variables. Moreover, if we go modulo the augmentation ideal in this ring, the resulting sl 2 [ t ] -module is isomorphic to the dual of a local Weyl module. [ABSTRACT FROM AUTHOR]

For any finite-dimensional simple Lie algebra g and finitely generated commutative associative algebra S , we give a complete classification of the simple weight modules of g ⊗ S with bounded weight multiplicities. [ABSTRACT FROM AUTHOR]

An affine Kac-Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac-Moody algebras give it for two-dimensional conformal field theory. [ABSTRACT FROM AUTHOR]

An affine Kac-Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac-Moody algebras give it for two-dimensional conformal field theory. [ABSTRACT FROM AUTHOR]