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Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally nilpotent representations

Daniel Simson (2007)

Colloquium Mathematicae

We prove that the study of the category C-Comod of left comodules over a K-coalgebra C reduces to the study of K-linear representations of a quiver with relations if K is an algebraically closed field, and to the study of K-linear representations of a K-species with relations if K is a perfect field. Given a field K and a quiver Q = (Q₀,Q₁), we show that any subcoalgebra C of the path K-coalgebra K◻Q containing K ◻ Q ₀ ⊕ K ◻ Q ₁ is the path coalgebra K ◻ ( Q , ) of a profinite bound quiver (Q,), and the category C-Comod...

Perfect rings for which the converse of Schur's lemma holds.

Abdelfattah Haily, Mostafa Alaoui (2001)

Publicacions Matemàtiques

If M is a simple module over a ring R then, by the Schur's lemma, the endomorphism ring of M is a division ring. However, the converse of this result does not hold in general, even when R is artinian. In this short note, we consider perfect rings for which the converse assertion is true, and we show that these rings are exactly the primary decomposable ones.

Phantom maps and purity in modular representation theory, I

D. Benson, G. Gnacadja (1999)

Fundamenta Mathematicae

Let k be a field and G a finite group. By analogy with the theory of phantom maps in topology, a map f : M → ℕ between kG-modules is said to be phantom if its restriction to every finitely generated submodule of M factors through a projective module. We investigate the relationships between the theory of phantom maps, the algebraic theory of purity, and Rickard's idempotent modules. In general, adding one to the pure global dimension of kG gives an upper bound for the number of phantoms we need...

Piecewise hereditary algebras under field extensions

Jie Li (2021)

Czechoslovak Mathematical Journal

Let A be a finite-dimensional k -algebra and K / k be a finite separable field extension. We prove that A is derived equivalent to a hereditary algebra if and only if so is A ⊗ k K .

P-injective group rings

Liang Shen (2020)

Czechoslovak Mathematical Journal

A ring R is called right P-injective if every homomorphism from a principal right ideal of R to R R can be extended to a homomorphism from R R to R R . Let R be a ring and G a group. Based on a result of Nicholson and Yousif, we prove that the group ring RG is right P-injective if and only if (a) R is right P-injective; (b) G is locally finite; and (c) for any finite subgroup H of G and any principal right ideal I of RH , if f ∈ Hom R ( I R , R R ) , then there exists g ∈ Hom R ( RH R , R R ) such that g | I = f . Similarly, we also obtain equivalent characterizations...

Poincaré duality and commutative differential graded algebras

Pascal Lambrechts, Don Stanley (2008)

Annales scientifiques de l'École Normale Supérieure

We prove that every commutative differential graded algebra whose cohomology is a simply-connected Poincaré duality algebra is quasi-isomorphic to one whose underlying algebra is simply-connected and satisfies Poincaré duality in the same dimension. This has applications in rational homotopy, giving Poincaré duality at the cochain level, which is of interest in particular in the study of configuration spaces and in string topology.

Poisson-Lie groupoids and the contraction procedure

Kenny De Commer (2015)

Banach Center Publications

On the level of Lie algebras, the contraction procedure is a method to create a new Lie algebra from a given Lie algebra by rescaling generators and letting the scaling parameter tend to zero. One of the most well-known examples is the contraction from 𝔰𝔲(2) to 𝔢(2), the Lie algebra of upper-triangular matrices with zero trace and purely imaginary diagonal. In this paper, we will consider an extension of this contraction by taking also into consideration the natural bialgebra structures on these...

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