Calculus of Finite Differences - eDisk

– p. 1/?Calculus of Finite DifferencesWe study the difference operator ∆:∆f(x) = f(x + h) − f(x)This is supposed to be the discrete analogue of thedifferential df.

– p. 1/?Calculus of Finite DifferencesWe study the difference operator ∆:∆f(x) = f(x + h) − f(x)This is supposed to be the discrete analogue of thedifferential df.I learned the topic from Samuel Goldberg at Oberlin.He wrote the book Introduction to DifferenceEquations, with illustrative examples from economics,psychology, and sociology.

– p. 2/?There are many similarities between ∆ and the usualcalculus operator d.The similarities are more striking if we set h = 1, andif we definex (n) = x(x − 1)(x − 2) · · · (x − n + 1);the x (n) s form a more natural basis for the space ofpolynomials when working with ∆.

– p. 3/?ThenThis should look familiar.∆x (n) = nx (n−1) .There is also the indefinite summation operator ∆ −1 ,the inverse of ∆, similar to the indefinite integral.

– p. 3/?ThenThis should look familiar.∆x (n) = nx (n−1) .There is also the indefinite summation operator ∆ −1 ,the inverse of ∆, similar to the indefinite integral.We have∆ −1 x (n) = x(n+1)n + 1 + C,where C is a constant (provided we work in C[x].)

The calculus of finite differences stayed with me,although I did not see any more of it.– p. 4/?

– p. 4/?The calculus of finite differences stayed with me,although I did not see any more of it.There were occasional encounters with this calculus,mainly in the context of difference equations (innumber theory or combinatorics).

– p. 4/?The calculus of finite differences stayed with me,although I did not see any more of it.There were occasional encounters with this calculus,mainly in the context of difference equations (innumber theory or combinatorics).But I did not encounter it in my work. I never had todiscrete-integrate a polynomial, and I never saw thecalculus of finite differences “in the wild”.

Twenty years later. . .– p. 5/?

– 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);

– 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))”;

– 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));

– 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;

– 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. 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.

– p. 8/?Research aim: determine the primitive ideals of thesealgebras.Recall: a primitive ideal of an algebra A is theannihilator of an irreducible A-module.Primitive ideals provide a glimpse (often a revealingone!) of the representation theory of the algebra A.

Side remark: the theory of primitive ideals inenveloping algebras of semisimple Lie algebras is abeautiful and fascinating topic, touching uponharmonic polynomials, the combinatorics of Youngtableaux, and other neat stuff. But you don’t see theseesoteric material in low dimensions—often bruteforce calculations will suffice.– p. 9/?

– p. 9/?Side remark: the theory of primitive ideals inenveloping algebras of semisimple Lie algebras is abeautiful and fascinating topic, touching uponharmonic polynomials, the combinatorics of Youngtableaux, and other neat stuff. But you don’t see theseesoteric material in low dimensions—often bruteforce calculations will suffice.Down-up algebras have low dimensions, so there ishope that I can actually calculate their primitiveideals.

– p. 9/?Side remark: the theory of primitive ideals inenveloping algebras of semisimple Lie algebras is abeautiful and fascinating topic, touching uponharmonic polynomials, the combinatorics of Youngtableaux, and other neat stuff. But you don’t see theseesoteric material in low dimensions—often bruteforce calculations will suffice.Down-up algebras have low dimensions, so there ishope that I can actually calculate their primitiveideals.(And students can do some of the work too.)

– p. 10/?Let’s take the case r = s = 1 and γ ≠ 0. The definingrelations becomehu − uh = γu;dh − hd = γd;du − ud = f(h).This is very close to the relations defining U(sl(2)).

– p. 11/?Verma modulesLet λ ∈ C. The Verma module V (λ) has basis {v i } i≥0whereuv i = v i+1 , dv 0 = 0, hv 0 = λv 0 ;other relations are derived from the defining relations.Thus, for i ≥ 0,hv i = (λ + iγ)v i ,dv i+1 = [f(λ) + f(λ + γ) + · · · + f(λ + iγ)]v i .(You can see why u is the “up” operator, d is the“down” operator, and h is the “horizontal” operator.)

Annihilators of simple Verma modules provide uswith a large portion of all primitive ideals.– p. 11/?Verma modulesLet λ ∈ C. The Verma module V (λ) has basis {v i } i≥0whereuv i = v i+1 , dv 0 = 0, hv 0 = λv 0 ;other relations are derived from the defining relations.Thus, for i ≥ 0,hv i = (λ + iγ)v i ,dv i+1 = [f(λ) + f(λ + γ) + · · · + f(λ + iγ)]v i .(You can see why u is the “up” operator, d is the“down” operator, and h is the “horizontal” operator.)

First step: find a central element of the formud − p(h), where p(x) ∈ C[x].– p. 12/?

– p. 12/?First step: find a central element of the formud − p(h), where p(x) ∈ C[x].The center of the algebra plays an important role,because of the following result.Lemma. Let a be an element in the center of L, andlet M be an irreducible L-module. Then a acts as ascalar on M.

– p. 12/?First step: find a central element of the formud − p(h), where p(x) ∈ C[x].The center of the algebra plays an important role,because of the following result.Lemma. Let a be an element in the center of L, andlet M be an irreducible L-module. Then a acts as ascalar on M.This is a version of Schur’s Lemma, and in this case itis true because C is an uncountable algebraicallyclosed field.

– p. 13/?Say ud − p(h) commutes with u. Then[ud − p(h)]u= u(ud + f(h)) − p(h)u= u[ud − p(h)] + up(h) + uf(h) − p(h)u= u[ud − p(h)] + u[p(h) + f(h) − p(h + γ)].

– p. 14/?Thus ud − p(h) commutes with u if and only ifp(h) + f(h) − p(h + γ) = 0, i.e.,p(h + γ) − p(h) = f(h).This should look familiar!

– p. 15/?The polynomial p is more useful than the polynomialf. For example, the Verma module formuladv i = [p(λ + iγ) − p(λ)]v i−1 , i > 0,is easier to work with thandv i+1 = [f(λ) + f(λ + γ) + · · · + f(λ + iγ)]v i .

– p. 15/?The polynomial p is more useful than the polynomialf. For example, the Verma module formuladv i = [p(λ + iγ) − p(λ)]v i−1 , i > 0,is easier to work with thandv i+1 = [f(λ) + f(λ + γ) + · · · + f(λ + iγ)]v i .Results are also more elegantly stated in terms of prather than f.Theorem. V (λ) and V (λ ′ ) have the same annihilatorif and only if p(λ) = p(λ ′ ).

– p. 16/?Here are results about the primitive ideals of ourdown-up algebra.Theorem. Suppose p(x) is not the constantpolynomial. Let λ ∈ C. Then the annihilator of V (λ)is a primitive ideal. All other primitive ideals areannihilators of finite-dimensional simple modules. Allfinite-dimensional simple modules are quotients ofVerma modules.So the situation in this case is quite similar to the caseof enveloping algebras of semisimple Lie algebras.

– p. 17/?The special case where p(x) is constant is most easilydescribed by an explicit list.Theorem. Suppose p(x) is the constant polynomial.The primitive ideals in this case are 〈u〉, 〈d〉,〈u,d,h−λ〉, where λ ∈ C, and 〈ud −c〉, where c ≠ 0.