Central extensions of the algebra of formal pseudo-differential symbols via Hochschild (co)homology and quadratic symplectic Lie algebras

J. Beltran, M. Farinati, E.G. Reyes

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Abstract

We describe the space of central extensions of the associative algebra Ψn of formal pseudo-differential symbols in n≥1 independent variables using Hochschild (co)homology groups: we prove that the first Hochschild (co)homology group HH1(Ψn) is 2n-dimensional and we use this fact to calculate the first Lie (co)homology group HLie 1(Ψn) of Ψn equipped with the Lie bracket induced by its associative algebra structure. As an application, we use our calculations to provide examples of infinite-dimensional quadratic symplectic Lie algebras. © 2017 Elsevier B.V.
LanguageEnglish
Pages2006-2021
Number of pages16
JournalJournal of Pure and Applied Algebra
Volume222
Issue number8
DOIs
Publication statusPublished - 2018

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title = "Central extensions of the algebra of formal pseudo-differential symbols via Hochschild (co)homology and quadratic symplectic Lie algebras",
abstract = "We describe the space of central extensions of the associative algebra Ψn of formal pseudo-differential symbols in n≥1 independent variables using Hochschild (co)homology groups: we prove that the first Hochschild (co)homology group HH1(Ψn) is 2n-dimensional and we use this fact to calculate the first Lie (co)homology group HLie 1(Ψn) of Ψn equipped with the Lie bracket induced by its associative algebra structure. As an application, we use our calculations to provide examples of infinite-dimensional quadratic symplectic Lie algebras. {\circledC} 2017 Elsevier B.V.",
author = "J. Beltran and M. Farinati and E.G. Reyes",
note = "Export Date: 12 April 2018 CODEN: JPAAA References: Alder, M., On trace functional for formal pseudodifferential operators and the symplectic structure of the Korteweg de Vries type equations (1978) Invent. Math., 50 (3), pp. 219-248; Bajo, I., Benayadi, S., Medina, A., Symplectic structures on quadratic Lie algebras (2007) J. Algebra, 316 (1), pp. 174-188; Beltran, J., Reyes, E.G., Formal pseudodifferential operators in one and several variables, central extensions and integrable systems (2015) Adv. Math. Phys., 2015. , 16 pages; Brylinski, J.-L., Getzler, E., The homology of algebras of pseudo-differential symbols and the non-commutative residue (1987) K-Theory, 1, pp. 385-403; Coutinho, S.C., A Primer of Algebraic D-Modules (1995) Lond. Math. Soc. Stud. Texts, 33. , Cambridge University Press; Dickey, L.A., Soliton Equations and Hamiltonian Systems (2003) Adv. Ser. Math. Phys., 26. , second edition World Scientific Publishing Co., Inc. River Edge, NJ; Dzhumadil'daev, A.S., Differentiations and central extensions of the lie algebra of formal pseudodifferential operators (1994) Algebra Anal., 6 (1), pp. 140-158; Fuks, D.B., Cohomology of Infinite-Dimensional Lie Algebras (1986) Contemp. Sov. Math., , translated from the Russian by A.B. Sosinskii Consultants Bureau New York; Gel'fand, I.M., Dikii, L.A., Asymptotic properties of the resolvent of Sturm–Liouville equations and the algebra of Korteweg–de Vries equations (1975) Russ. Math. Surv., 30 (5), pp. 77-113; Guieu, L., Roger, C., L'Alg{\`e}bre et le groupe de Virasoro. Aspects g{\'e}om{\'e}triques et alg{\'e}briques, g{\'e}n{\'e}ralisations (2007), Les Publications CRM Montreal; Khesin, B., Wendt, R., The Geometry of Infinite-Dimensional Groups (2009) Ergeb. Math. Grenzgeb., 51. , Springer-Verlag Berlin; Lam, T.Y., A First Course in Non-Commutative Rings (1991), Springer-Verlag; Lesch, M., On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols (1999) Ann. Glob. Anal. Geom., 17, pp. 151-187; Lesch, M., Neira Jim{\'e}nez, C., Classification of traces and hypertraces on spaces of classical pseudodifferential operators (2013) J. Noncommut. Geom., 7, pp. 457-498; Manin, Y.I., Algebraic aspects of nonlinear differential equations (1979) J. Sov. Math., 11, pp. 1-122; Majewski, M., Rational Homotopical Models and Uniqueness (2000) Mem. Am. Math. Soc., vol. MEMO/143/682; Mulase, M., Solvability of the super KP equation and a generalization of the Birkhoff decomposition (1988) Invent. Math., 92, pp. 1-46; Olver, P.J., Sanders, J., Wang, J.P., Classification of symmetry-integrable evolution equations (2001) B{\"a}cklund and Darboux Transformations. The Geometry of Solitons, CRM Proc. Lecture Notes, 29, pp. 363-372. , A. Coley D. Levi R. Milson C. Rogers P. Winternitz; Parshin, A.N., On a ring of formal pseudo-differential operators (1999) Proc. Steklov Inst. Math., Tr. Mat. Inst. Steklova, 224 (224), pp. 266-280. , (in Russian), Algebra. Topol. Differ. Uravn. i ikh Prilozh., 291–305; translation in; Radul, A.O., Lie algebras of differential operators, their central extensions, and W-algebras (1991) Funct. Anal. Appl., 25, pp. 25-39; Reyman, A.G., Semenov-Tian-Shansky, M.A., Algebras of flows and nonlinear partial differential equations (1980) Dokl. Akad. Nauk SSSR, 251 (6), pp. 1310-1314. , (in Russian); Scott, S., Traces and Determinants of Pseudodifferential Operators (2010), Oxford Science Publications; Van den Bergh, M., A relation between Hochschild homology and cohomology for Gorenstein rings (1998) Proc. Am. Math. Soc., 126 (5), pp. 1345-1348. , Erratum; Weibel, C., An Introduction to Homological Algebra (1994) Camb. Stud. Adv. Math., 38. , Cambridge University Press Cambridge; Wodzicki, M., Noncommutative Residue. I. Fundamentals, K-Theory, Arithmetic and Geometry, Moscow, 1984–1986 (1987) Lect. Notes Math., 1298, pp. 320-399. , Springer Berlin",
year = "2018",
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TY - JOUR

T1 - Central extensions of the algebra of formal pseudo-differential symbols via Hochschild (co)homology and quadratic symplectic Lie algebras

AU - Beltran, J.

AU - Farinati, M.

AU - Reyes, E.G.

N1 - Export Date: 12 April 2018 CODEN: JPAAA References: Alder, M., On trace functional for formal pseudodifferential operators and the symplectic structure of the Korteweg de Vries type equations (1978) Invent. Math., 50 (3), pp. 219-248; Bajo, I., Benayadi, S., Medina, A., Symplectic structures on quadratic Lie algebras (2007) J. Algebra, 316 (1), pp. 174-188; Beltran, J., Reyes, E.G., Formal pseudodifferential operators in one and several variables, central extensions and integrable systems (2015) Adv. Math. Phys., 2015. , 16 pages; Brylinski, J.-L., Getzler, E., The homology of algebras of pseudo-differential symbols and the non-commutative residue (1987) K-Theory, 1, pp. 385-403; Coutinho, S.C., A Primer of Algebraic D-Modules (1995) Lond. Math. Soc. Stud. Texts, 33. , Cambridge University Press; Dickey, L.A., Soliton Equations and Hamiltonian Systems (2003) Adv. Ser. Math. Phys., 26. , second edition World Scientific Publishing Co., Inc. River Edge, NJ; Dzhumadil'daev, A.S., Differentiations and central extensions of the lie algebra of formal pseudodifferential operators (1994) Algebra Anal., 6 (1), pp. 140-158; Fuks, D.B., Cohomology of Infinite-Dimensional Lie Algebras (1986) Contemp. Sov. Math., , translated from the Russian by A.B. Sosinskii Consultants Bureau New York; Gel'fand, I.M., Dikii, L.A., Asymptotic properties of the resolvent of Sturm–Liouville equations and the algebra of Korteweg–de Vries equations (1975) Russ. Math. Surv., 30 (5), pp. 77-113; Guieu, L., Roger, C., L'Algèbre et le groupe de Virasoro. Aspects géométriques et algébriques, généralisations (2007), Les Publications CRM Montreal; Khesin, B., Wendt, R., The Geometry of Infinite-Dimensional Groups (2009) Ergeb. Math. Grenzgeb., 51. , Springer-Verlag Berlin; Lam, T.Y., A First Course in Non-Commutative Rings (1991), Springer-Verlag; Lesch, M., On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols (1999) Ann. Glob. Anal. Geom., 17, pp. 151-187; Lesch, M., Neira Jiménez, C., Classification of traces and hypertraces on spaces of classical pseudodifferential operators (2013) J. Noncommut. Geom., 7, pp. 457-498; Manin, Y.I., Algebraic aspects of nonlinear differential equations (1979) J. Sov. Math., 11, pp. 1-122; Majewski, M., Rational Homotopical Models and Uniqueness (2000) Mem. Am. Math. Soc., vol. MEMO/143/682; Mulase, M., Solvability of the super KP equation and a generalization of the Birkhoff decomposition (1988) Invent. Math., 92, pp. 1-46; Olver, P.J., Sanders, J., Wang, J.P., Classification of symmetry-integrable evolution equations (2001) Bäcklund and Darboux Transformations. The Geometry of Solitons, CRM Proc. Lecture Notes, 29, pp. 363-372. , A. Coley D. Levi R. Milson C. Rogers P. Winternitz; Parshin, A.N., On a ring of formal pseudo-differential operators (1999) Proc. Steklov Inst. Math., Tr. Mat. Inst. Steklova, 224 (224), pp. 266-280. , (in Russian), Algebra. Topol. Differ. Uravn. i ikh Prilozh., 291–305; translation in; Radul, A.O., Lie algebras of differential operators, their central extensions, and W-algebras (1991) Funct. Anal. Appl., 25, pp. 25-39; Reyman, A.G., Semenov-Tian-Shansky, M.A., Algebras of flows and nonlinear partial differential equations (1980) Dokl. Akad. Nauk SSSR, 251 (6), pp. 1310-1314. , (in Russian); Scott, S., Traces and Determinants of Pseudodifferential Operators (2010), Oxford Science Publications; Van den Bergh, M., A relation between Hochschild homology and cohomology for Gorenstein rings (1998) Proc. Am. Math. Soc., 126 (5), pp. 1345-1348. , Erratum; Weibel, C., An Introduction to Homological Algebra (1994) Camb. Stud. Adv. Math., 38. , Cambridge University Press Cambridge; Wodzicki, M., Noncommutative Residue. I. Fundamentals, K-Theory, Arithmetic and Geometry, Moscow, 1984–1986 (1987) Lect. Notes Math., 1298, pp. 320-399. , Springer Berlin

PY - 2018

Y1 - 2018

N2 - We describe the space of central extensions of the associative algebra Ψn of formal pseudo-differential symbols in n≥1 independent variables using Hochschild (co)homology groups: we prove that the first Hochschild (co)homology group HH1(Ψn) is 2n-dimensional and we use this fact to calculate the first Lie (co)homology group HLie 1(Ψn) of Ψn equipped with the Lie bracket induced by its associative algebra structure. As an application, we use our calculations to provide examples of infinite-dimensional quadratic symplectic Lie algebras. © 2017 Elsevier B.V.

AB - We describe the space of central extensions of the associative algebra Ψn of formal pseudo-differential symbols in n≥1 independent variables using Hochschild (co)homology groups: we prove that the first Hochschild (co)homology group HH1(Ψn) is 2n-dimensional and we use this fact to calculate the first Lie (co)homology group HLie 1(Ψn) of Ψn equipped with the Lie bracket induced by its associative algebra structure. As an application, we use our calculations to provide examples of infinite-dimensional quadratic symplectic Lie algebras. © 2017 Elsevier B.V.

U2 - 10.1016/j.jpaa.2017.08.017

DO - 10.1016/j.jpaa.2017.08.017

M3 - Article

VL - 222

SP - 2006

EP - 2021

JO - Journal of Pure and Applied Algebra

T2 - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 8

ER -