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Quasitriangular Structure of Myhill–Nerode Bialgebras

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Quasitriangular Structure of Myhill–Nerode Bialgebras Axioms 2012, 1, 155-172; doi:10.3390/axioms1020155 OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Article Quasitriangular Structure of Myhill–Nerode Bialgebras Robert G. Underwood Department of Mathematics/Informatics Institute, Auburn University Montgomery, P.O. Box 244023, Montgomery, AL 36124, USA; E-Mail: [email protected]; Tel.: +1-334-244-3325; Fax: +1-334-244-3826 Received: 20 June 2012; in revised form: 15 July 2012 / Accepted: 17 July 2012 / Published: 24 July 2012 Abstract: In computer science the Myhill–Nerode Theorem states that a set L of words in a finite alphabet is accepted by a finite automaton if and only if the equivalence relation ∼L, defined as x ∼L y if and only if xz 2 L exactly when yz 2 L; 8z, has finite index. The Myhill–Nerode Theorem can be generalized to an algebraic setting giving rise to a collection of bialgebras which we call Myhill–Nerode bialgebras. In this paper we investigate the quasitriangular structure of Myhill–Nerode bialgebras. Keywords: algebra; coalgebra; bialgebra; Myhill–Nerode theorem; Myhill–Nerode bialgebra; quasitriangular structure 1. Introduction ^ Let Σ0 be a finite alphabet and let Σ0 denote the set of words formed from the letters in Σ0. Let ^ L ⊆ Σ0 be a language, and let ∼L be the equivalence relation defined as x ∼L y if and only if xz 2 L ^ exactly when yz 2 L; 8z 2 Σ0. The Myhill–Nerode Theorem of computer science states that L is accepted by a finite automaton if and only if ∼L has finite index (cf. [1, 1, Chapter III, x9, Proposition 9.2], [2, x3.4, Theorem 3.9]). In [3, Theorem 5.4] the authors generalize the Myhill–Nerode theorem to an algebraic setting in which a finiteness condition involving the action of a semigroup on a certain function plays the role of the finiteness of the index of ∼L, while a bialgebra plays the role of the finite automaton which accepts the language. We call these bialgebras Myhill–Nerode bialgebras. The purpose of this paper is to investigate the quasitriangular structure of Myhill–Nerode bialgebras. By construction, a Myhill–Nerode bialgebra B is cocommutative and finite dimensional over its base field. Thus B admits (at least) the trivial quasitriangular structure (B; 1 ⊗ 1). We ask: does B (or its linear dual B∗) have any non-trivial quasitriangular structures? Axioms 2012, 1 156 Towards a solution to this problem, we construct a class of commutative Myhill–Nerode bialgebras and give a complete account of the quasitriangular structure of one of them. We begin with some background information regarding algebras, coalgebras, and bialgebras. 2. Algebras, Coalgebras and Bialgebras Let K be an arbitrary field of characteristic 0 and let A be a vector space over K with scalar product ra for all r 2 K, a 2 A. Scalar product defines two maps s1 : K ⊗ A ! A with r ⊗ a 7! ra and s2 : A ⊗ K ! A with a ⊗ r 7! ra, for a 2 A, r 2 K. Let IA : A ! A denote the identity map. A K-algebra is a triple (A; mA; ηA) where mA : A ⊗ A ! A is a K-linear map which satisfies mA(IA ⊗ mA)(a ⊗ b ⊗ c) = mA(mA ⊗ IA)(a ⊗ b ⊗ c) (1) and ηA : K ! A is a K-linear map for which mA(IA ⊗ ηA)(a ⊗ r) = ra = mA(ηA ⊗ IA)(r ⊗ a) (2) for all r 2 K, a; b; c 2 A. The map mA is the multiplication map of A and ηA is the unit map of A. Condition (1) is the associative property and Condition (2) is the unit property. We write mA(a ⊗ b) as ab. The element 1A = ηA(1K ) is the unique element of A for which a1A = a = 1Aa for all a 2 A. Let A; B be algebras. An algebra homomorphism from A to B is a K-linear map φ : A ! B such that φ(mA(a1 ⊗ a2)) = mB(φ(a1) ⊗ φ(a2)) for all a1; a2 2 A; and φ(1A) = 1B: In particular, for A to be a subalgebra of B we require 1A = 1B. For any two vector spaces V , W let τ : V ⊗ W ! W ⊗ V denote the twist map defined as τ(a ⊗ b) = b ⊗ a, for a 2 V , b 2 W . For K-algebras A; B, we have that A ⊗ B is a K-algebra with multiplication mA⊗B :(A ⊗ B) ⊗ (A ⊗ B) ! A ⊗ B defined by mA⊗B((a ⊗ b) ⊗ (c ⊗ d)) = (mA ⊗ mB)(IA ⊗ τ ⊗ IB)(a ⊗ (b ⊗ c) ⊗ d) = (mA ⊗ mB)((a ⊗ c) ⊗ (b ⊗ d)) = ac ⊗ bd for a; c 2 A, b; d 2 B. The unit map ηA⊗B : K ! A ⊗ B given as ηA⊗B(r) = ηA(r) ⊗ 1B for r 2 K. Let C be a K-vector space. A K-coalgebra is a triple (C; ∆C ; C ) in which ∆C : C ! C ⊗ C is K-linear and satisfies (IC ⊗ ∆C )∆C (c) = (∆C ⊗ IC )∆C (c) (3) and C : C ! K is K-linear with s1(C ⊗ IC )∆C (c) = c = s2(IC ⊗ C )∆C (c) (4) Axioms 2012, 1 157 for all c 2 C. The maps ∆C and C are the comultiplication and counit maps, respectively, of the coalgebra C. Condition (3) is the coassociative property and Condition (4) is the counit property. We use the notation of M. Sweedler [4, x1.2] to write X ∆C (c) = c(1) ⊗ c(2) (c) Note that Condition (4) implies that X X C (c(1))c(2) = c = C (c(2))c(1) (5) (c) (c) Let C be a K-coalgebra. A nonzero element c of C for which ∆C (c) = c ⊗ c is a grouplike element of C. If c is grouplike, then c = s1(C ⊗ IC )∆C (c) = s1(C ⊗ IC )(c ⊗ c) = C (c)c and so, C (c) = 1. The grouplike elements of C are linearly independent [4, Proposition 3.2.1]. Let C; D be coalgebras. A K-linear map φ : C ! D is a coalgebra homomorphism if (φ⊗φ)∆C (c) = ∆D(φ(c)) and C (c) = D(φ(c)) for all c 2 C. The tensor product C ⊗ D of two coalgebras is again a coalgebra with comultiplication map ∆C⊗D : C ⊗ D ! (C ⊗ D) ⊗ (C ⊗ D) defined by ∆C⊗D(c ⊗ d) = (IC ⊗ τ ⊗ ID)(∆C ⊗ ∆D)(c ⊗ d) = (IC ⊗ τ ⊗ ID)(∆C (c) ⊗ ∆D(d)) X = (IC ⊗ τ ⊗ ID)( c(1) ⊗ c(2) ⊗ d(1) ⊗ d(2)) (c);(d) X = c(1) ⊗ d(1) ⊗ c(2) ⊗ d(2) (c);(d) for c 2 C, d 2 D. The counit map C⊗D : C ⊗ D ! K is defined as C⊗D(c ⊗ d) = C (c)D(d) for c 2 C, d 2 D. A K-bialgebra is a K-vector space B together with maps mB, ηB, ∆B, B for which (B; mB; ηB) is a K-algebra and (B; ∆B; B) is a K-coalgebra and for which ∆B and B are algebra homomorphisms. Let B; B0 be bialgebras. A K-linear map φ : B ! B0 is a bialgebra homomorphism if φ is both an algebra and coalgebra homomorphism. A K-Hopf algebra is a bialgebra H together with an additional K-linear map σH : H ! H that satisfies mH (IH ⊗ σH )∆H (h) = H (h)1H = mH (σH ⊗ IH )∆H (h) (6) Axioms 2012, 1 158 for all h 2 H. The map σH is the coinverse (or antipode) map and property Condition (6) is the coinverse (or antipode) property. Though we will not consider Hopf algebras here, more details on the subject can be found in [5–8]. An important example of a K-bialgebra is given as follows. Let G be a semigroup with unity, 1. Let KG denote the semigroup algebra. Then KG is a bialgebra with comultiplication map ∆KG : KG ! KG ⊗ KG defined by x 7! x ⊗ x, for all x 2 G, and counit map KG : KG ! K given by x 7! 1, for all x 2 G. The bialgebra KG is the semigroup bialgebra on G. Let B be a bialgebra, and let A be an algebra which is a left B-module with action denoted by “·”. Suppose that 0 X 0 b · (aa ) = (b(1) · a)(b(2) · a ) (b) and b · 1A = B(b)1A for all a; a0 2 A, b 2 B. Then A is a left B-module algebra.A K-linear map φ : A ! A0 is a left B-module algebra homomorphism if φ is both an algebra and a left B-module homomorphism. Let C be a coalgebra and a right B-module with action denoted by “·”. Suppose that for all c 2 C, b 2 B, X ∆C (c · b) = c(1) · b(1) ⊗ c(2) · b(2) (c);(b) and C (c · b) = C (c)B(b) Then C is a right B-module coalgebra.A K-linear map φ : C ! C0 is a right B-module coalgebra homomorphism if φ is both a coalgebra and a right B-module homomorphism. ∗ Let C be a coalgebra and let C = HomK (C; K) denote the linear dual of C. Then the coalgebra structure of C induces an algebra structure on C∗. Proposition 2.1 If C is a coalgebra, then C∗ is an algebra. Proof. Recall that C is a triple (C; ∆C ; C ) where ∆C : C ! C ⊗ C is K-linear and satisfies the coassociativity property, and C : C ! K is K-linear and satisfies the counit property. The dual map of ∆C is a K-linear map ∗ ∗ ∗ ∆C :(C ⊗ C) ! C ∗ ∗ ∗ ∗ Since C ⊗ C ⊆ (C ⊗ C) , we define the multiplication map of C , denoted as mC∗ , to be the ∗ ∗ ∗ ∗ restriction of ∆C to C ⊗ C .
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