A Lie coalgebra is a coalgebra whose comultiplication Δ : M → M ⊗ M satisfies the Lie conditions. Just as any algebra A whose multiplication ϕ : A ⊗ A → A ... an associated Lie algebra L(A), so any coalgebra C whose comultiplication Δ : C → C ⊗ C is associative gives rise to an associated Lie coalgebra Lc(C). The assignment C ↦ Lc(C) is functorial. A universal coenveloping coalgebra Uc(M) is defined for any Lie Lie coalgebra M by asking for a right adjoint Uc ... functor pair Lc ⊣ Uc. Theorem. For any Lie coalgebra M, the natural mapLc(UcM) → M is surjective ...
Tópico(s): Synthetic Organic Chemistry Methods
1980 - Elsevier BV | Advances in Mathematics
... Lie algebra structure on g, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra T′ on the suspension of g and write the perturbed coalgebra as T″. Suppose, furthermore, given a contraction of ... Lie algebra structure on M, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra S′ on the suspension of M, a Lie algebra twisting cochain from the perturbed coalgebra S″ to the loop Lie algebra L on ...
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2010 - De Gruyter | Forum Mathematicum
Robert G Heyneman, David E. Radford,
... papes we study various finiteness conditions on a coalgebra C and the dual algebra C* of all ... functionals on C. C is a ru$~k~ coalgebra if every finite dirne~jon~ C*-module is ... theorem we show in Section 3 that a coalgebra C is reflexive if the algebra C* contains ... that the cofree coaIgebra I(r’)% the shuffle coalgebra Sh(F) and the ~‘Bir~hoff-~~itt~~ coalgebra of divided powers B(V), are all reAexive ... a finite dimensional vector space. Since any connected coalgebra C may be imbedded in the shufXe coalgebra ...
Tópico(s): Algebraic structures and combinatorial models
1974 - Elsevier BV | Journal of Algebra
... counterpart of a unitary algebra is a unitary coalgebra. The counterpart of an associative algebra is an associative coalgebra. The counterpart of Lie algebra is Lie coalgebra. The underlying vector space, V, of algebra is ... A; the underlying vector space, V, of a coalgebra is denoted by C; the underlying vector space ... and the underlying vector space of a Lie coalgebra is denoted by M. The detailed definition of an associative coalgebra is provided, the associativity of the multiplication of ... defined by taking the defining diagrams for a coalgebra and reversing arrows. The formal, technical device that ...
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2003 - Elsevier BV | Handbook of algebra
... x]), and let M=(W1)∘=the dual Lie coalgebra of W1. By definition (W1)∘ is the largest subspace of (W1)∗ carrying a Lie coalgebra structure induced by the restriction to (W1)∘ of ... is an example of a non-zero Lie coalgebra in which no element of M other than 0 lies in a finite-dimensional sub Lie coalgebra. Thus Loc(M)=0 where Loc(M) denotes ...
Tópico(s): Advanced Topics in Algebra
1990 - Elsevier BV | Journal of Pure and Applied Algebra
... a well-known construction which gives the cofree coalgebra over a vector space as a certain completion of the tensor coalgebra. In the case of a one-dimensional vector space this is the coalgebra of recursive sequences. In this paper, it is ... rings. As should be the case, the cofree coalgebra over a finitely generated free module over a ... representation theorem for coalgebras, that is representing a coalgebra as a subcoalgebra of a matrix-like coalgebra.
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2003 - Elsevier BV | Journal of Pure and Applied Algebra
Xiaowu Chen, Hua-Lin Huang, Pu Zhang,
... the quiver of a bicomodule over a cosemisimple coalgebra. Applying this to the coradical C 0 of an arbitrary coalgebra C, we give an alternative definition of the ... of C. The dual Gabriel theorem for a coalgebra with a separable coradical is obtained, which generalizes the corresponding result for a pointed coalgebra. We also give a new description of C 1 = C 0 ∧C C 0 of any coalgebra C, which can be regarded as a generalization ... of the Gabriel quiver of a quasi-coFrobenius coalgebra.
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2006 - Springer Nature | Science in China Series A Mathematics
Filippo Bonchi, Ugo Montanari,
... the compositionality of the abstract semantics (bisimilarity). Universal Coalgebra provides a categorical framework where bisimilarity can be ... e., as the unique morphism to the final coalgebra. Moreover, if lifting a coalgebra to a structured setting is possible, then bisimilarity ... for every reactive system we can build a coalgebra. Furthermore, if bisimilarity is compositional in the reactive system, then we can lift this coalgebra to a structured coalgebra.
Tópico(s): Advanced Software Engineering Methodologies
2008 - Elsevier BV | Electronic Notes in Theoretical Computer Science
... ik, k) when C is a nice fc-coalgebra.Throughout this note we employ a theory of ... definition (2.2) of niceness for a homology coalgebra C comes essentially from Moore-Smith [12, §4] ... and we show (4.1) that a homology coalgebra Cis nice if and only if £n£(C) = ... e.Chas primitive dimension^ 1.For a homology coalgebra C over a field k, £*£(C) is closely ... k be a fixed field.A homology k-coalgebra is a connected positively graded A>coalgebra with commutative comultiplication [11].We let tfjk denote ...
Tópico(s): Advanced Algebra and Logic
1970 - American Mathematical Society | Transactions of the American Mathematical Society
... ik, k) when C is a nice fc-coalgebra.Throughout this note we employ a theory of ... definition (2.2) of niceness for a homology coalgebra C comes essentially from Moore-Smith [12, §4] ... and we show (4.1) that a homology coalgebra Cis nice if and only if £n£(C) = ... e.Chas primitive dimension^ 1.For a homology coalgebra C over a field k, £*£(C) is closely ... k be a fixed field.A homology k-coalgebra is a connected positively graded A>coalgebra with commutative comultiplication [11].We let tfjk denote ...
Tópico(s): Homotopy and Cohomology in Algebraic Topology
1970 - American Mathematical Society | Transactions of the American Mathematical Society
Ángel Ballesteros, Alfonso Blasco, Francisco J. Herranz, Fabio Musso, O. Ragnisco,
The coalgebra approach to the construction of classical integrable systems from Poisson coalgebras is reviewed, and the essential role ... of Hamiltonians with either undeformed or q-deformed coalgebra symmetry are given, and their Liouville superintegrability is ... are analysed in detail. Further generalizations of the coalgebra approach that make use of comodule and loop algebras are presented. The generalization of such a coalgebra symmetry framework to quantum mechanical systems is straightforward.
Tópico(s): Advanced Topics in Algebra
2009 - IOP Publishing | Journal of Physics Conference Series
... every tree t (an element of the final coalgebra) turns out to represent a new coalgebra At. The universal property of this family of ... of every system S there exists a unique coalgebra homomorphism from a unique At which takes the ... underlying objects are λ-presentable. Consequently, every F coalgebra is a λ-filtered colimit of λ-presentable ...
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2003 - Elsevier BV | Theoretical Computer Science
Ekaterina Komendantskaya, John Power,
... logic program induces a P f P f -coalgebra on the set of atomic formulae in the program. The coalgebra p sends an atomic formula A to the ... logic program with a P f P f -coalgebra on Set and showed that, if C(P ... logic program P qua P f P f -coalgebra, the corresponding C(P f P f )-coalgebra structure describes the parallel and-or derivation trees ...
Tópico(s): Formal Methods in Verification
2011 - Springer Science+Business Media | Lecture notes in computer science
Jiřı́ Adámek, Stefan Milius, Jiřı́ Velebil,
... a number of properties suitable for algebra and coalgebra: every endofunctor is set-based, it has an initial algebra and a terminal coalgebra, the categories of algebras and coalgebras are complete ... completely iterative monad. A description of a terminal coalgebra for the power-set functor is provided.
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2004 - Elsevier BV | Theoretical Computer Science
Lawrence S. Moss, Ignacio Viglizzo,
This paper connects coalgebra with a long discussion in the foundations of game theory on the modeling of type spaces. We argue that ... have not been explored so deeply in the coalgebra literature, we have some new results. We show ... the probability measure space functor has a final coalgebra. Moreover, we construct this final coalgebra from the relevant version of coalgebraic modal logic. ... and endow this set with a measurable and coalgebra structure.
Tópico(s): Economic theories and models
2004 - Elsevier BV | Electronic Notes in Theoretical Computer Science
Sava Krstić, John Launchbury, Duško Pavlović,
... between processes can be understood in terms of coalgebra homomorphisms, with homomorphisms to the final coalgebra exactly identifying bisimilar processes. The elements of the final coalgebra are thus natural representatives of bisimilarity classes, and ... of processes can be developed in a final-coalgebra- enriched category where arrows are processes, canonically represented. ... we describe a general framework for building final- coalgebra-enriched categories. Every such category is constructed from ...
Tópico(s): Logic, Reasoning, and Knowledge
2001 - Springer Science+Business Media | Lecture notes in computer science
... crossed product of an al¬gebra by a coalgebra C, which generalises the notion of a crossed ... find the necessary and sufficient conditions for two coalgebra crossed products be equivalent. We show that the two-dimensional quantum Euclidean group is a coalgebra crossed product. The paper is completed with an appendix describing the dualisation of construction of coalgebra crossed products.
Tópico(s): Homotopy and Cohomology in Algebraic Topology
1997 - Taylor & Francis | Communications in Algebra
J.R.B. Cockett, Jean-Simon Pacaud Lemay,
... A , in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the ... sense, is always a calculus category. When the coalgebra modality is monoidal, it is natural to demand an extra coherence between integration and the coalgebra modality. In the presence of this extra coherence, ... show that a calculus category with a monoidal coalgebra modality has its integral transformation given by antiderivatives ... calculus categories based on both monoidal and (mere) coalgebra modalities are presented. In addition, differential categories which ...
Tópico(s): Logic, programming, and type systems
2018 - Cambridge University Press | Mathematical Structures in Computer Science
Richard Blute, J.R.B. Cockett, Jean-Simon Pacaud Lemay, R. A. G. Seely,
... particularly relevant to linear logic settings, where the coalgebra modality is monoidal and the Seely isomorphisms give ... of differentiation, in the presence of a monoidal coalgebra modality, are completely equivalent. Thus, for linear logic ... also presents a number of separating examples for coalgebra modalities including examples which are and are not ... a differential category although they provide a monoidal coalgebra modality. On the other hand, Rota–Baxter algebras— ... a differential category which has a non-monoidal coalgebra modality.
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2019 - Springer Science+Business Media | Applied Categorical Structures
... K of characteristic different from 2, the Lie coalgebra dual to the Lie algebra of derivations of ... is algebraically closed, the structure of that Lie coalgebra is determined explicitly.
Tópico(s): Nonlinear Waves and Solitons
1990 - Elsevier BV | Journal of Pure and Applied Algebra
Coalgebra automata, introduced by the second author, generalize the well-known automata that operate on infinite words/ ... this paper we prove a complementation lemma for coalgebra automata. More specifically, we provide a construction that transforms a given coalgebra automaton with parity acceptance condition into a device ...
Tópico(s): Formal Methods in Verification
2009 - Springer Science+Business Media | Lecture notes in computer science
Sebastian Nowak, Daniel Simson,
ABSTRACT Let C be an indecomposable hereditary K-coalgebra, where K is an algebraically closed field. We ... of Science ®] , [Google Scholar]) with a path K-coalgebra , where Q is a pure semisimple locally Dynkin ... Corollaries 2.5 and 2.6 a K-coalgebra analogue of Gabriel's theorem [11] Gabriel, P. ... algebras (see also [[6] Doi, Y. 1981. Homological Coalgebra. J. Math. Soc. Japan, 33: 31–50. [Crossref], [ ...
Tópico(s): Advanced Topics in Algebra
2002 - Taylor & Francis | Communications in Algebra
Ángel Ballesteros, Alberto Enciso, Francisco J. Herranz, O. Ragnisco,
... family of ND spaces endowed with sl(2)-coalgebra symmetry is introduced. For all these spaces the ... integrals is obtained from the underlying sl(2)-coalgebra structure. In particular, ND spherically symmetric spaces with Euclidean signature are shown to be sl(2)-coalgebra spaces. As a byproduct of this construction we ...
Tópico(s): Advanced Differential Geometry Research
2007 - Elsevier BV | Physics Letters B
Pascual Jara, Luis Merino, Gabriel Navarro, Juan Francisco Ruíz López,
... colocalizing subcategories of a comodule category of a coalgebra C over a field, using the correspondence between ... coalgebras and obtain, in particular, that the "localized" coalgebra of a path coalgebra is again a path coalgebra.
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2006 - Taylor & Francis | Communications in Algebra
... et$ . Thus we can associate with every T-coalgebra A a certain filter-coalgebra $A_{\mathbb F}$ . Precisely, when T (weakly) preserves ... functor ${\mathbb P}$ , thus providing for every T-coalgebra A a Kripke structure $A_{\mathbb P}$ . We ... properties of T. In particular, for any T-coalgebra ${\mathcal A}$ there is a transition system ${\mathcal ...
Tópico(s): Geometric and Algebraic Topology
2005 - Springer Science+Business Media | Lecture notes in computer science
Abstract The notions of group coalgebra Galois extension and group entwining structure are defined. It is proved that any group coalgebra Galois extension induces a unique group-entwining map ... and Hajac [Brzeziński, T., Hajac, P. M. (1999). Coalgebra extensions and algebra coextensions of Galois type. Comm. ...
Tópico(s): Nonlinear Waves and Solitons
2004 - Taylor & Francis | Communications in Algebra
Juan Cuadra, José Gómez-Torrecillas,
... over an algebraically closed field k the basic coalgebra of a serial indecomposable coalgebra is a subcoalgebra of a path coalgebra kΓ where the quiver Γ is either a ... direct sum of subcoalgebras of the divided power coalgebra.
Tópico(s): Nonlinear Waves and Solitons
2004 - Elsevier BV | Journal of Pure and Applied Algebra
... of Tannaka duality include the reconstruction of a coalgebra in the category of vector spaces from its ... equivalent to the category of representations of some coalgebra. This paper generalizes these results by replacing the ... to the category of representations of some R-coalgebra. In particular we construct an R-coalgebra whose category of representations is equivalent to the ...
Tópico(s): Advanced Topics in Algebra
2000 - Elsevier BV | Advances in Mathematics
The concepts of a K-coalgebra C of tame comodule type and of wild comodule type over an algebraically closed field K are introduced in Simson [Colloq. ... present paper, we introduce the concepts fc-tame coalgebra and fc-wild coalgebra, and we prove that any Homcomputable coalgebra over an algebraically closed field K is either ...
Tópico(s): Rings, Modules, and Algebras
2008 - Wiley | Journal of the London Mathematical Society
Florencio Castaño Iglesias, S. Dăscălescu, C. Năstăsescu,
... ring with local units on a co-Frobenius coalgebra. We study a special class of co-Frobenius ... call symmetric coalgebras. We prove that any semiperfect coalgebra can be embedded in a symmetric coalgebra. We use a dual version of Brauer's ... with local units constructed from a co-Frobenius coalgebra, which we call the Nakayama automorphism. This is ...
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2004 - Elsevier BV | Journal of Algebra