×

Basic coalgebras. (English) Zbl 0920.16018

Chari, Vyjayanthi (ed.) et al., Modular interfaces. Modular Lie algebras, quantum groups, and Lie superalgebras. A conference in honor of Richard E. Block on the occassion of his retirement, Riverside, CA, USA, February 18–20, 1995. Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 4, 41-47 (1997).
A coalgebra \(B\) over a field \(k\) is said to be basic if every simple subcoalgebra of \(B\) is the dual of a finite-dimensional division algebra over \(k\). This is the case (Theorem 2.4) for the basic coalgebra \(B\) of any coalgebra \(C\), which is defined as the co-endomorphism coalgebra of the minimal injective cogenerator of the category of left \(C\)-comodules. Moreover, as a consequence of the theory developed by M. Takeuchi [J. Fac. Sci., Univ. Tokyo, Sect. I A 24, 629-644 (1977; Zbl 0385.18007)], \(C\) is Morita equivalent to its associated basic coalgebra \(B\). As a consequence, if \(k\) is algebraically closed, then the Ext quiver \(\Gamma(C)\) occurs as the quiver \(\Gamma(B)\) of the pointed coalgebra \(B\); and this last can be described in terms of group-like and non-trivial skew-primitive elements. A description of the Ext quiver is given in a non-trivial example. From the introduction: ‘We then discuss the path coalgebra \(C(\Gamma)\) of an arbitrary quiver \(\Gamma\). We prove that, when \(k\) is algebraically closed, any coalgebra \(C\) is equivalent to a “large” subcoalgebra of \(C(\Gamma(C))\); this is the analog of a well-known fact about finite-dimensional algebras due to Gabriel’. To be exact, this ‘large subcoalgebra’ contains the degree one piece \(C(\Gamma(C))_1\) of the coradical filtration (Theorem 4.3).
For the entire collection see [Zbl 0899.00030].

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)

Citations:

Zbl 0385.18007
PDFBibTeX XMLCite