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A I Molev - One of the best experts on this subject based on the ideXlab platform.
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casimir elements and Sugawara operators for takiff algebras
Journal of Mathematical Physics, 2021Co-Authors: A I MolevAbstract:For every simple Lie algebra g, we consider the associated Takiff algebra gl defined as the truncated polynomial current Lie algebra with coefficients in g. We use a matrix presentation of gl to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra U(gl). A similar matrix presentation for the affine Kac–Moody algebra gl is then used to prove an analog of the Feigin–Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. The proof relies on an explicit construction of a complete set of Segal–Sugawara vectors for the Lie algebra gl.
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casimir elements and Sugawara operators for takiff algebras
arXiv: Representation Theory, 2020Co-Authors: A I MolevAbstract:For every simple Lie algebra $\mathfrak{g}$ we consider the associated Takiff algebra $\mathfrak{g}^{}_{\ell}$ defined as the truncated polynomial current Lie algebra with coefficients in $\mathfrak{g}$. We use a matrix presentation of $\mathfrak{g}^{}_{\ell}$ to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra ${\rm U}(\mathfrak{g}^{}_{\ell})$. A similar matrix presentation for the affine Kac--Moody algebra $\widehat{\mathfrak{g}}^{}_{\ell}$ is then used to prove an analogue of the Feigin--Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. The proof relies on an explicit construction of a complete set of Segal--Sugawara vectors for the Lie algebra $\mathfrak{g}^{}_{\ell}$.
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segal Sugawara vectors for the lie algebra of type g2
Journal of Algebra, 2016Co-Authors: A I Molev, E Ragoucy, Natasha RozhkovskayaAbstract:Explicit formulas for Segal-Sugawara vectors associated with the simple Lie algebra $\mathfrak{g}$ of type $G_2$ are found by using computer-assisted calculations. This leads to a direct proof of the Feigin-Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. As an application, we give an explicit solution of Vinberg's quantization problem by providing formulas for generators of maximal commutative subalgebras of $U(\mathfrak{g})$. We also calculate the eigenvalues of the Hamiltonians on the Bethe vectors in the Gaudin model associated with $\mathfrak{g}$.
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pfaffian type Sugawara operators
arXiv: Representation Theory, 2011Co-Authors: A I MolevAbstract:We show that the Pfaffian of a generator matrix for the affine Kac--Moody algebra hat o_{2n} is a Segal--Sugawara vector. Together with our earlier construction involving the symmetrizer in the Brauer algebra, this gives a complete set of Segal--Sugawara vectors in type D.
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the macmahon master theorem for right quantum superalgebras and higher Sugawara operators for widehat mathfrak gl m n
arXiv: Representation Theory, 2009Co-Authors: A I Molev, E RagoucyAbstract:We prove an analogue of the MacMahon Master Theorem for the right quantum superalgebras. In particular, we obtain a new and simple proof of this theorem for the right quantum algebras. In the super case the theorem is then used to construct higher order Sugawara operators for the affine Lie superalgebra $\widehat{\mathfrak{gl}}(m|n)$ in an explicit form. The operators are elements of a completed universal enveloping algebra of $\widehat{\mathfrak{gl}}(m|n)$ at the critical level. They occur as the coefficients in the expansion of a noncommutative Berezinian and as the traces of powers of generator matrices. The same construction yields higher Hamiltonians for the Gaudin model associated with the Lie superalgebra $\mathfrak{gl}(m|n)$. We also use the Sugawara operators to produce algebraically independent generators of the algebra of singular vectors of any generic Verma module at the critical level over the affine Lie superalgebra.
E Ragoucy - One of the best experts on this subject based on the ideXlab platform.
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segal Sugawara vectors for the lie algebra of type g2
Journal of Algebra, 2016Co-Authors: A I Molev, E Ragoucy, Natasha RozhkovskayaAbstract:Explicit formulas for Segal-Sugawara vectors associated with the simple Lie algebra $\mathfrak{g}$ of type $G_2$ are found by using computer-assisted calculations. This leads to a direct proof of the Feigin-Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. As an application, we give an explicit solution of Vinberg's quantization problem by providing formulas for generators of maximal commutative subalgebras of $U(\mathfrak{g})$. We also calculate the eigenvalues of the Hamiltonians on the Bethe vectors in the Gaudin model associated with $\mathfrak{g}$.
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higher Sugawara operators for the quantum affine algebras of type a
Communications in Mathematical Physics, 2016Co-Authors: L Frappat, Naihuan Jing, E RagoucyAbstract:We give explicit formulas for the elements of the center of the completed quantum affine algebra in type A at the critical level that are associated with the fundamental representations. We calculate the images of these elements under a Harish-Chandra-type homomorphism. These images coincide with those in the free field realization of the quantum affine algebra and reproduce generators of the q-deformed classical \({{\mathcal W}}\)-algebra of Frenkel and Reshetikhin.
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the macmahon master theorem for right quantum superalgebras and higher Sugawara operators for widehat mathfrak gl m n
arXiv: Representation Theory, 2009Co-Authors: A I Molev, E RagoucyAbstract:We prove an analogue of the MacMahon Master Theorem for the right quantum superalgebras. In particular, we obtain a new and simple proof of this theorem for the right quantum algebras. In the super case the theorem is then used to construct higher order Sugawara operators for the affine Lie superalgebra $\widehat{\mathfrak{gl}}(m|n)$ in an explicit form. The operators are elements of a completed universal enveloping algebra of $\widehat{\mathfrak{gl}}(m|n)$ at the critical level. They occur as the coefficients in the expansion of a noncommutative Berezinian and as the traces of powers of generator matrices. The same construction yields higher Hamiltonians for the Gaudin model associated with the Lie superalgebra $\mathfrak{gl}(m|n)$. We also use the Sugawara operators to produce algebraically independent generators of the algebra of singular vectors of any generic Verma module at the critical level over the affine Lie superalgebra.
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the macmahon master theorem for right quantum superalgebras and higher Sugawara operators for hat gl m n
arXiv: Representation Theory, 2009Co-Authors: A I Molev, E RagoucyAbstract:We prove an analogue of the MacMahon Master Theorem for the right quantum superalgebras. In particular, we obtain a new and simple proof of this theorem for the right quantum algebras. In the super case the theorem is then used to construct higher order Sugawara operators for the affine Lie superalgebra \hat gl(m|n) in an explicit form. The operators are elements of a completed universal enveloping algebra of \hat gl(m|n) at the critical level. They occur as the coefficients in the expansion of a noncommutative Berezinian and as the traces of powers of generator matrices. The same construction yields higher Hamiltonians for the Gaudin model associated with the Lie superalgebra gl(m|n). We also use the Sugawara operators to produce algebraically independent generators of the algebra of singular vectors of any generic Verma module at the critical level over the affine Lie superalgebra.
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Sugawara and vertex operator constructions for deformed virasoro algebras
Annales Henri Poincaré, 2006Co-Authors: D Arnaudon, L Frappat, E Ragoucy, Jean Avan, Junichi ShiraishiAbstract:From the defining exchange relations of the \(\mathcal{A}_{{q,p}} {\left( {\widehat{{gl}}_{N} } \right)}\) elliptic quantum algebra, we construct subalgebras which can be characterized as q-deformed W N algebras. The consistency conditions relating the parameters p, q, N and the central charge c are shown to be related to the singularity structure of the functional coefficients defining the exchange relations of specific vertex operators representations of \(\mathcal{A}_{{q,p}} {\left( {\widehat{{gl}}_{N} } \right)}\) available when N = 2.
Francesco Toppan - One of the best experts on this subject based on the ideXlab platform.
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an n 8 superaffine malcev algebra and its n 8 Sugawara
Physics Letters A, 2001Co-Authors: H L Carrion, M Rojas, Francesco ToppanAbstract:Abstract A supersymmetric affinization of the algebra of octonions is introduced. It satisfies a super-Malcev property and is N =8 supersymmetric. Its Sugawara construction recovers, in a special limit, the non-associative N =8 superalgebra of Englert et al. This Letter extends to supersymmetry the results obtained by Osipov in the bosonic case.
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an n 8 superaffine malcev algebra and its n 8 Sugawara
arXiv: High Energy Physics - Theory, 2001Co-Authors: H L Carrion, M Rojas, Francesco ToppanAbstract:A supersymmetric affinization of the algebra of octonions is introduced. It satisfies a super-Malcev property and is N=8 supersymmetric. Its Sugawara construction recovers, in a special limit, the non-associative N=8 superalgebra of Englert et al. This paper extends to supersymmetry the results obtained by Osipov in the bosonic case.
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n 4 Sugawara construction on widehat rm sl 2 1 widehat rm sl 3 and mkdv type superhierarchies
Modern Physics Letters A, 1999Co-Authors: E Ivanov, S Krivonos, Francesco ToppanAbstract:The local Sugawara constructions of the "small" N=4 SCA in terms of supercurrents of N=2 extensions of the affine $\widehat{{\rm sl}(2|1)}$ and $\widehat{{\rm sl}(3)}$ algebras are investigated. The associated super mKdV type hierarchies induced by N=4 SKdV ones are defined. In the $\widehat{{\rm sl}(3)}$ case the existence of two nonequivalent Sugawara constructions is found. The "long" one involves all the affine $\widehat{{\rm sl}(3)}$ currents, while the "short" one deals only with those from the subalgebra $\widehat{{\rm sl}(2)\oplus {\rm u}(1)}$. As a consequence, the $\widehat{{\rm sl}(3)}$-valued affine superfields carry two nonequivalent mKdV type super-hierarchies induced by the correspondence between "small" N=4 SCA and N=4 SKdV hierarchy. However, only the first hierarchy possesses genuine global N=4 supersymmetry. We discuss peculiarities of the realization of this N=4 supersymmetry on the affine supercurrents.
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n 4 Sugawara construction on affine sl 2 1 sl 3 and mkdv type superhierarchies
arXiv: Exactly Solvable and Integrable Systems, 1999Co-Authors: E Ivanov, S Krivonos, Francesco ToppanAbstract:The local Sugawara constructions of the "small" N=4 SCA in terms of supercurrents of N=2 extensions of the affinization of the sl(2|1) and sl(3) algebras are investigated. The associated super mKdV type hierarchies induced by N=4 SKdV ones are defined. In the sl(3) case the existence of two inequivalent Sugawara constructions is found. The long one involves all the affine sl(3)-valued currents, while the "short" one deals only with those from the subalgebra sl(2)\oplus u(1). As a consequence, the sl(3)-valued affine superfields carry two inequivalent mKdV type super hierarchies induced by the correspondence between "small" N=4 SCA and N=4 SKdV hierarchy. However, only the first hierarchy posseses genuine global N=4 supersymmetry. We discuss peculiarities of the realization of this N=4 supersymmetry on the affine supercurrents.
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n 4 Sugawara construction on hat sl 2 1 hat sl 3 and mkdv type superhierarchies
arXiv: Exactly Solvable and Integrable Systems, 1999Co-Authors: E Ivanov, S Krivonos, Francesco ToppanAbstract:The local Sugawara constructions of the ``small'' N=4 SCA in terms of supercurrents of N=2 extensions of the affine $\hat{sl(2|1)}$ and $\hat{sl(3)}$ algebras are investigated. The associated super mKdV type hierarchies induced by N=4 SKdV ones are defined. In the $\hat{sl(3)}$ case the existence of two inequivalent Sugawara constructions is found. The ``long'' one involves all the affine $\hat{sl(3)}$ currents, while the ``short'' one deals only with those from the subalgebra $\hat{sl(2)\oplus u(1)}$. As a consequence, the $\hat{sl(3)}$-valued affine superfields carry two inequivalent mKdV type super hierarchies induced by the correspondence between ``small'' N=4 SCA and N=4 SKdV hierarchy. However, only the first hierarchy posseses genuine global N=4 supersymmetry. We discuss peculiarities of the realization of this N=4 supersymmetry on the affine supercurrents.
Hermann Nicolai - One of the best experts on this subject based on the ideXlab platform.
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Sugawara-Type Constraints in Hyperbolic Coset Models
Communications in Mathematical Physics, 2011Co-Authors: Thibault Damour, Axel Kleinschmidt, Hermann NicolaiAbstract:In the conjectured correspondence between supergravity and geodesic models on infinite-dimensional hyperbolic coset spaces, and E-10/K (E-10) in particular, the constraints play a central role. We present a Sugawara-type construction in terms of the E-10 Noether charges that extends these constraints infinitely into the hyperbolic algebra, in contrast to the truncated expressions obtained in Damour et al. (Class. Quant. Grav. 24: 6097, 2007) that involved only finitely many generators. Our extended constraints are associated to an infinite set of roots which are all imaginary, and in fact fill the closed past light-cone of the Lorentzian root lattice. The construction makes crucial use of the E-10 Weyl group and of the fact that the E-10 model contains both D = 11 supergravity and D = 10 IIB supergravity. Our extended constraints appear to unite in a remarkable manner the different canonical constraints of these two theories. This construction may also shed new light on the issue of 'open constraint algebras' in traditional canonical approaches to gravity.
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Sugawara type constraints in hyperbolic coset models
arXiv: High Energy Physics - Theory, 2009Co-Authors: Thibault Damour, Axel Kleinschmidt, Hermann NicolaiAbstract:In the conjectured correspondence between supergravity and geodesic models on infinite-dimensional hyperbolic coset spaces, and E10/K(E10) in particular, the constraints play a central role. We present a Sugawara-type construction in terms of the E10 Noether charges that extends these constraints infinitely into the hyperbolic algebra, in contrast to the truncated expressions obtained in arXiv:0709.2691 that involved only finitely many generators. Our extended constraints are associated to an infinite set of roots which are all imaginary, and in fact fill the closed past light-cone of the Lorentzian root lattice. The construction makes crucial use of the E10 Weyl group and of the fact that the E10 model contains both D=11 supergravity and D=10 IIB supergravity. Our extended constraints appear to unite in a remarkable manner the different canonical constraints of these two theories. This construction may also shed new light on the issue of `open constraint algebras' in traditional canonical approaches to gravity.
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the Sugawara generators at arbitrary level
arXiv: High Energy Physics - Theory, 1996Co-Authors: Reinhold W Gebert, Kilian Koepsell, Hermann NicolaiAbstract:We construct an explicit representation of the Sugawara generators for arbitrary level in terms of the homogeneous Heisenberg subalgebra, which generalizes the well-known expression at level 1. This is achieved by employing a physical vertex operator realization of the affine algebra at arbitrary level, in contrast to the Frenkel--Kac--Segal construction which uses unphysical oscillators and is restricted to level 1. At higher level, the new operators are transcendental functions of DDF ``oscillators'' unlike the quadratic expressions for the level-1 generators. An essential new feature of our construction is the appearance, beyond level 1, of new types of poles in the operator product expansions in addition to the ones at coincident points, which entail (controllable) non-localities in our formulas. We demonstrate the utility of the new formalism by explicitly working out some higher-level examples. Our results have important implications for the problem of constructing explicit representations for higher-level root spaces of hyperbolic Kac--Moody algebras, and $E_{10}$ in particular.
Michael Forger - One of the best experts on this subject based on the ideXlab platform.
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the algebra of the energy momentum tensor and the noether currents in classical non linear sigma models
Communications in Mathematical Physics, 1994Co-Authors: Michael Forger, J Laartz, U SchaperAbstract:The recently derived current algebra of classical non-linear sigma models on arbitrary Riemannian manifolds is extended to include the energy-momentum tensor. It is found that in two dimensions the energy-momentum tensor θμυ, the Noether currentjμ associated with the global symmetry of the theory and the composite fieldj appearing as the coefficient of the Schwinger term in the current algebra, together with the derivatives ofjμ andj, generate a closed algebra. The subalgebra generated by the light-cone components of the energy-momentum tensor consists of two commuting copies of the Virasoro algebra, with central chargec=0, reflecting the classical conformal invariance of the theory, but the current algebra part and the semidirect product structure are quite different from the usual Kac-Moody/Sugawara type construction.
