Isomorphism problems for Hopf–Galois structures on separable field extensions

Let $L/K$ be a finite separable extension of fields whose Galois closure $E/K$ has Galois group G. Greither and Pareigis use Galois descent to show that a Hopf algebra giving a Hopf–Galois structure on $L/K$ has the form $E{[N]}^G$ for some group N of order $[L:K]$. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as K-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and K-algebras that appear in the classification of Hopf–Galois structures on a cyclic extension of degree $p^n$, for p an odd prime number.     

On actions of Drinfel'd doubles on finite dimensional algebras

Let q be an nth root of unity for $n>2$ and let $T_n(q)$ be the Taft (Hopf) algebra of dimension $n^2$. In 2001, Susan Montgomery and Hans-Jürgen Schneider classified all non-trivial $T_n(q)$-module algebra structures on an n-dimensional associative algebra A. They further showed that each such module structure extends uniquely to make A a module algebra over the Drinfel'd double of $T_n(q)$. We explore what it is about the Taft algebras that leads to this uniqueness, by examining actions of (the Drinfel'd double of) Hopf algebras H “close” to the Taft algebras on finite-dimensional algebras analogous to A above. Such Hopf algebras H include the Sweedler (Hopf) algebra of dimension 4, bosonizations of quantum linear spaces, and the Frobenius–Lusztig kernel $u_q({\mathfrak{sl}}_2)$.     

On finite-dimensional copointed Hopf algebras over dihedral groups

We classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero such that its coradical is isomorphic to the algebra of functions ${\mathbb{k}}^{{\mathbb{D}}_m}$ over a dihedral group ${\mathbb{D}}_m$, with $m=4a\ge 12$. We obtain this classification by means of the lifting method, where we use cohomology theory to determine all possible deformations. Our result provides an infinite family of new examples of finite-dimensional copointed Hopf algebras over dihedral groups.     

Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies

We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra $T_p$ for any integer $p>2$ which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of $T_p$, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. Also, we find a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber's original formula for Hochschild cohomology.     

On the automorphisms of quantum Weyl algebras

Motivated by Weyl algebra analogues of the Jacobian conjecture and the tame generators problem, we prove quantum versions of these problems for a family of analogues to the Weyl algebras. In particular, our results cover the Weyl–Hayashi algebras and tensor powers of a quantization of the first Weyl algebra which arises as a primitive factor algebra of $U_q^{+}({\mathfrak{so}}_5)$.     

The classification of 3-dimensional noetherian cubic Calabi–Yau algebras

It is known that every 3-dimensional noetherian Calabi–Yau algebra generated in degree 1 is isomorphic to a Jacobian algebra of a superpotential. Recently, S.P. Smith and the first author classified all superpotentials whose Jacobian algebras are 3-dimensional noetherian quadratic Calabi–Yau algebras. The main result of this paper is to classify all superpotentials whose Jacobian algebras are 3-dimensional noetherian cubic Calabi–Yau algebras. As an application, we show that if S is a 3-dimensional noetherian cubic Calabi–Yau algebra and σ is a graded algebra automorphism of S, then the homological determinant of σ can be calculated by the formula $\mathrm{hdet}\sigma ={(\det \sigma )}^2$ with one exception.     

Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension p3 and pq2

Let p and q be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension $p^3$ and of dimension $p{q}^2$. We obtain that the $p+1$ non-isomorphic self-dual semisimple Hopf algebras of dimension $p^3$ classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac–Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension $p^3$, established by the third-named author in an appendix.     

Basic r-symmetric tropical polynomials

We consider tropical polynomials in nr variables, divided into n blocks of r variables, and especially r-symmetric tropical polynomials, which are invariant under the action of the symmetric group $S_n$ on the blocks. We define a set of basic r-symmetric tropical polynomials and show that the basic 2-symmetric tropical polynomials give coordinates on ${\mathbb{R}}^{2n}/S_n$ more efficiently than known polynomials. Moreover, we present special cases for $r\ge 3$ where the basic polynomials separate orbits.     

The structure of DGA resolutions of monomial ideals

Let $I?\mathbb{k}[x_1,\dots ,x_n]$ be a squarefree monomial ideal in a polynomial ring. In this paper we study multiplications on the minimal free resolution $\mathbb{F}$ of $\mathbb{k}[x_1,\dots ,x_n]/I$. In particular, we characterize the possible vectors of total Betti numbers for such ideals which admit a differential graded algebra (DGA) structure on $\mathbb{F}$. We also show that under these assumptions the maximal shifts of the graded Betti numbers are subadditive.On the other hand, we present an example of a strongly generic monomial ideal which does not admit a DGA structure on its minimal free resolution. In particular, this demonstrates that the Hull resolution and the Lyubeznik resolution do not admit DGA structures in general.Finally, we show that it is enough to modify the last map of $\mathbb{F}$ to ensure that it admits the structure of a DG algebra.     

Boolean constant degree functions on the slice are juntas

We show that a Boolean degree $d$ function on the “slice” $\left(\genfrac{}{}{0ex}{}{\left[n\right]}{k}\right)?\{(x_1,\dots ,x_n)\in \{0,1\}:{\sum }_{i=1}^n{x}_i=k\}$ is a junta (depends on a constant number $\gamma \left(d\right)$ of coordinates), assuming that $k,n?k$ are large enough. This generalizes a classical result of Nisan and Szegedy on the hypercube ${\{0,1\}}^n$. Moreover, we show that the maximum number of coordinates that a Boolean degree $d$ function can depend on is the same on the slice and on the hypercube.     

Gaps in the number of generators of monomial Togliatti systems

Let $I_{d,n}?k[x_0,?,x_n]$ be a minimal monomial Togliatti system of forms of degree d. In [4], Mezzetti and Miró-Roig proved that the minimal number of generators $\mu (I_{d,n})$ of $I_{d,n}$ lies in the interval $[2n+1,\left(\begin{array}{c}n+d?1\\ n?1\end{array}\right)]$. In this paper, we prove that for $n\ge 4$ and $d\ge 3$, the integer values in $[2n+3,3n?1]$ cannot be realized as the number of minimal generators of a minimal monomial Togliatti system. We classify minimal monomial Togliatti systems $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{d,n})=2n+2$ or 3n (i.e. with the minimal number of generators reaching the border of the non-existence interval). Finally, we prove that for $n=4$, $d\ge 3$ and $\mu \in [9,\left(\begin{array}{c}d+3\\ 3\end{array}\right)]?\{11\}$ there exists a minimal monomial Togliatti system $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{n,d})=\mu$.     

Linkage of sets of cyclic algebras

Let p be a prime integer and F the function field in two algebraically independent variables over a smaller field $F_0$. We prove that if $\mathrm{char}(F_0)=p?3$ then there exist $p^2?1$ cyclic algebras of degree p over F that have no maximal subfield in common, and if $\mathrm{char}(F_0)=0$ then there exist $p^2$ cyclic algebras of degree p over F that have no maximal subfield in common.     

The Frobenius complexity of Hibi rings

We study the Frobenius complexity of Hibi rings over fields of characteristic $p>0$. In particular, for a certain class of Hibi rings (which we call ${\omega }^{(?1)}$-level), we compute the limit of the Frobenius complexity as $p\to \infty$.