Miodrag Cristian Iovanov

Generalized Frobenius Algebras and Hopf Algebras

”Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. Recently, it was shown in [I] that they admit left-right symmetric characterizations analogue to those of Frobenius algebras: a coalgebra C is co-Frobenius if and only if it is isomorphic to its rational dual. We consider the more general quasi-co-Frobenius (QcF) coalgebras; in the first main result we show that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or equivalently right) rational dual Rat(C∗), in the sense that certain coproduct or product powers of these objects are isomorphic. These show that QcF coalgebras can be viewed as generalizations of both co-Frobenius coalgebras and Frobenius algebras. Surprisingly, these turn out to have many applications to fundamental results of Hopf algebras. The equivalent characterizations of Hopf algebras with left (or right) nonzero integrals as left (or right) co-Frobenius, or QcF, or semiperfect or with nonzero rational dual all follow immediately from these results. Also, the celebrated uniqueness of integrals follows at the same time as just another equivalent statement. Moreover, as a by-product of our methods, we observe a short proof for the bijectivity of the antipode of a Hopf algebra with nonzero integral. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, the results on coalgebras allow the introduction of a general concept of Frobenius algebra, which makes sense for infinite dimensional and topological algebras, and specializes to the classical notion in the finite case: this will be a topological algebra A which is isomorphic to its complete topological dual A∨. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer q-homological algebra, topology or graph theory, showing the importance of the concept. Introduction A K algebra A over a field K is called Frobenius if A is isomorphic to A∗ as right Amodules. This is equivalent to there being an isomorphism of left A-modules between A and A∗. This is the modern algebra language formulation for an old question posed by Frobenius. Given a finite dimensional algebra with a basis x1, . . . , xn, the left multiplication by an element a induces a representation A 7→ EndK(A) = Mn(K), a 7→ (aij)i,j (aij ∈ K), where a · xi = n ∑ j=1 aijxj . Similarly, the right multiplication produces a matrix aij by writing xi · a = n ∑ j=1 ajixj , a ′ ij ∈ K, and this induces another representation A 3 a 7→ (aij)i,j . Frobenius’ problem came as the natural question of when the two representations are equivalent. Frobenius algebras occur in many different fields of mathematics, such as topology (the cohomology ring of a compact oriented manifold with coefficients in a field is a Frobenius algebra by Poincare duality), topological quantum field theory (there is a one-to-one correspondence between 2-dimensional quantum field theories and

The Splitting Problem for Coalgebras: A Direct Approach

In this note we give a new and elementary proof of a result of Năstăsescu and Torrecillas (J. Algebra, 281:144–149, 2004) stating that a coalgebra C is finite dimensional if and only if the rational part of any right module M over the dual algebra

Triangular matrix coalgebras and applications

C^*

When ∏ is Isomorphic to ⊕

is a direct summand in M (the splitting problem for coalgebras).

Infinite-dimensional diagonalization and semisimplicity

We formally introduce and study generalized comatrix coalgebras and upper triangular comatrix coalgebras, which are not only a dualization but also an extension of classical generalized matrix algebras. We use these to answer several open questions on Noetherian and Artinian type notions in the theory of coalgebras, and to give complete connections between these. We also solve completely the so called finite splitting problem for coalgebras: we show that is a coalgebra such that the rational part of every left finitely generated -module splits off if and only if is an upper triangular matrix coalgebra, for a serial coalgebra whose Ext-quiver is a finite union of cycles, a finite dimensional coalgebra and a finite dimensional bicomodule .

Generators for comonoids and universal constructions

For a category 𝒞, we investigate the problem of when the coproduct ⊕ and the product functor ∏ from 𝒞 I to 𝒞 are isomorphic for a fixed set I, or, equivalently, when the two functors are Frobenius functors. We show that for an Ab category 𝒞 this happens if and only if the set I is finite (and even in a much general case, if there is a morphism in 𝒞 that is invertible with respect to addition). However, we show that ⊕ and ∏ are always isomorphic on a suitable subcategory of 𝒞 I which is isomorphic to 𝒞 I but is not a full subcategory. If 𝒞 is only a preadditive category, then we give an example that shows that the two functors can be isomorphic for infinite sets I. For the module category case, we provide a different proof to display an interesting connection to the notion of Frobenius corings.

Path subcoalgebras, finiteness properties and quantum groups

We characterize the diagonalizable subalgebras of End(V), the full ring of linear operators on a vector space V over a field, in a manner that directly generalizes the classical theory of diagonalizable algebras of operators on a finite-dimensional vector space. Our characterizations are formulated in terms of a natural topology (the “finite topology”) on End(V), which reduces to the discrete topology in the case where V is finite-dimensional. We further investigate when two subalgebras of operators can and cannot be simultaneously diagonalized, as well as the closure of the set of diagonalizable operators within End(V). Motivated by the classical link between diagonalizability and semisimplicity, we also give an infinite-dimensional generalization of the Wedderburn–Artin theorem, providing a number of equivalent characterizations of left pseudocompact, Jacoboson semisimple rings that parallel various characterizations of artinian semisimple rings. This theorem unifies a number of related results in the literature, including the structure of linearly compact, Jacobson semsimple rings and cosemisimple coalgebras over a field.

Frobenius Extensions of Corings

We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We find concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. This answers an open question in Agore (Proc Am Math Soc 139:855–863, 2011) on the existence of a cofree coring, and constructs the cofree (co)module coalgebra on a B-(co)module, for a bialgebra B.

Quiver algebras, path coalgebras and coreflexivity

We study subcoalgebras of path coalgebras that are spanned by paths (called path subcoalgebras) and subcoalgebras of incidence coalgebras, and propose a unifying approach for these classes. We discuss the left quasi-co-Frobenius and the left co-Frobenius properties for these coalgebras. We classify the left co-Frobenius path subcoalgebras, showing that they are direct sums of certain path subcoalgebras arising from the infinite line quiver or from cyclic quivers. We also discuss the coreflexive property for the considered classes of coalgebras. Finally, we investigate which of the co-Frobenius path subcoalgebras can be endowed with Hopf algebra structures, in order to produce some quantum groups with non-zero integrals, and classify all these structures over a field with primitive roots of unity of any order. These turn out to be liftings of quantum lines over certain not necessarily abelian groups.

The Bijectivity of the Antipode Revisited

Let 𝒞 and 𝒟 be two corings over a ring A and be a morphism of corings. We investigate the situation when the associated induced (“corestriction of scalars”) functor ℳ 𝒞→ ℳ 𝒟 is a Frobenius functor, and call these morphisms Frobenius extensions of corings. The characterization theorem generalizes notions such as Frobenius corings and is applied to several situations; in particular, provided some (general enough) flatness conditions hold, the notion proves to be dual to that of Frobenius extensions of rings (algebras). Several finiteness theorems are given for each case we consider; these theorems extend existing results from Frobenius extensions of rings or from Frobenius corings, showing that a certain finiteness property almost always occur for many instances of Frobenius functors.