arXiv subjects

Ana-Loredana Agore

Publications and source records attributed to Ana-Loredana Agore.

4 recordsLinked to original sources

arXiv2018

Hopf Algebras which Factorize through the Taft Algebra $T_{m^{2}}(q)$ and the Group Hopf Algebra $K[C_{n}]$

We completely describe by generators and relations and classify all Hopf algebras which factorize through the Taft algebra $T_{m^{2}}(q)$ and the group Hopf algebra $K[C_{n}]$: they are $nm^{2}$-dimensional quantum groups $T_{nm^{2}}^ ω(q)$ associated to an $n$-th root of unity $ω$. Furthermore, using Dirichlet's prime number theorem we are able to count the number of isomorphism types of such Hopf algebras. More precisely, if $d = {\rm gcd}(m,ν(n))$ and $\frac{ν(n)}{d} = p_1^{α_1} \cdots p_r^{α_r}$ is the prime decomposition of $\frac{ν(n)}{d}$ then the number of types of Hopf algebras that factorize through $T_{m^{2}}(q)$ and $K[C_n]$ is equal to $(α_1 + 1)(α_2 + 1) \cdots (α_r + 1)$, where $ν(n)$ is the order of the group of $n$-th roots of unity in $K$. As a consequence of our approach, the automorphism groups of these Hopf algebras are described as well.

math.RA↗

arXiv2014

Bicrossed Products, Matched Pair Deformations and the Factorization Index for Lie Algebras

For a perfect Lie algebra $\mathfrak{h}$ we classify all Lie algebras containing $\mathfrak{h}$ as a subalgebra of codimension $1$. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product $\mathfrak{h} \ltimes (k^* \times {\rm Aut}_{\rm Lie} (\mathfrak{h}))$. In the non-perfect case the classification of these Lie algebras is a difficult task. Let $\mathfrak{l} (2n+1, k)$ be the Lie algebra with the bracket $[E_i, G] = E_i$, $[G, F_i] = F_i$, for all $i = 1, \dots, n$. We explicitly describe all Lie algebras containing $\mathfrak{l} (2n+1, k)$ as a subalgebra of codimension $1$ by computing all possible bicrossed products $k \bowtie \mathfrak{l} (2n+1, k)$. They are parameterized by a set of matrices ${\rm M}_n (k)^4 \times k^{2n+2}$ which are explicitly determined. Several matched pair deformations of $\mathfrak{l} (2n+1, k)$ are described in order to compute the factorization index of some extensions of the type $k \subset k \bowtie \mathfrak{l} (2n+1, k)$. We provide an example of such extension having an infinite factorization index.

math.RA↗

arXiv2014

The Classification of All Crossed Products $H_4 \# k[C_{n}]$

Using the computational approach introduced in [Agore A.L., Bontea C.G., Militaru G., J. Algebra Appl. 12 (2013), 1250227, 24 pages, arXiv:1207.0411] we classify all coalgebra split extensions of $H_4$ by $k[C_n]$, where $C_n$ is the cyclic group of order $n$ and $H_4$ is Sweedler's $4$-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras $H_4 \# k[C_{n}]$ by explicitly computing two classifying objects: the cohomological 'group' ${\mathcal H}^{2} ( k[C_{n}], H_4)$ and $\text{CRP}( k[C_{n}], H_4):=$ the set of types of isomorphisms of all crossed products $H_4 \# k[C_{n}]$. More precisely, all crossed products $H_4 \# k[C_n]$ are described by generators and relations and classified: they are $4n$-dimensional quantum groups $H_{4n, λ, t}$, parameterized by the set of all pairs $(λ, t)$ consisting of an arbitrary unitary map $t : C_n \to C_2$ and an $n$-th root $λ$ of $\pm 1$. As an application, the group of Hopf algebra automorphisms of $H_{4n, λ, t}$ is explicitly described.

math.RA↗

arXiv2009

Crossed Product of Cyclic Groups

All crossed products of two cyclic groups are explicitly described using generators and relations. A necessary and sufficient condition for an extension of a group by a group to be a cyclic group is given.

math.GR↗