PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2014 | 12 | 1 | 57-78
Tytuł artykułu

Left-right noncommutative Poisson algebras

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The notions of left-right noncommutative Poisson algebra (NPlr-algebra) and left-right algebra with bracket AWBlr are introduced. These algebras are special cases of NLP-algebras and algebras with bracket AWB, respectively, studied earlier. An NPlr-algebra is a noncommutative analogue of the classical Poisson algebra. Properties of these new algebras are studied. In the categories AWBlr and NPlr-algebras the notions of actions, representations, centers, actors and crossed modules are described as special cases of the corresponding wellknown notions in categories of groups with operations. The cohomologies of NPlr-algebras and AWBlr (resp. of NPr-algebras and AWBr) are defined and the relations between them and the Hochschild, Quillen and Leibniz cohomologies are detected. The cases P is a free AWBr, the Hochschild or/and Leibniz cohomological dimension of P is ≤ n are considered separately, exhibiting interesting possibilities of representations of the new cohomologies by the well-known ones and relations between the corresponding cohomological dimensions.
Wydawca
Czasopismo
Rocznik
Tom
12
Numer
1
Strony
57-78
Opis fizyczny
Daty
wydano
2014-01-01
online
2013-10-30
Twórcy
autor
  • Andrea Razmadze Mathematical Institute at the Ivane Javakhishvili Tbilisi State University, tamar@rmi.ge
autor
Bibliografia
  • [1] Borceux F., Janelidze G., Kelly G.M., Internal object actions, Comment. Math. Univ. Carolin., 2005, 46(2), 235–255
  • [2] Borceux F., Janelidze G., Kelly G.M., On the representability of actions in a semi-abelian category, Theory Appl. Categ., 2005, 14(11), 244–286
  • [3] Bourn D., Janelidze G., Centralizers in action accessible categories, Cah. Topol. Géom. Différ. Catég., 2009, 50(3), 211–232
  • [4] Casas J.M., Datuashvili T., Noncommutative Leibniz-Poisson algebras, Comm. Algebra, 2006, 34(7), 2507–2530 http://dx.doi.org/10.1080/00927870600651091
  • [5] Casas J.M., Datuashvili T., Ladra M., Actor of a precrossed module, Comm. Algebra, 2009, 37(12), 4516–4541 http://dx.doi.org/10.1080/00927870902829130
  • [6] Casas J.M., Datuashvili T., Ladra M., Universal strict general actors and actors in categories of interest, Appl. Categ. Structures, 2010, 18(1), 85–114 http://dx.doi.org/10.1007/s10485-008-9166-z
  • [7] Casas J.M., Datuashvili T., Ladra M., Actor of an alternative algebra, preprint available at http://arxiv.org/abs/0910.0550v1
  • [8] Casas J.M., Datuashvili T., Ladra M., Actors in categories of interest, preprint available at http://arxiv.org/abs/math0702574v2
  • [9] Casas J.M., Pirashvili T., Algebras with bracket, Manuscripta Math., 2006, 119(1), 1–15 http://dx.doi.org/10.1007/s00229-005-0551-8
  • [10] Cornish W.H., Amalgamating commutative regular rings, Comment. Math. Univ. Carolin., 1977, 18(3), 423–436
  • [11] Datuashvili T., Cohomologically trivial internal categories in categories of groups with operations, Appl. Categ. Structures, 1995, 3(3), 221–237 http://dx.doi.org/10.1007/BF00878442
  • [12] Dotsenko V., Khoroshkin A., Gröbner bases for operads, Duke Math. J., 2010, 153(2), 363–396 http://dx.doi.org/10.1215/00127094-2010-026
  • [13] Fresse B., Homologie de Quillen pour les algèbres de Poisson, C. R. Acad. Sci. Paris Sér. I Math., 1998, 326(9), 1053–1058 http://dx.doi.org/10.1016/S0764-4442(98)80061-X
  • [14] Fresse B., Théorie des opérades de Koszul et homologie des algèbres de Poisson, Ann. Math. Blaise Pascal, 2006, 13(2), 237–312 http://dx.doi.org/10.5802/ambp.219
  • [15] Higgins P.J., Groups with multiple operators, Proc. London Math. Soc., 1956, 6(3), 366–416 http://dx.doi.org/10.1112/plms/s3-6.3.366
  • [16] Hochschild G., Cohomology and representations of associative algebras, Duke Math. J., 1947, 14(4), 921–948 http://dx.doi.org/10.1215/S0012-7094-47-01473-7
  • [17] Hoffbeck E., Poincaré-Birkhoff-Witt criterion for Koszul operads, Manuscripta Math., 2010, 131(1–2), 87–110 http://dx.doi.org/10.1007/s00229-009-0303-2
  • [18] Huebschmann J., Poisson cohomology and quantization, J. Reine Angew. Math., 1990, 408, 57–113
  • [19] Kanatchikov I.V., On field-theoretic generalizations of a Poisson algebra, Rep. Math. Phys., 1997, 40(2), 225–234 http://dx.doi.org/10.1016/S0034-4877(97)85919-8
  • [20] Kubo F., Finite-dimensional non-commutative Poisson algebras, J. Pure Appl. Algebra, 1996, 113(3), 307–314 http://dx.doi.org/10.1016/0022-4049(95)00151-4
  • [21] Kubo F., Non-commutative Poisson algebra structures on affine Kac-Moody algebras, J. Pure Appl. Algebra, 1998, 126(1–3), 267–286 http://dx.doi.org/10.1016/S0022-4049(96)00141-7
  • [22] Loday J.-L., Cyclic Homology, Grundlehren Math. Wiss., 301, Springer, Berlin, 1992
  • [23] Loday J.-L., Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math., 1993, 39(3–4), 269–293
  • [24] Loday J.-L., Algèbres ayant deux opérations associatives (digèbres), C. R. Acad. Sci. Paris Sér. I Math., 1995, 321(2), 141–146
  • [25] Loday J.-L., Dialgebras, In: Dialgebras and Related Operads, Lecture Notes in Math., 1763, Springer, Berlin, 2001, 7–66 http://dx.doi.org/10.1007/3-540-45328-8_2
  • [26] Loday J.-L., Pirashvili T., Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann., 1993, 296(1), 139–158 http://dx.doi.org/10.1007/BF01445099
  • [27] Loday J.-L., Ronco M., Trialgebras and families of polytopes, In: Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-theory, Contemp. Math., 346, American Mathematical Society, Providence, 2004, 369–398 http://dx.doi.org/10.1090/conm/346/06296
  • [28] Loday J.-L., Vallette B., Algebraic Operads, Grundlehren Math. Wiss., 346, Springer, Heidelberg, 2012 http://dx.doi.org/10.1007/978-3-642-30362-3
  • [29] Montoli A., Action accessibility for categories of interest, Theory Appl. Categ., 2010, 23(1), 7–21
  • [30] Orzech G., Obstruction theory in algebraic categories, I, II, J. Pure Appl. Algebra, 1972, 2(4), 287–340 http://dx.doi.org/10.1016/0022-4049(72)90008-4
  • [31] Porter T., Extensions, crossed modules and internal categories in categories of groups with operations, Proc. Edinburgh Math. Soc., 1987, 30(3), 373–381 http://dx.doi.org/10.1017/S0013091500026766
  • [32] Quillen D., On the (co-)homology of commutative rings, In: Applications of Categorical Algebra, New York, 1968, American Mathematical Society, Providence, 1970, 65–87
  • [33] Tong J., Jin Q., Non-commutative Poisson algebra structures on the Lie algebra \(so_n \left( {\widetilde{C_\mathbb{Q} }} \right)\) , Algebra Colloq., 2007, 14(3), 521–536
  • [34] Xu P., Noncommutative Poisson algebras, Amer. J. Math., 1994, 116(1), 101–125 http://dx.doi.org/10.2307/2374983
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-013-0321-x
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.