FPSAC 2010, San Francisco, USA DMTCS proc. (subm.), by the authors, 1–6

Radford theorem for generalized deformations of Hoffmann type

Gérard H. E. Duchamp^{1}^{†} and Hoang Ngoc Minh^{1}

1LIPN - UMR 7030, CNRS - Université Paris 13, 93430 Villetaneuse, France

Abstract. Résumé.

Keywords: Comultiplicatioon, Hopf algebra

1 Introduction

With the advent of Quantum theories emerged the need for deforming not only parameters or formulas but structures and laws (see [3]). On the other hand, computer science provided us with laws the revealed later to be better understood as dual as the shuffle product recursively defined by

au^{⊔⊔}bv = a(u^{⊔⊔}bv) + b(au^{⊔⊔}v)  (1)

and the q-infiltration product [5] by

au ↑ bv = a(u ↑ bv) + b(au ↑ v) + qδ_{a,b}a(u ↑ v).  (2)

In fact all these laws are particular cases of diagonal deformations [7] and such deformations can be applied to stuffles.

.../...

Dual laws of theoretical computer science : shuffle, infiltration, q-infiltration (work with Luque), LDIAG(q_{c},_{q} s) and explanationn of the different natures of the parameters.

2 Background

Twisted products of graded algebras, diagonal deformations.

†The authors wish to acknowledge support from Agence Nationale de la Recherche (Paris, France) under Program No. ANR-08-BLAN-0243-2. subm. to DMTCS \begin{matrix}c \\ ◯ \end{matrix} by the authors Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

3 Colour factors and products

Colours factors were introduced by [10] and the theory was developped or used in [6, 4, 9, 11].

Let A = ⊕_{α∈D}A_{α} and B = ⊕_{β∈D}B_{β} be two D-graded associative algebras (D is a commutative semigroup whose law is denoted additively). Readers that are not familiar with graded algebras can think of D = ℕ^{(X)}, the free commutative monoid over X and A_{α} = K[X]_{α}, the space of homogeneous polynomials of multidegree α.

We suppose given a mapping χ : D × D ⟶ K and define a law of algebra on A ⊗ B by

(x_{1} ⊗ y_{1})(x_{2} ⊗ y_{2}) = χ(β_{1}, α_{2})(x_{1}x_{2} ⊗ y_{1}y_{2})  (3)

for (x_{i}) ∈ A_{α_{i}} and (y_{i}) ∈ B_{β_{i}} (i = 1, 2). The computations of ((x_{1} ⊗ y_{1})(x_{2} ⊗ y_{2})) (x_{3} ⊗ y_{3}) and (x_{1} ⊗ y_{1}) ((x_{2} ⊗ y_{2})(x_{3} ⊗ y_{3})) using (3) both lead to the following proposition the second part – converse – of which relies on the existence of free elements.

Proposition 1 [11] Let χ : D × D ⟶ K. The following are equivalent

i) For A, B D-graded associative algebras, the product defined by (3) is associative. ii) (∀α_{1}, α_{2}, α_{3}, β_{1}, β_{2}, β_{3}) ∈ D

χ(β_{1}, α_{2})χ(β_{1} + β_{2}, α_{3}) = χ(β_{2}, α_{3})χ(β_{1}, α_{2} + α_{3})  (4)

Definition 2 Every mapping χ : D × D ⟶ K which fulfills the equivalent conditions of proposition (1) will be called a colour factor.

Remark 3 i) If χ is bilinear, which means in this context that the following equations are satisfied (for all α, α′, β, β′ ∈ D)

χ(α + α′, β) = χ(α, β)χ(α′, β)
χ(α, β + β′) = χ(α, β)χ(α, β′)  (5)

then, the two members of (3) amount to

χ(β_{1}, α_{2})χ(β_{1}, α_{3})χ(β_{2}, α_{3}) = ∏_{1≤i<j≤3} χ(β_{i}, α_{j})  (6)

and hence χ is a colour factor. But the full class of colour factors is much larger than solutions of Eq. (5). Just observe that Eq.(4) is homogeneous in the classical sense i.e. for all λ ∈ k, if χ fulfills (4) then λχ still does. Hence, for example, any constant function on D × D is a colour factor. This shows the existence of colour factors that are not bilinear. ii) It may seem that one could generalize (5) to the case when D is noncommutative but, in fact, there is no gain of generality because, as K is commutative, the bicharacter factorizes through D^{ab} (the quotient of D by the finest congruence ≡ such that D/ ≡ is abelian).

Note(s) 4 i) The colour product of two algebras A = ⊕_{α∈D}A_{α} and B = ⊕_{β∈D}B_{β} comes also as a graded algebra by

(A ⊗ B)_{γ} = ⊕_{α+β=γ} A_{α} ⊗ B_{β}.  (7)

The usual identification

(A ⊗ B) ⊗ C ≃ A ⊗ (B ⊗ C)  (8)

holds for the coloured products. ii) Moreover, if A →^{f} A′ (resp. B →^{g} B′) are two morphisms of (graded) algebras, then A ⊗ B ⟶^{f⊗g} B ⊗ B′ is a morphism of algebras (colour products).

4 Special classes of laws

4.1 Dual laws

4.1.1 Algebras and coalgebras in duality

An algebra (A, μ) and a coalgebra (C, Δ) are called in duality iff there is a non-degenerate pairing ⟨−|−⟩ such that for all x, y ∈ A, z ∈ C

⟨μ(x, y)|z⟩ = ⟨x ⊗ y|Δ(z)⟩^{⊗2}  (9)

In the following, we will call dual law a law K⟨A⟩ ⊗ K⟨A⟩ ⟶^{∗} K⟨A⟩ on the free algebra which is the dual of a comultiplication, the pairing being given on the basis of words by ⟨u|v⟩ = δ_{u,v}.

Our first examples are essential in modern and not-so-modern research ([?, ?]). Firstly, we have the dual of the Cauchy product

Δ_{Cauchy}(w) = ∑_{uv=w} u ⊗ v .  (10)

Contrary to this one (10), which is not a morphism of algebras^{(i)}

K⟨A⟩ ⟶ K⟨A⟩ ⊗ K⟨A⟩ ,  (11)

one has three very well-known examples being so, namely duals of the shuffle^{⊔⊔}, the Hadamard ⊙ and the infiltration product ↑. As they are morphisms between the algebras (11), they are well defined by their values on the letters. Respectively

Δ_{⊔⊔} (x) = x ⊗ 1 + 1 ⊗ x ; Δ_{⊙}(x) = x ⊗ x ; Δ_{↑}(x) = x ⊗ 1 + 1 ⊗ x + x ⊗ x .  (12)

One can prove that the deformations Δ_{q} = Δ_{⊔⊔} (x) + qΔ_{⊙}(x) are also co-associative and that they are the unique solutions of the problem of bialgebra comultiplications on K⟨A⟩ that are compatible with subalphabets [5].

In the sequel, we will make use several time of the following lemma the proof of which is left to the reader.

Lemma 5 Let A be an algebra and C be a coalgebra in (non-degenerate) duality, then A is associative iff C is coassociative.

(i) Unless A = ∅.

4.1.2 Duality between grouplike elements and unities

Let (C, Δ) be a coalgebra with counit ϵ. We call group-like an element u such that

ϵ(u) = 1 ; Δ(u) = u ⊗ u .  (13)

One then has C = ker(ϵ) ⊕ K.u and

Δ(y) = Δ^{+}(y) + y ⊗ u + u ⊗ y − ϵ(y)u ⊗ u .  (14)

where Δ^{+} is a comultiplication on C for which ker(ϵ) = C^{+} is a subcoalgebra (i. e. Δ^{+}(C^{+}) ⊂ C^{+} ⊗C^{+}) [?].

Proposition 6 Let (C, Δ, ϵ) be a coalgebra with counit, u a group-like element in C and (C^{+}, Δ^{+}) be as in (14). On the other hand, let A be an algebra and A^{(1)} = A ⊕ K.v be the algebra with unit constructed from A by adjunction of the unity v. Then, if C^{+} and A are in duality by ⟨ | ⟩, so are C and A^{(1)} by ⟨ | ⟩_{•} defined as follows

⟨x + αv|y + βu⟩_{•} = ⟨x|y⟩ + βα .  (15)

Proof — Let

⟨(x_{1} + α_{1}v) ⊗ (x_{2} + α_{2}v)|Δ(y + βu)⟩_{•}^{⊗2} =
⟨(x_{1} + α_{1}v) ⊗ (x_{2} + α_{2}v)|Δ^{+}(y) + y ⊗ u + u ⊗ y + βu ⊗ u)⟩_{•}^{⊗2}  (16)

but, according to the fact that

⟨x_{i}|u⟩ = ⟨x_{1} ⊗ v|Δ^{+}(y)⟩ = ⟨v ⊗ x_{2}|Δ^{+}(y)⟩ = ⟨v ⊗ v|Δ^{+}(y)⟩ = ⟨v|y⟩ = 0

one has from (16)

⟨(x_{1} + α_{1}v) ⊗ (x_{2} + α_{2}v)|Δ(y + βu)⟩_{•}^{⊗2} =
⟨x_{1} ⊗ x_{2}|Δ^{+}(y)⟩^{⊗2} + α_{2}⟨x_{1}|y⟩ + α_{1}⟨x_{2}|y⟩ + α_{1}α_{2}β =
⟨x_{1}x_{2} + α_{2}x_{1} + α_{1}x_{2} + α_{1}α_{2}v|y + βu⟩_{•} = ⟨(x_{1} + α_{1}v)(x_{2} + α_{2}v)|y + βu⟩_{•}  (17)

which proves the claim. □

4.2 Deformed laws

Let S be a semigroup graded on a semigroup of degrees D and A = K[S] its algebra. A colour factor χ : D × D → K being given, we endow the algebra A ⊗ A with the coloured tensor product structure. Notice that the diagonal subspace D_{S} = ⊕_{x∈S}Kx ⊗ x is a subalgebra as

(x ⊗ x)(y ⊗ y) = χ(|x|, |y|)xy ⊗ xy .  (18)

Carrying (18) back to A by means of the isomorphism of vector spaces, A → D_{S}, on sees immediately that the deformed product on A given by

x._{χ}y = χ(|x|, |y|)xy  (19)

Radford theorem for generalized deformations of Hoffmann type 5 is associative. If A is endowed with the scalar product for which the basis (s)_{s∈S} is orthonormal, the pairing is nondegenerate and the dual comultiplucation is given by

Δ(z) = ∑_{xy=z} χ(|x|, |y|)x ⊗ y .  (20)

The construction together with lemma (5) proves that this comultiplication on A is coassociative.

5 Generalized shuffles and stuffles

Discussion of the commutation factor

6 Conclusion

Grading polyzeta values

References

[1] N. Bourbaki. Éléments de Mathématique - Algèbre, chap. 1 à 3, 2nd edition, Springer, 2007.

[2] N. Bourbaki. Éléments de Mathématique - Algèbre commutative, chap. 1 à 4, Springer, 2006.

[3] V. Chari, A. Pressley, A guide to quantum groups. Cambridge Univ. Press, 1994.

[4] Désarménien J., Duchamp G., Krob D., Melançon G, Quelques remarques sur les superalgèbres de Lie libres C.R.A.S. n^{0}5 (1994).

[5] G. Duchamp, M. Flouret, É. Laugerotte, J.-G. Luque, Direct and dual laws for automata with multiplicities, Theoret. Comp. Sci. 267 (2001) 105-120.

[6] G H E Duchamp, A Klyachko, D Krob, J Y Thibon, Noncommutative symmetric functions III: Deformations of Cauchy and convolution algebras. Discrete Mathematics and Theoretical Computer Science Vol. 2 (1998).

[7] Duchamp G. H. E., Tollu C., Sweedler’s duals and Schützenberger’s calculus arXiv:0712.0125 (to be published).

[8] S. Eilenberg. Automata, Languages and Machines, vol. A., Academic Press, 1974.

[9] A. A. Mikhalev and A. A. Zolotykh, Combinatorial Aspects of Lie Superalgebras. CRC Press, Boca Raton, New York, 1995.

[10] R. Ree, Generalized Lie elements, Canad. J. Math. 12 (1960), 493–502.

[11] J. Zhou, Combinatoire des derivations, Thèse de l’Université de Marne-la-Vallée (1996).

6 Gérard H. E. Duchamp and Hoang Ngoc Minh