S. Dăscălescu

Lifting of Nichols algebras of typeB 2

We compute liftings of the Nichols algebra of a Yetter-Drinfeld module of Cartan typeB2 subject to the small restriction that the diagnonal elements of the braiding matrix are primitiventh roots of 1 with oddn≠5. As well, we compute the liftings of a Nichols algebra of Cartan typeA2 if the diagonal elements of the braiding matrix are cube roots of 1; this case was not completely covered in previous work of Andruskiewitsch and Schneider. We study the problem of when the liftings of a given Nichols algebra are quasi-isomorphic. The Appendix (with I. Rutherford) contains a generalization of the quantum binomial formula. This formula was used in the computation of liftings of typeB2 but is also of interest independent of these results.

GRADINGS OF MATRIX ALGEBRAS BY CYCLIC GROUPS

Let k be a field and G be a group. In this paper we are concerned with special cases of the general problem of finding the isomorphism types of G-graded algebra structures on the k-algebra of matrices MmðkÞ, where m is a positive integer. We propose an elementary approach, which relies on linear algebra and the duality between group actions and gradings for finite cyclic groups with appropriate roots of unity. In the Hopf algebra language, this duality means that the group algebra of a finite cyclic group is a selfdual Hopf algebra, provided that the field has enough roots of unity. Among the G-gradings of MmðkÞ there are some which are easy to understand, called good gradings. A grading is good if all the matrix units eij; 1 i; j m, are homogeneous elements. In fact a good grading is just ENDðVÞ for some G-graded vector space V of dimension m (see [4, 1]). It is important to study good gradings, since under certain assumptions on k, G and m, any grading is isomorphic to a good one. In this case, the classification of the isomorphism types of G-gradings reduces to the classification of the isomorphism types of good G-gradings, which is a more combinatorial problem. For example if the group G is torsion free, then by using ingredients of graded Clifford

ON GRADINGS OF MATRIX ALGEBRAS AND DESCENT THEORY

We classify gradings on matrix algebras by a finite abelian group. A grading is called good if all elementary matrices are homogeneous. For cyclic groups, all gradings on a matrix algebra over an algebraically closed field are good. We can count the number of good gradings by a cyclic group. Using descent theory, we classify non-good gradings on a matrix algebra that become good after a base extension.

Hopf Algebras of Dimension 14

Let H be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field k of characteristic 0. If H has no nontrivial skew-primitive elements, we find some bounds for the dimension of H_1, the second term in the coradical filtration of H. Using these results, we are able to show that every Hopf algebra of dimension 14 is semisimple and thus isomorphic to a group algebra or the dual of a group algebra. Also a Hopf algebra of dimension pq where p and q are odd primes with p<q and q less than or equal to 1 + 3p, and also less than or equal to 13, is semisimple and thus a group algebra or the dual of a group algebra. We also have some partial results in the classification problem for dimension 16.

Isomorphism of generalized triangular matrix-rings and recovery of tiles

We prove an isomorphism theorem for generalized triangular matrix-rings, over rings having only the idempotents 0and 1, in particular, over indecomposable commutative rings or over local rings (not necessarily commutative). As a consequence, we obtain a recovery result for the tile in a tiled matrix-ring.

Relative Regular Objects in Categories

We define the concept of a regular object with respect to another object in an arbitrary category. We present basic properties of regular objects and we study this concept in the special cases of abelian categories and locally finitely generated Grothendieck categories. Applications are given for categories of comodules over a coalgebra and for categories of graded modules, and a link to the theory of generalized inverses of matrices is presented. Some of the techniques we use are new, since dealing with arbitrary categories allows us to pass to the dual category.

Good Gradings on Upper Block Triangular Matrix Algebras

We investigate group gradings of upper block triangular matrix algebras over a field such that all the matrix units lying there are homogeneous elements. We describe these gradings as endomorphism algebras of graded flags and classify them as orbits of a certain biaction of a Young subgroup and the group G on the set G n , where G is the grading group and n is the size of the matrix algebra. In particular, the results apply to algebras of upper triangular matrices.

Isomorphisms between Morita context rings

Let (R, S, R M S , S N R , f, g) be a general Morita context, and let be the ring associated with this context. Similarly, let be another Morita context ring. We study the set Iso(T, T ′) of ring isomorphisms from T to T ′. Our interest in this problem is motivated by: (i) the problem to determine the automorphism group of the ring T, and (ii) the recovery of the non-diagonal tiles problem for this type of generalized matrix rings. We introduce two classes of isomorphisms from T to T ′, the disjoint union of which is denoted by Iso0(T, T ′). We describe Iso0(T, T ′) by using the ℤ-graded ring structure of T and T ′. Our main result characterizes Iso0(T, T ′) as the set consisting of all semigraded isomorphisms and all anti-semigraded isomorphisms from T to T ′, provided that the rings R′ and S′ are indecomposable and at least one of M′ and N′ is nonzero; in particular, Iso0(T, T ′) contains all graded isomorphisms and all anti-graded isomorphisms from T to T ′. We also present a situation where Iso0(T, T ′) = Iso(T, T ′). This is in the case where R, S, R′ and S′ are rings having only trivial idempotents and all the Morita maps are zero. In particular, this shows that the group of automorphisms of T is completely determined.

Group gradings of M 2 ( K )

We describe all group gradings of the matrix algebra M 2 ( k ), where k is an arbitrary field. We prove that any such grading reduces to a grading of type C 2 , a grading of type C 2 × C 2 , or to a good grading. We give new simple proofs for the description of C 2 -gradings and C 2 × C 2 -gradings on M 2 ( K ).

Group Gradings on M 3(k)

We describe and classify all group gradings on the matrix algebra M 3(k), where k is an arbitrary field. We show that any such grading is either isomorphic to a good grading, for which all the matrix units are homogeneous elements, or reduces to a C 3-grading or to a C 3 × C 3-grading. We show that a grading which is not isomorphic to a good grading is a graded division ring. The isomorphism types of non-good C 3-gradings are in a bijective correspondence to cubic Galois extensions of k. The non-good C 3 × C 3-gradings which do not reduce to C 3-gradings are fine gradings, and their description also depends on cubic Galois extensions of k.