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Given [MATH] put [MATH] [MATH] and [MATH] Given a limit ordinal [MATH] put [MATH] where [MATH] is the least congruence with all [MATH] [MATH] , forming one class. Call that class [MATH] |
Now to obtain a clique-like collection [MATH] of algebras in [MATH] , let [MATH] be the free algebra on one generator (in which all ranks are finite). For every ordinal [MATH] let [MATH] be the above algebra [MATH] for [MATH] (which has an element of rank [MATH] but none of rank [MATH] ). |
Analogously a clique-like class in [MATH] is given: just change [MATH] to be the cofree unary algebra on a 2-element set X. (This can be described as the algebra [MATH] |
with the operation assigning to every map [MATH] the map [MATH] , whereas the couniversal morphism is the evaluation at [MATH] .) |
5.11 Remark Let [MATH] be a concrete category for which a faithful functor [MATH] to the category of sets is given. Recall that [MATH] is called |
almost alg-universal if there exists an embedding [MATH] which is almost full . This means that every morphism [MATH] of [MATH] with [MATH] non-constant has the form [MATH] for a unique graph homomorphism [MATH] |
Given a clique-like collection [MATH] in [MATH] , it then follows that [MATH] is clique-like in [MATH] : choose [MATH] to be the class of all morphisms [MATH] with [MATH] nonconstant. Analogously, every clique-like collection in [MATH] yields one in [MATH] |
5.12 Example For the following categories [MATH] the colimit completions [MATH] and [MATH] are not cowellpowered: as proved in , each of these categories is almost alg-universal. |
(1) Monoids and homomorphisms. (2) Posets and monotone functions. (3) Lattices and homomorphisms. (4) Topological spaces and continuous functions. |
(5) Small categories and functors. Here [MATH] assigns to every small category the set of all morphisms. Under the assumption that no cardinal is measurable, further examples are |
(6) Compact Hausdorff spaces and continuous functions. (7) Metric spaces and (uniformly) continuous functions. 5.13 Example For the category [MATH] of abelian groups we also have that |
[MATH] and [MATH] are not cowellpowered. This follows from the result of Przeździecki that there exists an embedding [MATH] such that for every pair of graphs [MATH] and [MATH] the homomorphism group [MATH] |
is the free abelian group on [MATH] . In particular, if [MATH] is a clique on [MATH] vertices in [MATH] , then the abelian groups |
[MATH] fulfil for every ordinal [MATH] that (a) there exist infinitely many homomorphisms from [MATH] to [MATH] , and (b) if [MATH] , there exist no non-zero homomorphism from [MATH] to [MATH] |
Thus, we get a clique-like class in [MATH] w.r.t. the collection [MATH] of zero homomorphisms. Analogously, a clique-like class on [MATH] yields a clique-like class in [MATH] |
In all of the above concrete examples except [MATH] the categories [MATH] are finitely complete and cocomplete. Thus, the fact that the colimit completions of those categories (and their duals) are not cowellpowered implies that they are also not wellpowered, see Proposition 5.1. |
# Source: arxiv 1806.02656 # Title: Little and Big $q-$Jacobi Polynomials and the Askey-Wilson algebra # Sections: all # Downloaded: 2026-03-03T02:41:43.716080+00:00 |
Little and Big [MATH] Jacobi Polynomials and the Askey-Wilson algebra (Date: May 31, 2018) Abstract. The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey-Wilson algebra. The operators that these polynomials respectively diagonalize are identified within the Askey-... |
MSC: 81R50; 81R10; 81U15; 39A70; 33D50; 39A13. Keywords : Askey-Wilson algebra; Tridiagonalisation: Orthogonal polynomials 1. Introduction |
This paper indicates how the little and big q-Jacobi polynomials KS96 occur in the context of representations of the Askey-Wilson algebra. This structure was originally identified in Z91 as the algebra |
The little q-Jacobi polynomials KS96 have received algebraic interpretations some time ago, either as matrix elements of co-representations of the quantum group [MATH] |
VS88 K89 MMNNU88 M91 or equivalently FV93 , as matrix elements of q-exponentials in the generators of the quantum algebra [MATH] |
FV93b . The big q-Jacobi polynomials have been connected to quantum 2-spheres NM90 . We shall provide here a rather different algebraic setting for these q-polynomials as basis vectors for modules of the Askey-Wilson algebra. |
On the one hand, consider twisted primitive elements in [MATH] K93 KJ98 . It is known that such elements provide an embedding of the Askey-Wilson algebra in [MATH] |
GZ93 . Take the holomorphic realization of [MATH] and a specialization of the generators, it will be seen that the operators of which the little and big q-Jacobi polynomials are eigenfunctions belong to this realization of the Askey-Wilson algebra. |
On the other hand, the polynomials of the Askey scheme and especially those under consideration here, are solutions of bispectral problems. It has been shown recently how algebraic Heun operators can be associated to any such problems GVZ17 . For some choices of the generic parameters, the resulting Heun operator can b... |
As a matter of fact, it has already been shown that the recurrence coefficients of the Askey-Wilson polynomials can be obtained from those of the big q-Jacobi polynomials from the tridiagonalization of the recurrence operator of the latter polynomials TVZ17 |
It will be seen that the tridiagonalization of the q-difference operator, of which the little q-Jacobi polynomials are eigenfunctions, leads to the Askey-Wilson algebra in the fashion described above. It will be further observed that if the equitable presentation of [MATH] is called upon ITW06 T15 , the little q-Jacobi... |
The paper will be organized as follows. After having provided some relevant facts about [MATH] , the Askey-Wilson algebra and its embedding in [MATH] are presented in Section 2. In Section 3, the little q-Jacobi polynomials are introduced via their q-difference equation, which is identified under special choices of the... |
1.1. Notations In this paper, we fix a nonzero complex number [MATH] which is not a root of unity. We will use the standard [MATH] -shifted factorials (also called [MATH] Pochhammer functions) KS96 |
[EQUATION] 2. The Askey-Wilson algebra and its embedding in [MATH] We shall recap in this section the basic tools that shall be used in the remainder of the paper: first the standard Chevalley presentation of [MATH] and second the equitable one, together with isomorphisms between the two. The realization of [MATH] in t... |
2.1. The Chevalley presentation of [MATH] The Chevalley presentation of [MATH] consists of three generators denoted [MATH] . They satisfy |
[EQUATION] The central element of [MATH] is the Casimir operator: [EQUATION] 2.2. The equitable presentation of [MATH] The equitable presentation of [MATH] consists of three generators denoted [MATH] |
ITW06 [EQUATION] In T15 , Lemma 5.1] , an isomorphism with the presentation ( 2.1 ) is given. Here we will use a special case. In our notation, it reads: |
[EQUATION] 2.3. The [MATH] difference operators realization of [MATH] An irreducible infinite dimensional representation [MATH] with [MATH] can be realized by [MATH] difference operators acting on the space of formal power series [MATH] in the variable [MATH] . The lowest weight vector corresponds to [MATH] . Define [M... |
[EQUATION] On [MATH] , the eigenvalue of [MATH] 2.2 ) is given by: [EQUATION] Note that if [MATH] , the representation becomes reducible as [MATH] . The corresponding invariant subspace of polynomials of degree [MATH] |
2.4. The Askey-Wilson algebra We now turn to the Askey-Wilson algebra and its embedding in [MATH] Let [MATH] be arbitrary scalars. Define: |
[EQUATION] A third operator, namely [EQUATION] is constructed. By straightforward calculations, one finds: [EQUATION] where [MATH] and [MATH] are expressed in terms of [MATH] |
[EQUATION] Proposition 2.1 [MATH] satisfy the Askey-Wilson algebra: [EQUATION] where [EQUATION] Proof. Straightforward, using ( 2.1 ) and ( 2.2 ).∎ |
In the text below, we consider successively special values of the structure constants of the AW algebra. In each case, a realization of [MATH] in terms of [MATH] Chevalley generators is given. For a first choice of [MATH] namely [MATH] the operator [MATH] will be denoted by [MATH] . It is diagonalized by the little q-J... |
[EQUATION] where the structure constants are specializations of [MATH] as given above. 3. The little [MATH] Jacobi polynomials We shall now make precise the circumstances under which the little q-Jacobi polynomials are eigenfunctions of [MATH] and form a representation basis of a specialized Askey-Wilson algebra. Given... |
3.1. The second order q-difference operator [MATH] If we now compare the operator [MATH] of ( 2.9 ) using ( 2.5 ) with (3.12.4) of KS96 , one has: |
[EQUATION] Proposition 3.1 For the specialization [MATH] , the ‘reduced’ operator [MATH] is diagonalized by the little [MATH] Jacobi polynomials. |
Proof. Denote the little q-Jacobi polynomials as: [EQUATION] They satisfy the second-order q-difference equation: [EQUATION] where |
[EQUATION] The exact relation between the parameters [MATH] and the parameters entering in the little q-Jacobi polynomials is as follows: |
[EQUATION] and [EQUATION] 3.2. Eigenfunctions and the explicit expression of the little [MATH] Jacobi polynomials First, we construct the eigenfunctions of [MATH] in ( 2.7 ) for [MATH] |
Lemma 3.1 For [MATH] , one has: [EQUATION] Proof. Considering ( 2.7 ) for [MATH] and using ( 2.5 ), the action on the monomials [MATH] reads: |
[EQUATION] Let [MATH] be such that [MATH] . Define [MATH] . The action of [MATH] on [MATH] gives the recurrence relation: [EQUATION] |
The solution reads [EQUATION] from which, setting [MATH] and using the [MATH] binomial theorem KS96 , we finally obtain ( 3.3 ). |
On the eigenfunctions of [MATH] , we now consider the action of [MATH] Lemma 3.2 One has: [EQUATION] with [EQUATION] Proof. Recall ( 3.1 ). The l.h.s. of ( 3.5 ) reads: |
[EQUATION] where we have denoted [MATH] . The r.h.s reads: [EQUATION] Equating both sides of the equation ( 5.6 ), the coefficients [MATH] and [MATH] are determined uniquely. |
Remark 3.1 Observe that the coefficient [MATH] coincides with the spectrum of [MATH] , as expected ( [MATH] is lower triangular in the basis [MATH] ). |
Next, we are interested in the overlap coefficients between the eigenfunctions of [MATH] (the [MATH] ) and the eigenfunctions of [MATH] (the little q-Jacobi polynomials). |
Proposition 3.2 One has: [EQUATION] where [EQUATION] Proof. By Prop. 3.1 recall that [MATH] are eigenfunctions of [MATH] with the identification ( 3.2 ). Consider the expansion ( 3.6 ). The action of [MATH] on [MATH] gives the recurrence relation: |
[EQUATION] The solution reads [EQUATION] By straightforward calculations, one finds: [EQUATION] Setting the normalization such that: |
[EQUATION] we get ( 3.7 ). Remark 3.2 The expansion formula ( 3.6 ) coincides with (2.46) of K94 with [MATH] [EQUATION] Remark 3.3 |
Note that the little q-Jabobi operator [MATH] is also bidiagonal in the monomial basis [MATH] . By analogy with the proof of Lemma 3.1 one can retrieve the following familar explicit expression for the little q-Jacobi polynomial KS96 , eq. (3.12.1)] (see also K94 , eq. (2.45)] ): |
[EQUATION] 4. Tridiagonalization of the little [MATH] Jacobi operator and the equitable presentation of [MATH] Calling upon the equitable embedding of the Askey-Wilson algebra T11 which will be recalled next, we shall now make the observation that the little q-Jacobi operator and a tridiagonalization of this operator r... |
Remark 4.1 Define [EQUATION] Then [EQUATION] where [MATH] and [MATH] are the equitable [MATH] generators introduced in ( 2.4 ). Using the equitable presentation of [MATH] 2.4 ), by Proposition 1.1 and Lemma 3.4 in T11 , one gets: |
Proposition 4.1 Define [EQUATION] The generators [MATH] satisfy the Askey-Wilson algebra: [EQUATION] where [MATH] denotes the ‘normalized’ Casimir element |
[EQUATION] Remark 4.2 In terms of the Chevalley generators of [MATH] , one has [MATH] with ( 4.1 ) and [EQUATION] For the analysis to follow, let us introduce the operators [MATH] and [MATH] |
[EQUATION] where [MATH] is understood as a power series in the Cartan element of [MATH] . Note that [MATH] . Also, the action of each operator on the space of monomials in [MATH] , according to ( 2.5 ), is such that: |
[EQUATION] Lemma 4.1 Consider [MATH] as the [MATH] difference operators obtained from ( 4.3 ) with ( 2.4 ). One has: [EQUATION] with |
[EQUATION] Proof. Recall that [MATH] is given by ( 4.1 ). Consider the r.h.s. of ( 4.7 ). The action of [MATH] on the monomials [MATH] is such that: |
[EQUATION] By straightforward calculations, we obtain: [EQUATION] Compare the expression above with ( 4.4 ), we obtain the constraints: |
[EQUATION] from which we get ( 4.8 ). This lemma thus states that the generator [MATH] of the equitable presentation of the Askey-Wilson algebra is obtained from [MATH] by tridiagonalization. Stated differently, the upshot is that [MATH] i.e. [MATH] together with its tridiagonalization ( 4.7 generate the Askey-Wilson a... |
The tridiagonalized form of [MATH] can be inverted to express [MATH] in terms of [MATH] as follows. Lemma 4.2 Consider [MATH] as the [MATH] difference operators obtained from ( 4.3 ) with ( 2.4 ). One has: |
[EQUATION] with [EQUATION] Proof. Recall [MATH] is given by ( 4.4 ). Consider the r.h.s. of ( 4.9 ). The action of [MATH] on the monomials [MATH] is such that: |
[EQUATION] By straightforward calculations, we obtain: [EQUATION] Compare the expression above with ( 4.1 ), we obtain the constraints: |
[EQUATION] from which we get ( 4.10 ). 5. The big [MATH] Jacobi polynomials We shall now carry for the big q-Jacobi polynomials an analysis similar to the one that was given in Section 3 for the little q-Jacobi polynomials; namely, identify the specialization of the parameters in [MATH] that will lead to the big q-Jaco... |
5.1. The second order q-difference operator [MATH] Recall ( 2.5 ). By straightforward replacements, from ( 2.9 ) we get for the specialization |
[MATH] [EQUATION] Proposition 5.1 For [MATH] , the operator [MATH] is diagonalized by the big q-Jacobi polynomials. Proof. Compare the operator [MATH] written as ( 5.1 ) with the second-order q-difference operator (3.5.4) in KS96 . Denote the big q-Jacobi polynomials as: |
[EQUATION] They satisfy the second-order q-difference equation: [EQUATION] where [EQUATION] The exact relation between the parameters [MATH] and the parameters entering in the big q-Jacobi polynomials is as follows: |
[EQUATION] and [EQUATION] Remark 5.1 Note that if instead we would like to consider the spectral problem for the big q-Jacobi polynomials |
[EQUATION] the substitution [MATH] into ( 5.2 ) gives the following identification: [EQUATION] For the special choice [MATH] , note that the big q-Jacobi polynomial [MATH] can be expressed in terms of the little q-Jacobi polynomial [MATH] . See KS96 for details. |
5.2. Eigenfunctions and the explicit expression of the big [MATH] Jacobi polynomials Consider the eigenfunctions of [MATH] in ( 2.7 ) for [MATH] . They coincide with the ones given in Lemma 3.1 On these eigenfunctions of [MATH] , we now consider the action of [MATH] |
Lemma 5.1 One has: [EQUATION] with [EQUATION] Proof. Recall ( 5.1 ). The l.h.s. of ( 5.6 ) reads: [EQUATION] where [MATH] . The r.h.s reads: |
[EQUATION] Equating both sides of the equation ( 3.5 ), the coefficients [MATH] and [MATH] are determined uniquely. Remark 5.2 Observe that the coefficient [MATH] coincides with the spectrum of [MATH] , as expected ( [MATH] is lower triangular in the basis [MATH] ). |
Next, we are interested in the overlap coefficients between the eigenfunctions of [MATH] (the [MATH] ) and the eigenfunctions of [MATH] (the big q-Jacobi polynomials). The proof of the following proposition is straightforward, by analogy with the proof of Prop. 3.2 |
Proposition 5.2 One has: [EQUATION] where [EQUATION] Remark 5.3 In particular, note that: [EQUATION] 6. A connection between some big [MATH] Jacobi polynomials and special little [MATH] Jacobi polynomials by tridiagonalization |
Much as in IK11 and IK12 we shall supplement the observations of the preceding section by noting that a large class of big q-Jacobi polynomials can be obtained from a particular tridiagonalization of the special little q-Jacobi operator with ( 3.2 ) for [MATH] |
Lemma 6.1 Consider the little q-Jacobi operator [MATH] and the big q-Jacobi operator [MATH] of the form: [EQUATION] For [MATH] , one has: |
[EQUATION] with ( 4.5 ) and [EQUATION] Proof. Consider the l.h.s. of ( 6.3 ). The monomial [MATH] is easily reduced in terms of [MATH] , namely: |
[EQUATION] Consider the r.h.s. of ( 6.3 ). Using: [EQUATION] by straightforward calculations, for a slightly more general combination we obtain: |
[EQUATION] Comparing the r.h.s and l.h.s of ( 6.3 ), we identify: [EQUATION] together with the constraints: [EQUATION] The first constraint gives [MATH] . Inserting in the second constraint, one gets [MATH] . The other equations imply 6.4 ). |
Note from eq. ( 3.2 ) that the condition [MATH] implies that [MATH] . Thus, for a large parameter set, big q-Jacobi polynomials can be constructed from little q- Jacobi polynomials with that special value of the parameter [MATH] by using the one-sided tridiagonalization introduced in IK11 and IK12 |
7. Conclusion This paper has shown how the little and big q-Jacobi polynomials form bases for representations of the (specialized) Askey-Wilson algebra. The starting point has been the embedding of the Askey-Wilson algebra in [MATH] realized in terms of q-difference operators. It then proved possible to identify within... |
It should similarly prove interesting to adopt the somewhat opposite viewpoint, that is to start from the little and big q-Jacobi operators and their tridiagonalizations and to look at the conditions for these operators to form Askey-Wilson algebras. Recall that the most general tridiagonalization of the ordinary Jacob... |
Acknowledgements: We thank Paul Terwilliger for comments. L.V. would like to express his gratitude for the hospitality extended to him by the Institut Denis-Poisson of the Université François-Rabelais de Tours as Chercheur Invité where most of this research was carried out. The research of L.V. is funded in part by a d... |
# Source: arxiv 1806.02661 # Title: New mechanism for repeated posted price auction with a strategic buyer without discounting # Sections: all # Downloaded: 2026-03-02T09:23:08.105676+00:00 |
New mechanism for repeated posted price auction without discounting Abstract. On ad exchange platforms the place for advertisement is sold through different kinds of auctions. However, it is not uncommon the situation where the seller repeatedly encounters only one buyer, thus the posted price auction degenerates into ... |
I learned this problem from a discussion with members of Yandex research team and my main motivation was to find an incentive-compatible seller’s strategy. In this short paper such a strategy is proposed and a corresponding distortion at the top type lower bound (Spence-Mirrlees property, actually) for the surplus of t... |
The key ingredients are the following. The main leash that the buyer has is the frequency of accepted deals. Once this frequency (as a function on the buyer’s type) is fixed, the strategy randomly chooses between the rewarding price which incentivises the buyer to reveal his type (the higher the type, the more average ... |
National Research University Higher School of Economics, Soyuza Pechatnikov str., 16, St. Petersburg, Russian Federation. Support from the Basic Research Program of the National Research University Higher School of Economics is gratefully acknowledged. Supported in part by Young Russian Mathematics award. |
I foresee the progress of game theory as depending on successive reductions in the base of common knowledge required to conduct useful analyses of practical problems. |
Only by repeated weakening of common knowledge assumptions will the theory approximate reality. R. Wilson, Game-Theoretic Analyses of Trading Processes |
1. Setup 1.1. The game The repeated posted price auction , also known as the fishmonger problem, is an archetypical repeated monopoly-monopsony game with asymmetric information. This is a game with two players: the seller ( fisher ) and the buyer ( cook ). In each round [MATH] the fisher proposes to the cook a unit of ... |
The repeated posted price auction appears in ad exchange conducted by the leading Internet companies, see and the references therein. It is common to assume that [MATH] is drawn from a known distribution. In line with |
, recently there have appeared a bunch of papers (e.g. ) which weaken this assumption since it is practically unrealistic if the buyer is unique of their kind. We will study the case when the fisher has no information at all about [MATH] (it is even worse than the worst case in terminology of |
where [MATH] was at least in the interval [MATH] ). Consequently, to make sense of the problem we must further assume that the game has an infinite number of rounds. |
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