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Acknowledgements: We are grateful to the anonymous reviewer for their helpful corrections that have improved the clarity of our exposition. The second author was supported by a London Mathematical Society Early Career Fellowship at the University of Oxford., Let $A$ be a finite-dimensional algebra over a field of characteristic $p>0$. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective $A$--modules $P$ into those of the torsion submodules of $P$. As an application, we show that blocks of both the classical and quantum Schur algebras $S(2,r)$ and $S_q(2,r)$ are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain $2p^k$ simple modules for some $k$.

Let k be an algebraically closed field of characteristic p > 0 . We show that if X ⊆ P k n is an equidimensional subscheme with Hilbert–Kunz multiplicity less than λ at all points x ∈ X , then for a general hyperplane H ⊆ P k n , the Hilbert–Kunz multiplicity of X ∩ H is less than λ at all points x ∈ X ∩ H . This answers a conjecture and generalizes a result of Carvajal-Rojas, Schwede and Tucker, whose conclusion is the same as ours when X ⊆ P k n is normal. In the process, we substantially generalize certain uniform estimates on Hilbert–Kunz multiplicities of fibers of maps obtained by the aforementioned authors that should be of independent interest.

Motivated by the recent progress towards classification of simple finite-dimensional Lie algebras over an algebraically closed field of characteristic 2, we investigate such 15-dimensional Skryabin algebras.

We introduce the principal representation category $\mathscr{O}({\bf G})$ of reductive algebraic groups with Frobenius maps and put forward a conjecture that this category is a highest weight category. When $\Bbbk$ is complex field $\mathbb{C}$, we provide some evidences of this conjecture. We also study certain kind of bound quiver algebras whose representations are related to the principal representation category $\mathscr{O}({\bf G})$ ., 14 pages

Let X denote a quasi-projective variety over a field on which a connected linear algebraic group G acts with finitely many orbits. Then, the G-orbits define a stratification of X. We establish several key properties of the category of equivariant perverse sheaves on X, which have locally constant cohomology sheaves on each of the orbits. Under the above assumptions, we show that this category comes close to being a highest weight category in the sense of Cline, Parshall and Scott and defines a quasi-hereditary algebra. We observe that the above hypotheses are satisfied by all toric varieties and by all spherical varieties associated to connected reductive groups over any algebraically closed field. Next we show that the odd dimensional intersection cohomology sheaves vanish on all spherical varieties defined over algebraically closed fields of positive characteristics, extending similar results for spherical varieties defined over the field of complex numbers by Michel Brion and the author in prior work. Assuming that the linear algebraic group G and the action of G on X are defined over a finite field F q , and where the odd dimensional intersection cohomology sheaves on the orbit closures vanish, we also establish several basic properties of the mixed category of mixed equivariant perverse sheaves so that the associated terms in the weight filtration are finite sums of the shifted equivariant intersection cohomology complexes on the orbit closures.

Let $ G $ be a connected standard simple algebraic group of type $ C $ or $ D $ over an algebraically closed field $ \Bbbk $ of positive characteristic $ p > 0 $, and $ \mathfrak{g}: = \mathrm{Lie}(G) $ be the Lie algebra of $ G $. Motivated by the variety of $ \mathbb{E}(r, \mathfrak{g}) $ of $ r $-dimensional elementary subalgebras of a restricted Lie algebra $ \mathfrak{g} $, in this paper we characterize the irreducible components of $ \mathbb{E}(\mathrm{rk}_{p}(\mathfrak{g})-1, \mathfrak{g}) $ where the $ p $-rank $ \mathrm{rk}_{p}(\mathfrak{g}) $ is defined to be the maximal dimension of an elementary subalgebra of $ \mathfrak{g} $.

Let w = [ [ x k , y l ] , [ x m , y n ] ] be a non-trivial double commutator word. We show that w is surjective on PSL 2 ( K ) , where K is an algebraically closed field of characteristic 0.

Let ( X , D X ) be a smooth pointed stable curve over an algebraically closed field k of characteristic p > 0 . Suppose that ( X , D X ) is generic. We give a necessary and sufficient condition for new-ordinariness of prime-to-p cyclic tame coverings of ( X , D X ) . This result generalizes a result of S. Nakajima concerning the ordinariness of prime-to-p cyclic etale coverings of generic curves to the case of tamely ramified coverings.

The purpose of this paper is to prove that the symbolic Rees rings of ideals defining certain finite sets of points in the projective plane over an algebraically closed field are finitely generated using a ring theoretical criterion which is known as Huneke's criterion.

We show that the Fargues--Fontaine curve associated to an algebraically closed field of characteristic p is geometrically simply connected; that is, its base extension from Q_p to any complete algebraically closed overfield admits no nontrivial connected finite etale covering. We then deduce from this an analogue for perfectoid spaces (and some related objects) of Drinfeld's lemma on the fundamental group of a product of schemes in characteristic p.

Let P be a lattice polytope with the $$h^{*}$$ -vector $$(1, h^*_1, \ldots , h^*_s)$$ . In this note we show that if $$h_s^* \le h_1^*$$ , then the Ehrhart ring $${\mathbb {k}}[P]$$ is generated in degrees at most $$s-1$$ as a $${\mathbb {k}}$$ -algebra. In particular, if $$s=2$$ and $$h_2^* \le h_1^*$$ , then P is IDP. To see this, we show the corresponding statement for semi-standard graded Cohen–Macaulay domains over algebraically closed fields.

We classify the finite dimensional irreducible modules for affine walled Brauer algebras over an algebraically closed field with arbitrary characteristic.

Let $K$ be an algebraically closed field of characteristic zero and let $G$ be a finitely generated subgroup of the multiplicative group of $K$. We consider $K$-valued sequences of the form $a_n:=f(\varphi^n(x_0))$, where $\varphi\colon X\to X$ and $f\colon X\to\mathbb{P}^1$ are rational maps defined over $K$ and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $\varphi$ and $f$. Many classical sequences from number theory and algebraic combinatorics fall under this dynamical framework, and we show that the set of $n$ for which $a_n\in G$ is a finite union of arithmetic progressions along with a set of Banach density zero. In addition, we show that if $a_n\in G$ for every $n$ and $X$ is irreducible and the $\varphi$ orbit of $x$ is Zariski dense in $X$ then there are a multiplicative torus $\mathbb{G}_m^d$ and maps $\Psi:\mathbb{G}_m^d \to \mathbb{G}_m^d$ and $g:\mathbb{G}_m^d \to \mathbb{G}_m$ such that $a_n = g\circ \Psi^n(y)$ for some $y\in \mathbb{G}_m^d$. We then obtain results about the coefficients of $D$-finite power series using these facts., Comment: 29 pages

Given a sequence d~ = (d1, . . . , dk) of natural numbers, we consider the Lie subalgebra h of gl(d, F), where d = d1 + · · · + dk and F is a field of characteristic 0, generated by two block upper triangular matrices D and E partitioned according to d~, and study the problem of computing the nilpotency degree m of the nilradical n of h. We obtain a complete answer when D and E belong to a certain family of matrices that arises naturally when attempting to classify the indecomposable modules of certain solvable Lie algebras. Our determination of m depends in an essential manner on the symmetry of E with respect to an outer automorphism of sl(d). The proof that m depends solely on this symmetry is long and delicate. As a direct application of our investigations on h and n we give a full classification of all uniserial modules of an extension of the free ℓ-step nilpotent Lie algebra on n generators when F is algebraically closed. Fil: Cagliero, Leandro Roberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina Fil: Levstein, Fernando. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina Fil: Szechtman, Fernando. University Of Regina; Canadá

The aim of this work is to provide a concrete list of non-isomorphic commutative power-associative superalgebras up to dimension 4 over an algebraically closed field of characteristic prime to 30. As a byproduct, we exhibit an example of a simple non-Jordan power-associative superalgebra whose even part is not semisimple.

We study asymptotic behaviour of graded and non-graded codimensions of simple Jordan superalgebras over a field of characteristic zero. It is known that the PI-exponent of any finite-dimensional associative or Jordan or Lie algebra A is a non-negative integer less than or equal to the dimension of algebra A. Moreover, the PI-exponent is equal to the dimension if and only if A is simple provided that the base field is algebraically closed. In the present paper we prove that for a Jordan superalgebra P(t) = H(Mt∣t, trp) its non-graded and ℤ2-graded exponents are strictly less than dim P(t). In particular, exp P(2) is fractional.

Let $D$ be a connected component of a possibly disconnected reductive group $G$ over an algebraic closed field. We define a partition of $D$ into finitely many Strata each of which is a union of $G^0$-conjugacy classes of fixed dimension. In the case where $D=G^0$ this recovers a known partition., Comment: 22 pages, some misprints corrected

Let G be a semisimple algebraic group defined over an algebraically closed field. We provide some criteria for normality and rational singularities of G-saturation under certain circumstances. Our results are applied to determine when the commuting variety over simple Lie algebra of low rank is normal and Cohen-Macaulay. We also present some interesting connections between injective modules and normality (resp. rational singularities) of their G-saturations. Finally, we generalize a machinery used to study singularities of nilpotent orbit closures.

Let Λ be a basic finite dimensional algebra over an algebraically closed field $\Bbbk $ , and let $\widehat {\Lambda }$ be the repetitive algebra of Λ. In this article, we prove that if $\widehat {V}$ is a left $\widehat {\Lambda }$ -module with finite dimension over $\Bbbk $ , then $\widehat {V}$ has a well-defined versal deformation ring $R(\widehat {\Lambda },\widehat {V})$ , which is a local complete Noetherian commutative $\Bbbk $ -algebra whose residue field is also isomorphic to $\Bbbk $ . We also prove that $R(\widehat {\Lambda }, \widehat {V})$ is universal provided that $\underline {\text {End}}_{\widehat {\Lambda }}(\widehat {V})=\Bbbk $ and that in this situation, $R(\widehat {\Lambda }, \widehat {V})$ is stable after taking syzygies. We apply the obtained results to finite dimensional modules over the repetitive algebra of the 2-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over $\mathbb {P}^{1}_{\Bbbk }$ .

Kaplansky’s famous conjectures about generalizing results from groups to Hopf al-gebras inspired many mathematicians to try to find solusions for them. Recently, Cohen and Westreich in [8] and [10] have generalized the concepts of nilpotency and solvability of groups to Hopf algebras under certain conditions and proved interesting results. In this article, we follow their work and give a detailed example by considering a finite group G and an algebraically closed field K. In more details, we construct the group Hopf algebra H = KG and examine its properties to see what of the properties of the original finite group can be carried out in the case of H.

We prove a coordinatization theorem for unital alternative algebras containing 2 x 2 matrix algebra with the same identity element 1. This solves an old problem announced by Nathan Jacobson on the description of alternative algebras containing a generalized quaternion algebra H with the same 1, for the case when the algebra H is split. In particular, this is the case when the basic field is finite or algebraically closed.

Given an n × n matrix c over a unitary ring R, the centralizer of c in the full n × n matrix ring M n ( R ) is called a principal centralizer matrix ring, denoted by S n ( c , R ) . We investigate its structure and prove: (1) If c is an invertible matrix with a c-free point, or if R has no zero-divisors and c is a Jordan-similar matrix with all eigenvalues in the center of R, then M n ( R ) is a separable Frobenius extension of S n ( c , R ) in the sense of Kasch. (2) If R is an integral domain and c is a Jordan-similar matrix, then S n ( c , R ) is a cellular R-algebra in the sense of Graham and Lehrer. In particular, if R is an algebraically closed field and c is an arbitrary matrix in M n ( R ) , then S n ( c , R ) is always a cellular algebra, and the extension S n ( c , R ) ⊆ M n ( R ) is always a separable Frobenius extension.

The study of deformations of Lie algebras is related to the problem of classification of simple Lie algebras over fields of small characteristics. The classification of finite-dimensional simple Lie algebras over algebraically closed fields of characteristic $p>3$ is completed. Over fields of characteristic 2, a large number of examples of Lie algebras are constructed that do not fit into previously known schemes. Description of deformations of classical Lie algebras gives new examples of simple Lie algebras and gives a possibility to describe known examples as deformations of classical Lie algebras. In this paper, we describe global deformations of Lie algebras of the type Dn and their quotient algebras $\overline{D}_n$ by the center in the case of a field of characteristic 2.

In this article, we give two different sufficient conditions for the irreducibility of a polynomial of more than one variable, over the field of complex numbers, that can be written as a sum of two polynomials which depend on mutually disjoint sets of variables. These conditions are derived from analyzing the Newton polytope of such a polynomial and then applying the `Irreducibility criterion' introduced by Gao., 10 pages, minor change in the statement of Proposition 1.4 and the proof of the same is rewritten for clarity, to appear in Indian Journal of Pure and Applied Mathematics

Let S be a semigroup, let M be a monoid with neutral element e, and let $$\mathbb {K}$$ be an algebraically closed field of characteristic $$\ne 2$$ with identity element 1. Inspired by Stetkaer’s procedure [19] we describe, in terms of multiplicative functions and characters of 2-dimensional representations of S, the solutions $$g:S\rightarrow \mathbb {K}$$ of the functional equation $$\begin{aligned} g(xy)+\mu (y)g(\psi (y)x)=2g(x)g(y),\quad x,y\in S, \end{aligned}$$ where $$\psi :S\rightarrow S$$ is an anti-endomorphism that need not be involutive and $$\mu :S\rightarrow \mathbb {K}$$ is a multiplicative function such that $$\mu (x\psi (x))=1$$ for all $$x\in S$$ . This enables us to find the solutions $$g:M\rightarrow \mathbb {K}$$ of the new functional equation $$\begin{aligned} g(x\sigma (y))+g(\psi (y)x) =2g(x)g(y),\quad x,y\in M, \end{aligned}$$ where $$\sigma :M\rightarrow M$$ is an involutive endomorphism.

We show that a weak version of the canonical bundle formula holds for fibrations of relative dimension one. We provide various applications thereof, for instance, using the recent result of Xu and Zhang, we prove the log non-vanishing conjecture for three-dimensional klt pairs over any algebraically closed field $k$ of characteristic $p>5$. We also show the log abundance conjecture for threefolds over $k$ when the nef dimension is not maximal, and the base point free theorem for threefolds over the algebraic closure of any finite field of characteristic $p>2$., 33 pages; the article has been substantially changed and the results are stronger: we now show log non-vanishing for all three-dimensional klt pairs of characteristic $p>5$, and log abundance when the nef dimension is not maximal. Further, we corrected a mistake in Lemma 4.3(1) (now Lemma 5.1(1)): the effectivity of adjoint divisors on surfaces holds only up to numerical equivalence

We give a new proof of the classification due to Peternell-Szurek-Wi\'{s}niewski of nef vector bundles on a projective space with the first Chern class less than three and on a smooth hyperquadric with the first Chern class less than two over an algebraically closed field of characteristic zero., Comment: 29 pages; v3: The title is changed, and sections 2 and 3 in v2 are simplified and unified; v2: An error in Remark 7.6 is fixed, and the corresponding corrections are made in Theorem 7.5. Lemmas 7.2 and 7.4 are improved and Lemma 7.4 becomes Theorem 7.4. Some proofs are replaced by shorter ones

Let $G$ be a finite group and $k$ an algebraically closed field of characteristic $p>0$. We define the notion of a Dade $kG$-module as a generalization of endo-permutation modules for $p$-groups. We show that under a suitable equivalence relation, the set of equivalence classes of Dade $kG$-modules forms a group under tensor product, and the group obtained this way is isomorphic to the Dade group $D(G)$ defined by Lassueur. We also consider the subgroup $D^{\Omega} (G)$ of $D(G)$ generated by relative syzygies $\Omega_X$, where $X$ is a finite $G$-set. If $C(G,p)$ denotes the group of superclass functions defined on the $p$-subgroups of $G$, there are natural generators $\omega_X$ of $C(G,p)$, and we prove the existence of a well-defined group homomorphism $\Psi_G:C(G,p)\to D^\Omega(G)$ that sends $\omega_X$ to $\Omega_X$. The main theorem of the paper is the verification that the subgroup of $C(G,p)$ consisting of the dimension functions of $k$-orientable real representations of $G$ lies in the kernel of $\Psi_G$., Comment: Minor revision, 40 pages

We introduce the notion of $$q^\prime $$ -compactness for Sheffer stroke basic algebras and Visser algebras. Our goal is to determine when induced lattice of a Sheffer stroke basic algebra and a Visser algebra is a strongly algebraically closed algebra, and we find the condition that the lattices of complete congruences relations on a Sheffer stroke basic algebra are weakly relatively pseudocomplemented. In particular, an open question proposed by A. Di-Nola, G. Georgescu and A. Iorgulescu about the connections of dually Brouwerian pseudo-BL-algebras with other algebraic structures in Di Nola et al. (Mult Val Logic 8:717–750, 2002) is answered.

Let $k$ be a field of characteristic $p>0$. Denote by $W_r(k)$ the ring of truntacted Witt vectors of length $r \geq 2$, built out of $k$. In this text, we consider the following question, depending on a given profinite group $G$. $Q(G)$: Does every (continuous) representation $G\longrightarrow GL_d(k)$ lift to a representation $G\longrightarrow GL_d(W_r(k))$? We work in the class of cyclotomic pairs (Definition 4.3), first introduced in [DCF] under the name "smooth profinite groups". Using Grothendieck-Hilbert' theorem 90, we show that the algebraic fundamental groups of the following schemes are cyclotomic: spectra of semilocal rings over $\mathbb{Z}[\frac{1}{p}]$, smooth curves over algebraically closed fields, and affine schemes over $\mathbb{F}_p$. In particular, absolute Galois groups of fields fit into this class. We then give a positive partial answer to $Q(G)$, for a cyclotomic profinite group $G$: the answer is positive, when $d=2$ and $r=2$. When $d=2$ and $r=\infty$, we show that any $2$-dimensional representation of $G$ stably lifts to a representation over $W(k)$: see Theorem 6.1. \\When $p=2$ and $k=\mathbb{F}_2$, we prove the same results, up to dimension $d=4$. We then give a concrete application to algebraic geometry: we prove that local systems of low dimension lift Zariski-locally (Corollary 6.3)., to appear in Math. Z

The notion of affine structures arises in many fields of mathematics, including convex homogeneous cones, vertex algebras, and affine manifolds. On the other hand, it is well known that Frobenius Lie algebras correspond to the research of homogeneous domains. Moreover, there are 16 isomorphism classes of 6-dimensional Frobenius Lie algebras over an algebraically closed field. The research studied the affine structures for the 6-dimensional Borel subalgebra of a simple Lie algebra. The Borel subalgebra was isomorphic to the first class of Csikós and Verhóczki’s classification of the Frobenius Lie algebras of dimension 6 over an algebraically closed field. The main purpose was to prove that the Borel subalgebra of dimension 6 was equipped with incomplete affine structures. To achieve the purpose, the axiomatic method was considered by studying some important notions corresponding to affine structures and their completeness, Borel subalgebras, and Frobenius Lie algebras. A chosen Frobenius functional of the Borel subalgebra helped to determine the affine structure formulas well. The result shows that the Borel subalgebra of dimension 6 has affine structures which are not complete. Furthermore, the research also gives explicit formulas of affine structures. For future research, another isomorphism class of 6-dimensional Frobenius Lie algebra still needs to be investigated whether it has complete affine structures or not.

Let $k$ be a field with char $k\neq 2$ and $k$ be not algebraically closed. Let $a\in k\setminus k^2$ and $L=k(\sqrt{a})(x,y)$ be a field extension of $k$ where $x,y$ are algebraically independent over $k$. Assume that $\sigma$ is a $k$-automorphism on $L$ defined by \[ \sigma: \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c(x+\frac{b}{x})+d}{y} \] where $b,c,d \in k$, $b\neq 0$ and at least one of $c,d$ is non-zero. Let $L^{\langle\sigma\rangle}=\{u\in L:\sigma(u)=u\}$ be the fixed subfield of $L$. We show that $L^{\langle\sigma\rangle}$ is isomorphic to the function field of a certain surface in $P^4_k$ which is given as the intersection of two quadrics. We give criteria for the $k$-rationality of $L^{\langle\sigma\rangle}$ by using the Hilbert symbol. As an appendix of the paper, we also give an alternative geometric proof of a part of the result which is provided to the authors by J.-L. Colliot-Th\'el\`ene., Comment: To appear in Manuscripta Math. The main theorems (old Theorem 1.7 and Theorem 1.8) incorporated into (new) Theorem 1.8. Section 3 and Section 4 interchanged

Let A be a representation-finite self-injective algebra over an algebraically closed field k. We give a new characterization for an orthogonal system in the stable module category A- $\underline {\text {mod}}$ to be a simple-minded system. As a by-product, we show that every Nakayama-stable orthogonal system in A- $\underline {\text {mod}}$ extends to a simple-minded system.

Let [Formula: see text] be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. In this paper we show that any finite-dimensional indecomposable [Formula: see text]-module is generated by one element. In particular, any indecomposable submodule of [Formula: see text] under the adjoint action is generated by a special element of [Formula: see text]. Using this result, we show that the Hopf algebra [Formula: see text] is a principal ideal ring, i.e., any two-sided ideal of [Formula: see text] is generated by one element. As an application, we give explicitly the generators of ideals, primitive ideals, maximal ideals and completely prime ideals of the Taft algebras.

Let [Formula: see text] be a fixed algebraically closed field of arbitrary characteristic, let [Formula: see text] be a finite dimensional self-injective [Formula: see text]-algebra, and let [Formula: see text] be an indecomposable non-projective left [Formula: see text]-module with finite dimension over [Formula: see text]. We prove that if [Formula: see text] is the Auslander–Reiten translation of [Formula: see text], then the versal deformation rings [Formula: see text] and [Formula: see text] (in the sense of F.M. Bleher and the second author) are isomorphic. We use this to prove that if [Formula: see text] is further a cluster-tilted [Formula: see text]-algebra, then [Formula: see text] is universal and isomorphic to [Formula: see text].

Let Hm,n be the mn2-dimensional Radford Hopf algebra over an algebraically closed field of characteristic zero. In this paper, we show that any finite dimensional indecomposable Hm,n-module is a cy...

Let $f\,:\,C\,\longrightarrow \,D$ be a nonconstant separable morphism between irreducible smooth projective curves defined over an algebraically closed field. We say that $f$ is genuinely ramified if ${\mathcal O}_D$ is the maximal semistable subbundle of $f_*{\mathcal O}_C$ (equivalently, the induced homomorphism $f_*\,:\, \pi _1^{\textrm{et}}(C)\,\longrightarrow \, \pi _1^{\textrm{et}}(D)$ of étale fundamental groups is surjective). We prove that the pullback $f^*E\,\longrightarrow \, C$ is stable for every stable vector bundle $E$ on $D$ if and only if $f$ is genuinely ramified.

Let F be an algebraically closed field of prime characteristic p. The modular McKay graph of G : = S L n ( p ) with respect to its standard FG-module W is the connected, directed graph whose vertices are the irreducible FG-modules and for which there is an edge from a vertex V 1 to V 2 if V 2 occurs as a composition factor of the tensor product V 1 ⊗ W . We show that the diameter of this modular McKay graph is 1 2 ( p − 1 ) ( n 2 − n ) .

The objects of study in this paper are Hopf algebras $H$ which are finitely generated algebras over an algebraically closed field and are extensions of a commutative Hopf algebra by a finite dimensional Hopf algebra. Basic structural and homological properties are recalled and classes of examples are listed. Bounds are obtained on the dimensions of simple $H$-modules, and the structure of $H$ is shown to be severely constrained when the finite dimensional extension is semisimple and cosemisimple., Comment: Preliminary version. Any comments are welcome

Let K be an algebraically closed field of characteristic 0, and let G be a divisible ordered Abelian group. Maclane [Bull. Am. Math. Soc., 45, 888-890 (1939)] showed that the Hahn field K((G)) is algebraically closed. Our goal is to bound the lengths of roots of a polynomial p(x) over K((G)) in terms of the lengths of its coefficients. The main result of the paper says that if 𝛾 is a limit ordinal greater than the lengths of all of the coefficients, then the roots all have length less than ωω𝛾.

The group of units modulo constants of an affine variety over an algebraically closed field is free abelian of finite rank. Computing this group is difficult but of fundamental importance in tropical geometry, where it is necessary in order to realize intrinsic tropicalizations. We present practical algorithms for computing unit groups of smooth curves of low genus. Our approach is rooted in divisor theory, based on interpolation in the case of rational curves and on methods from algebraic number theory in the case of elliptic curves.

After the classification of simple Lie algebras over a field of characteristic p > 3, the main problem not yet solved in the theory of finite dimensional Lie algebras is the classification of simple Lie algebras over a field of characteristic 2. The first result for this classification problem ensures that all finite dimensional Lie algebras of absolute toral rank 1 over an algebraically closed field of characteristic 2 are soluble. Describing simple Lie algebras (respectively, Lie 2-algebras) of finite dimension of absolute toral rank (respectively, toral rank) 3 over an algebraically closed field of characteristic 2 is still an open problem. In this paper we show that there are no classical type simple Lie 2-algebras with toral rank odd and furthermore that the simple contragredient Lie 2-algebra G(F4,a) of dimension 34 has toral rank 4. Additionally, we give the Cartan decomposition of G(F4,a).

Let $\mathcal {C}$ C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, $\mathcal {C}$ C is defined in k2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least two. We use this data to construct three affine pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, with the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring $\mathcal {O}(\mathcal {C})$ O ( C ) of $\mathcal {C}$ C as a quantum homogeneous space, and prove this when each of g and f has degree at most five or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree three A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(− 1,− 1,0). The final section of the paper lists some questions for future work.