A Functorial Approach to Gabriel k -quiver Constructions for Coalgebras and Pseudocompact Algebras
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A Functorial Approach to Gabriel k-quiver Constructions for Coalgebras and Pseudocompact Algebras Kostiantyn Iusenko1
· John William MacQuarrie2
· Samuel Quirino1,2
Received: 8 April 2020 / Accepted: 21 July 2020 © Sociedade Brasileira de Matemática 2020
Abstract We define the path coalgebra and Gabriel quiver constructions as functors between the category of k-quivers and the category of pointed k-coalgebras, for k a field. We define a congruence relation on the coalgebra side, show that the functors above respect this relation, and prove that the induced Gabriel k-quiver functor is left adjoint to the corresponding path coalgebra functor. We dualize, obtaining adjoint pairs of functors (contravariant and covariant) for pseudocompact algebras. Using these tools we describe precisely to what extent presentations of coalgebras and pseudocompact algebras in terms of path objects are unique, giving an application to homogeneous algebras. Keywords Adjoint functors · Path coalgebra · Complete path algebra · Gabriel k-quiver.
1 Introduction Let k be a field. A (k-)coalgebra is defined in the monoidal category of k-vector spaces by axioms dual to those of an associative, unital k-algebra. Coalgebras have been studied extensively since their introduction for at least two reasons: Firstly, they form “half the structure” of Hopf algebras, whose applications range from group theory to physics (we refer to Abe (1980); D˘asc˘alescu et al. (2001); Montgomery (1993) and
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Samuel Quirino [email protected] Kostiantyn Iusenko [email protected] http://www.ime.usp.br/∼iusenko John William MacQuarrie [email protected]
1
Instituto de Matemática e Estatística, Univ. de São Paulo, São Paulo, SP, Brazil
2
Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil
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references therein). And secondly, due to the fact that coalgebras have very strong finiteness properties, making them a natural context in which to generalize concepts and results from finite dimensional algebras and their representations (e.g. Green 1976; Simson 2004, 2011, and the references therein). While the formal properties of coalgebras are very pleasant to work with, explicit calculations can be unwieldy. For this reason, a standard trick when working with a coalgebra C is to pass to its vector space dual C ∗ , which inherits naturally the structure of a topological, associative, unital k-algebra. The class of algebras dual to the class of coalgebras is precisely the class of pseudocompact algebras (Brumer 1966; Simson 2001), and thus understanding pseudocompact algebras and their representations provides useful tools when working with coalgebras. But pseudocompact algebras are of independent interest, appearing for example as completed group algebras of profinite groups, so that the understanding of their structure and representations has applications in Galois theory, finite group theory, algebraic geometry and more. The combinatorial approach to the representation theory of finite dimensional algebras begins with two fundame
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