Calculus of Finite Differences - eDisk

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– p. 6/?Down-up AlgebrasDown-up algebras were introduced by Benkart andRoby (1998); generalized by Cassidy and Shelton(2004).They encompass many previously studied algebras:• The enveloping algebra of sl(2);• S. P. Smith’s “algebras similar to U(sl(2))”;• Witten’s deformation of U(sl(2));• Conformal sl(2) enveloping algebras;• The enveloping algebra of the Heisenbergalgebra;
– p. 7/?Definition of down-up algebrasUnfortunately these algebras are defined by generatorsand relations. We use C as our ground field. Fix apolynomial f ∈ C[x] and scalars r,s,γ ∈ C. Thedown-up algebra L = L(f,r,s,γ) is generated byelements d, u, and h, subject to the relations:hu − ruh = γudh − rhd = γddu − sud = f(h).Think of d as a down operator, u as an up operator,and h as a horizontal or sideways operator.
Page 1 and 2: - p. 1/?Calculus of Finite Differen
Page 3: - p. 2/?There are many similarities
Page 6 and 7: - p. 3/?ThenThis should look famili
Page 8 and 9: - p. 4/?The calculus of finite diff
Page 10 and 11: Twenty years later. . .- p. 5/?
Page 12 and 13: - p. 6/?Down-up AlgebrasDown-up alg
Page 14 and 15: - p. 6/?Down-up AlgebrasDown-up alg
Page 18 and 19: - p. 8/?Research aim: determine the
Page 20 and 21: - p. 9/?Side remark: the theory of
Page 22 and 23: - p. 10/?Let’s take the case r =
Page 24 and 25: Annihilators of simple Verma module
Page 26 and 27: - p. 12/?First step: find a central
Page 28 and 29: - p. 13/?Say ud − p(h) commutes w
Page 30 and 31: - p. 15/?The polynomial p is more u
Page 32 and 33: - p. 16/?Here are results about the

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