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Elements of Computer-Algebraic Analysis 2 G Operator algebras, partial classification Operator algebras, partial classification More general framework: G-algebras More general framework: G-algebras Recommended literature (textbooks) 1 J. C. McConnell, J. C. Robson, ”Noncommutative Noetherian Rings”, Graduate Studies in Mathematics, 30, AMS (2001) Elements of Computer-Algebraic Analysis 2 G. R. Krause and T. H. Lenagan, ”Growth of Algebras and Gelfand-Kirillov Dimension”, Graduate Studies in Mathematics, 22, AMS (2000) Viktor Levandovskyy 3 S. Saito, B. Sturmfels and N. Takayama, “Gr¨obner Lehrstuhl D f¨urMathematik, RWTH Aachen, Germany Deformations of Hypergeometric Differential Equations”, Springer, 2000 4 J. Bueso, J. G´omez–Torrecillas and A. Verschoren, 22.07. Tutorial at ISSAC 2012, Grenoble, France ”Algorithmic methods in non-commutative algebra. Applications to quantum groups”, Kluwer , 2003 5 H. Kredel, ”Solvable polynomial rings”, Shaker Verlag, 1993 6 H. Li, ”Noncommutative Gr¨obnerbases and filtered-graded transfer”, Springer, 2002 VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classification Operator algebras, partial classification More general framework: G-algebras More general framework: G-algebras Recommended literature (textbooks and PhD theses) Software D-modules and algebraic analysis: 1 V. Ufnarovski, ”Combinatorial and Asymptotic Methods of Algebra”, Springer, Encyclopedia of Mathematical Sciences 57 kan/sm1 by N. Takayama et al. (1995) D-modules package in Macaulay2 by A. Leykin and H. Tsai 2 F. Chyzak, ”Fonctions holonomes en calcul formel”, Risa/Asir by M. Noro et al. PhD. Thesis, INRIA, 1998 OreModules package suite for Maple by D. Robertz, 3 V. Levandovskyy, ”Non-commutative Computer Algebra for A. Quadrat et al. polynomial algebras: Gr¨obner bases, applications and Singular:Plural with a D-module suite; by V. L. et al. implementation” PhD. Thesis, TU Kaiserslautern, 2005 holonomic and D-finite functions: 4 C. Koutschan, ”Advanced Applications of the Holonomic Groebner, Ore algebra, Mgfun, ... by F. Chyzak Systems Approach”, PhD. Thesis, RISC Linz, 2009 HolonomicFunctions by C. Koutschan 5 K. Schindelar, ”Algorithmic aspects of algebraic system theory”, PhD. Thesis, RWTH Aachen, 2010 Singular:Locapal (partly under development) by V. L. et al. VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classification Operator algebras, partial classification More general framework: G-algebras More general framework: G-algebras Overview What is computer algebraic Analysis? Algebraic Analysis Operator algebras and their partial classification 1 As a notion, it arose in 1958 in the group of Mikio Sato More general: G-algebras and Gr¨obnerbases in G-algebras (Japan) Module theory; Dimension theory; Gel’fand-Kirillov dimension 2 Main objects: systems of linear partial DEs with variable Linear modeling with variable coefficients coefficients, generalized functions Elimination of variables and Gel’fand-Kirillov dimension 3 Main idea: study systems and generalized functions in a Ore localization; smallest Ore localizations coordinate-free way (i. e. by using modules, sheaves, Solutions via homological algebra categories, localizations, homological algebra, . ) The complete annihilator program 4 Keywords include D-Modules, (sub-)holonomic D-Modules, Some computational D-module theory, Weyl closure regular resp. irregular holonomic D-Modules Purity; pure modules, pure functions, preservation of purity 5 Interplay: singularity theory, special functions, . Purity filtration of a module; connection to solutions Other ingredients: symbolic algorithmic methods for discrete resp. continuous problems like symbolic summation, symbolic integration Jacobson normal form etc. VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classification Operator algebras, partial classification More general framework: G-algebras More general framework: G-algebras What is computer-algebraic Analysis? Some important names in computer-algebraic analysis Algebraization as a trend W. Gr¨obnerand B. Buchberger: Gr¨obnerbases and Algebra: Ideas, Concepts, Methods, Abstractions constructive ideal/module theory O. Ore: Ore Extension and Ore Localization Computer algebra works with algebraic concepts in a I. M. Gel’fand and A. Kirillov: GK-Dimension (semi-)algorithmic way at three levels: B. Malgrange: M. isomorphism, M. ideal, . 1 Theory: Methods of Algebra in a constructive way J. Bernstein, M. Sato, M. Kashiwara, C. Sabbah, 2 Algorithmics: Algorithms (or procedures) and their Z. Mebkhout, B. Malgrange et al.: D-module theory Correctness, Termination and Complexity results (if possible) N. Takayama, T. Oaku, B. Sturmfels, M. Saito, M. Granger, 3 Realization: Implementation, Testing, Benchmarking, U. Walther, F. Castro, H. Tsai, A. Leykin et al.: (not only) Challenges; Distribution, Lifecycle, Support and computational D-module theory software-technical aspects ... VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classification Operator algebras Operator algebras, partial classification Operator algebras More general framework: G-algebras Partial classification of operator algebras More general framework: G-algebras Partial classification of operator algebras Operator algebras: partial Classification Let K be an effective field, that is (+, , , :) can be performed − · algorithmically. Moreover, let be a K-vector space (”function space”). F Let x be a local coordinate in . It induces a K-linear map Part I. Operator algebras and their partial classification. F X : , i. e. X (f ) = x f for f . Moreover, let F → F · ∈ F o : be a K-linear map. x F → F Then, in general, o X = X o , that is x ◦ 6 ◦ x o (x f ) = x o (f ) for f . x · 6 · x ∈ F The non-commutative relation between ox and X can be often read off by analyzing the properties of ox like, for instance, the product rule. VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classification Operator algebras Operator algebras, partial classification Operator algebras More general framework: G-algebras Partial classification of operator algebras More general framework: G-algebras Partial classification of operator algebras Classical examples: Weyl algebra Classical examples: shift algebra Let g be a sequence in discrete argument k and s is the shift operator s(g(k)) = g(k + 1). Note, that s is multiplicative. Let f : C C be a differentiable function and ∂(f (x)) := ∂f . → ∂x Thus s(kg(k)) = (k + 1)g(k + 1) = (k + 1)s(g(k)) holds. Product rule tells us that ∂(x f (x)) = x ∂(f (x)) + f (x), what is translated into the following relation between operators The operator algebra, corr. to s is the 1st shift algebra (∂ x x ∂ 1) (f (x)) = 0. S1 = K k, s sk = (k + 1)s = ks + s . ◦ − ◦ − h | i Intermezzo The corresponding operator algebra is the 1st Weyl algebra For a function in differentiable argument x and in discrete argument k the natural operator algebra is D1 = K x, ∂ ∂x = x∂ + 1 . h | i A = D S = K x, k, ∂ , s ∂ x = x∂ + 1, s k = ks + s , 1 ⊗K 1 h x k | x x k k k xk = kx, xs = s x, ∂ k = k∂ , ∂ s = s ∂ . k k x x x k k x i VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classification Operator algebras Operator algebras, partial classification Operator algebras More general framework: G-algebras Partial classification of operator algebras More general framework: G-algebras Partial classification of operator algebras Examples form the q-World Two frameworks for bivariate operator algebras Let k K be fields and q K . ⊂ ∈ ∗ In q-calculus and in quantum algebra three situations are common Algebra with linear (affine) relation for a fixed k: (a) q k, (b) q is a root of unity over k, and Let q K and α, β, γ K. Define ∈ ∗ (c) q is transcendental over k and k(q) K. ∈ ∈ ⊆ (1)(q, α, β, γ) := K x, y yx q xy = αx + βy + γ f (qx) f (x) A h | − · i Let ∂q(f (x)) = (q −1)x be a q-differential operator. x − Because of integration operator (f (x)) := f (t)dt for a R, The corr. operator algebra is the 1st q-Weyl algebra I a ∈ obeying the relation x x = 2 we need yet more general I − I −I R D(q) = K x, ∂ ∂ x = q x∂ + 1 . framework. 1 h q | q · q i Algebra with nonlinear relation (2) The 1st q-shift algebra corresponds to the q-shift operator Let N N and c0,..., cN , α K. Then (q, c0,..., cN , α) is s (f (x)) = f (qx): ∈ n ∈ i A q K x, y yx q xy = i=1 ci y + αx + c0 or h | − · n i i K x, y yx q xy = i=1 ci x + αy + c0 . Kq[x, sq] = K x, sq sqx = q xsq . h | − · P i h | · i P VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classification Operator algebras Operator algebras, partial classification Operator algebras More general framework: G-algebras Partial classification of operator algebras More general framework: G-algebras Partial classification of operator algebras Theorem (L.–Koutschan–Motsak, 2011) Theorem (L.–Makedonsky–Petravchuk, new) (1)(q, α, β, γ) = K x, y yx q xy = αx + βy + γ , A h | − · i (2) where q K ∗ and α, β, γ K For N 2 and c ,..., c , α K, (q, c ,..., c , α) ∈ ∈ ≥ 0 N ∈ A 0 N is isomorphic to the 5 following model algebras: = K x, y yx q xy = N c y i + αx + c is isomorphic to ... h | − · i=1 i 0i (q) 1 K[x, y], 1 K [x, s] or D , if q = 1, q 1 P6 2 the 1st Weyl algebra D1 = K x, ∂ ∂x = x∂ + 1 , 2 S = K x, s sx = xs + s , if q = 1 and α = 0, h | i 1 h | i 6 3 the 1st shift algebra S1 = K x, s sx = xs + s , 3 K x, y yx = xy + f (y) , where f K[y] with deg(f ) = N, if h | i h | i ∈ 4 the 1st q-commutative algebra K [x, s] = K x, s sx = q xs , q = 1 and α = 0.
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