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We determine the 2-fusion systems of J-component type in which the centralizer of some fully centralized involution has a maximal J-component that is the 2-fusion system of U 3 ( 3 ) .

The submonoid of the 3-strand braid group B 3 generated by σ 1 and σ 1 σ 2 is known to yield an exotic Garside structure on B 3 . We introduce and study an infinite family ( M n ) n ≥ 1 of Garside monoids generalizing this exotic Garside structure, i.e., such that M 2 is isomorphic to the above monoid. The corresponding Garside group G ( M n ) is isomorphic to the ( n , n + 1 ) -torus knot group–which is isomorphic to B 3 for n = 2 and to the braid group of the exceptional complex reflection group G 12 for n = 3 . This yields a new Garside structure on ( n , n + 1 ) -torus knot groups, which already admit several distinct Garside structures. The ( n , n + 1 ) -torus knot group is an extension of B n + 1 , and the Garside monoid M n surjects onto the submonoid Σ n of B n + 1 generated by σ 1 , σ 1 σ 2 , … , σ 1 σ 2 ⋯ σ n , which is not a Garside monoid when n > 2 . Using a new presentation of B n + 1 that is similar to the presentation of G ( M n ) , we nevertheless check that Σ n is an Ore monoid with group of fractions isomorphic to B n + 1 , and give a conjectural presentation of it, similar to the defining presentation of M n . This partially answers a question of Dehornoy–Digne–Godelle–Krammer–Michel.

We introduce an algorithmic framework to investigate spherical and geodesic growth series of braid groups relatively to the Artin's or Birman–Ko–Lee's generators. We present our experimentations in the case of three and four strands and conjecture rational expressions for the spherical growth series with respect to the Birman–Ko–Lee's generators.

We study a quotient of the group algebra of the braid group in which the Artin generators satisfy a cubic relation. This quotient is maximal among the ones satisfying such a cubic relation. We also investigate the proper quotients of it that appear in the realm of quantum groups, and describe another maximal quotient related to the usual Hecke algebras. Finally, we describe the connection between this algebra and a quotient of the algebra of horizontal chord diagrams introduced by Vogel. We prove that these two are isomorphic for n ≤ 5 .

This paper proves the conjectures (previously proved in the case n = 0 ) that for any Coxeter system ( W , S ) and any natural number n, the set of n-small roots for W is bipodal and the set of n-low elements of W is a Garside shadow. The proof here uses special cases ( k = 3 ) of properties (notably, their existence) of certain maximal rank k reflection subgroups of W, for non-negative integers k.

We consider Artin-Tits groups of FC type and prove the two following results. If any two distinct elements of a standard parabolic subgroup are conjugated by a standard generator of the whole group, then this generator has to be in the subgroup. We also prove that classical transversals of standard parabolic subgroups of Artin-Tits groups of FC type are compatible with intersection with standard parabolic subgroups: If A X , A Y are two standard parabolic subgroups of an Artin-Tits groups A of FC type and T is the classical transversal of A X in A, then T ∩ A Y is a transversal of A X ∩ Y in A Y . So the two properties hold for Artin-Tits groups of spherical type and Right-angled Artin-Tits groups.

Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that contains a member of each conjugacy class of involutions of G.

Let $G$ be a finite group of order divisible by a prime $p$. The number of $p$-regular and $p'$-regular conjugacy classes of $G$ is at least $2\sqrt{p-1}$. Also, the number of $p$-rational and $p'$-rational irreducible characters of $G$ is at least $2\sqrt{p-1}$. Along the way we prove a uniform lower bound for the number of $p$-regular classes in a finite simple group of Lie type in terms of its rank and size of the underlying field., 46 pages

Several combinatorial actions of the affine Weyl group of type C ˜ n on triangulations, trees, words and permutations are compared. Addressing a question of David Vogan, we show that, modulo a natural involution, these permutation representations are multiplicity-free. The proof uses a general construction of Gelfand subgroups in the affine Weyl groups of types C ˜ n and B ˜ n .

Garside theory emerged from the study of Artin groups and their generalizations. Finite-type Artin groups admit two types of interval Garside structures corresponding to their standard and dual presentations. Concerning affine Artin groups, Digne established interval Garside structures for two families of these groups by using their dual presentations. Recently, McCammond established that none of the remaining dual presentations (except for one additional case) correspond to interval Garside structures. In this paper, shifting attention from dual presentations to other nice presentations for the affine Artin group of type A ˜ discovered by Shi and Corran–Lee–Lee, I will construct interval Garside structures related to this group. This construction is the first successful attempt to establish interval Garside structures not related to the dual presentations in the case of affine Artin groups.

By work of Belyĭ [2] , the absolute Galois group G Q = Gal ( Q ‾ / Q ) of the field Q of rational numbers can be embedded into A = Aut ( F ˆ 2 ) , the automorphism group of the free profinite group F ˆ 2 on two generators. The image of G Q lies inside G T ˆ , the Grothendieck-Teichmuller group. While it is known that every abelian representation of G Q can be extended to G T ˆ , Lochak and Schneps [13] put forward the challenge of constructing irreducible non-abelian representations of G T ˆ . We do this virtually, namely by showing that a rich class of arithmetically defined representations of G Q can be extended to finite index subgroups of G T ˆ . This is achieved, in fact, by extending these representations all the way to finite index subgroups of A = Aut ( F ˆ 2 ) . We do this by developing a profinite version of the work of Grunewald and Lubotzky [7] , which provided a rich collection of representations for the discrete group Aut ( F d ) .

An even Artin group is a group which has a presentation with relations of the form ( s t ) n = ( t s ) n with n ≥ 1 . With a group G we associate a Lie Z -algebra TG r ( G ) . This is the usual Lie algebra defined from the lower central series, truncated at the third rank. For each even Artin group G we determine a presentation for TG r ( G ) . By means of this presentation we obtain information about the diagram of G. We then prove an isomorphism criterion for Coxeter matrices that ensures that the diagram of G is uniquely determined by this information. Let d ≥ 2 . We show that, if two even Artin groups G and H having presentations with relations of the form ( s t ) d k = ( t s ) d k with k ≥ 0 are such that TG r ( G ) ≃ TG r ( H ) , then G and H have the same presentation up to permutation of the generators. On the other hand, we show an example of two non-isomorphic even Artin groups G and H such that TG r ( G ) ≃ TG r ( H ) .

In this paper we introduce and study some geometric objects associated to Artin monoids. The Deligne complex for an Artin group is a cube complex that was introduced by the second author and Davis (1995) to study the K(\pi,1) conjecture for these groups. Using a notion of Artin monoid cosets, we construct a version of the Deligne complex for Artin monoids. We show that for any Artin monoid this cube complex is contractible. Furthermore, we study the embedding of the monoid Deligne complex into the Deligne complex for the corresponding Artin group. We show that for any Artin group this is a locally isometric embedding. In the case of FC-type Artin groups this result can be strengthened to a globally isometric embedding, and it follows that the monoid Deligne complex is CAT(0) and its image in the Deligne complex is convex. We also consider the Cayley graph of an Artin group, and investigate properties of the subgraph spanned by elements of the Artin monoid. Our final results show that for a finite type Artin group, the monoid Cayley graph embeds isometrically, but not quasi-convexly, into the group Cayley graph., Comment: 21 pages

We lay the foundations of the first-order model theory of Coxeter groups. Firstly, with the exception of the $2$-spherical non-affine case (which we leave open), we characterize the superstable Coxeter groups of finite rank, which we show to be essentially the Coxeter groups of affine type. Secondly, we characterize the Coxeter groups of finite rank which are domains, a central assumption in the theory of algebraic geometry over groups, which in many respects (e.g. $\lambda$-stability) reduces the model theory of a given Coxeter system to the model theory of its associated irreducible components. In the second part of the paper we move to specific definability questions in right-angled Coxeter groups (RACGs) and $2$-spherical Coxeter groups. In this respect, firstly, we prove that RACGs of finite rank do not have proper elementary subgroups which are Coxeter groups, and prove further that reflection independent ones do not have proper elementary subgroups at all. Secondly, we prove that if the monoid $Sim(W, S)$ of $S$-self-similarities of $W$ is finitely generated, then $W$ is a prime model of its theory. Thirdly, we prove that in reflection independent RACGs of finite rank the Coxeter elements are type-determined. We then move to $2$-spherical Coxeter groups, proving that if $(W, S)$ is irreducible, $2$-spherical even and not affine, then $W$ is a prime model of its theory, and that if $W_{\Gamma}$ and $W_{\Theta}$ are as in the previous sentence, then $W_{\Gamma}$ is elementary equivalent to $W_{\Theta}$ if and only if $\Gamma \cong \Theta$, thus solving the elementary equivalence problem for most of the $2$-spherical Coxeter groups. In the last part of the paper we focus on model theoretic applications of the notion of reflection length from Coxeter group theory, proving in particular that affine Coxeter groups are not connected., Comment: 38 pages

Let $R$ be a local ring of positive characteristic and $X$ a complex with nonzero finitely generated homology and finite injective dimension. We prove that if derived base change of $X$ via the Frobenius (or more generally, via a contracting) endomorphism has finite injective dimension then $R$ is Gorenstein., 9 pages, comments are welcome

International audience; We show that there is an equivalence of categories between Lie-Rinehart algebras over a commutative algebra and homotopy equivalence classes of negatively graded Lie ∞-algebroids over their resolutions (=acyclic Lie ∞-algebroids). This extends to a purely algebraic setting the construction of the universal Q-manifold of a locally real analytic singular foliation of [26], [28]. In particular, it makes sense for the universal Lie ∞-algebroid of every singular foliation, without any additional assumption, and for Androulidakis-Zambon singular Lie algebroids. Also, to any ideal preserved by the anchor map of a Lie-Rinehart algebra , we associate a homotopy equivalence class of negatively graded Lie ∞-algebroids over complexes computing . Several explicit examples are given.

There is an embedding of affine vertex algebras $V^k(\mathfrak{gl}_n) \hookrightarrow V^k(\mathfrak{sl}_{n+1})$, and the coset $\mathcal{C}^k(n) = \text{Com}(V^k(\mathfrak{gl}_n), V^k(\mathfrak{sl}_{n+1}))$ is a natural generalization of the parafermion algebra of $\mathfrak{sl}_2$. It was called the algebra of generalized parafermions by the third author and was shown to arise as a one-parameter quotient of the universal two-parameter $\mathcal{W}_{\infty}$-algebra of type $\mathcal{W}(2,3,\dots)$. In this paper, we consider an analogous structure of orthogonal type, namely $\mathcal{D}^k(n) = \text{Com}(V^k(\mathfrak{so}_{2n}), V^k(\mathfrak{so}_{2n+1}))^{\mathbb{Z}_2}$. We realize this algebra as a one-parameter quotient of the two-parameter even spin $\mathcal{W}_{\infty}$-algebra of type $\mathcal{W}(2,4,\dots)$, and we classify all coincidences between its simple quotient $\mathcal{D}_k(n)$ and the algebras $\mathcal{W}_{\ell}(\mathfrak{so}_{2m+1})$ and $\mathcal{W}_{\ell}(\mathfrak{so}_{2m})^{\mathbb{Z}_2}$. As a corollary, we show that for the admissible levels $k = -(2n-2) + \frac{1}{2} (2 n + 2 m -1)$ for $\widehat{\mathfrak{so}}_{2n}$ the simple affine algebra $L_k(\mathfrak{so}_{2n})$ embeds in $L_k(\mathfrak{so}_{2n+1})$, and the coset is strongly rational. As a consequence, the category of ordinary modules of $L_k(\mathfrak{so}_{2n+1})$ at such a level is a braided fusion category., Comment: Minor corrections, final version to appear in J. Algebra

We determine the Lie subalgebra $\mathfrak{g}_{nil}$ of a Borcherds symmetrizable generalized Kac-Moody Lie algebra $\mathfrak{g}$ generated by $ad$-locally nilpotent elements and show that it is `essentially' the same as the Levi subalgebra of $\mathfrak{g}$ with its simple roots precisely the real simple roots of $\mathfrak{g}$., 5 pages

Let $\mathcal C$ be a class of Hausdorff topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup $X$ is called $\mathcal C$-$closed$ if $X$ is closed in each topological semigroup $Y\in \mathcal C$ containing $X$ as a discrete subsemigroup; $X$ is $projectively$ $\mathcal C$-$closed$ if for each congruence $\approx$ on $X$ the quotient semigroup $X/_\approx$ is $\mathcal C$-closed. A semigroup $X$ is called $chain$-$finite$ if for any infinite set $I\subseteq X$ there are elements $x,y\in I$ such that $xy\notin\{x,y\}$. We prove that a semigroup $X$ is $\mathcal C$-closed if it admits a homomorphism $h:X\to E$ to a chain-finite semilattice $E$ such that for every $e\in E$ the semigroup $h^{-1}(e)$ is $\mathcal C$-closed. Applying this theorem, we prove that a commutative semigroup $X$ is $\mathcal C$-closed if and only if $X$ is periodic, chain-finite, all subgroups of $X$ are bounded, and for any infinite set $A\subseteq X$ the product $AA$ is not a singleton. A commutative semigroup $X$ is projectively $\mathcal C$-closed if and only if $X$ is chain-finite, all subgroups of $X$ are bounded and the union $H(X)$ of all subgroups in $X$ has finite complement $X\setminus H(X)$., Comment: 19 pages

Let D be a domain and let S be a torsion-free monoid such that D has characteristic 0 or the quotient group of S satisfies the ascending chain condition on cyclic subgroups. We give a characterization of when the monoid algebra D [ S ] is weakly Krull. As corollaries, we reobtain the results on when D [ S ] is Krull resp. weakly factorial, due to Chouinard resp. Chang. Furthermore, we deduce a characterization of generalized Krull monoid algebras analogous to our main result and we characterize the weakly Krull domains among the affine monoid algebras.

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.

If $k$ is a field and $R$ is a commutative $k$-algebra, we explore the question of when the ring $D_{R|k}$ of $k$-linear differential operators on $R$ is isomorphic to its opposite ring. Under mild hypotheses, we prove this is the case whenever $R$ Gorenstein local or when $R$ is a ring of invariants. As a key step in the proof we show that in many cases of interest canonical modules admit right $D$-module structures. After this work was completed we realized that some of our results were already proved in higher generality by Yekutieli, albeit using more sophisticated methods., Comment: v2: some results are shifted to improve readability. 16 pages, comments welcome

Inspired by the work of Amitsur [1] on finite groups whose irreducible characters all have degree (multiplicity) 1 or 2, in this paper we study association schemes whose irreducible characters all have multiplicity 1 or 2. We will first show that the general case can be reduced to commutative association schemes. Then for commutative association schemes with multiplicities 1 or 2, we prove that their Krein parameters are all rational integers. Using automorphism groups of association schemes, we obtain a characterization and classification of those commutative association schemes all valencies and multiplicities of which are 1 or 2 in terms of Cayley schemes.

We study simple and projective modules of a certain class of Ehresmann semigroups, a well-studied generalization of inverse semigroups. Let S be a finite right (left) restriction Ehresmann semigroup whose corresponding Ehresmann category is an EI-category, that is, every endomorphism is an isomorphism. We show that the collection of finite right restriction Ehresmann semigroups whose categories are EI is a pseudovariety. We prove that the simple modules of the semigroup algebra k S (over any field k ) are formed by inducing the simple modules of the maximal subgroups of S via the corresponding Schutzenberger module. Moreover, we show that over fields with good characteristic the indecomposable projective modules can be described in a similar way but using generalized Green's relations instead of the standard ones. As a natural example, we consider the monoid PT n of all partial functions on an n-element set. Over the field of complex numbers, we give a natural description of its indecomposable projective modules and obtain a formula for their dimension. Moreover, we find certain zero entries in its Cartan matrix.

Let n ≥ 4 , 2 ≤ r ≤ n − 2 and e ≥ 1 . We show that the intersection of the locus of degree e morphisms from P 1 to G ( r , n ) with the restricted universal sub-bundles having a given splitting type and the locus of degree e morphisms with the restricted universal quotient-bundle having a given splitting type is non-empty and generically transverse along at least one component of the intersection. As a consequence, we get that the locus of degree e morphisms from P 1 to G ( r , n ) with the restricted tangent bundle having a given splitting type need not always be irreducible.

We study those pre-Hilbert absolute-valued algebras satisfying the identity ( x 2 , y , x 2 ) = 0 . We prove that such an algebra A is finite-dimensional in each one of the following two cases: (1) A satisfies the additional identity ( x , x 2 , x ) = 0 , (2) A contains a weak left-unit. In the first case A is flexible and isomorphic to either R , C , C ⁎ , H , H ⁎ , O , O ⁎ or P . In the second one A has a left-unit and is isomorphic to either R , C , C ⁎ , H , H ⁎ , O , or O ⁎ . We also prove the existence of infinite-dimensional pre-Hilbert absolute-valued algebras satisfying ( x 2 , y , x 2 ) = 0 and containing only a non-zero idempotent.

We show that the quantized flag manifold at a root of unity has natural affine open covering parametrized by the elements of the Weyl group. In particular, the quantized flag manifold turns out to be a quasi-scheme in the sense of Rosenberg [12] .

The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant $\gamma_G(M)$ for a finite dimensional module $M$ of a finite group $G$ which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for all the signed permutation modules of the symmetric group using tools from representation theory and combinatorics., Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:2012.00341

Let $\mathfrak{n}$ be a locally nilpotent infinite-dimensional Lie algebra over $\mathbb{C}$. Let $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$ be its universal enveloping algebra and its symmetric algebra respectively. Consider the Jacobson topology on the primitive spectrum of $\mathrm{U}(\mathfrak{n})$ and the Poisson topology on the primitive Poisson spectrum of $\mathrm{S}(\mathfrak{n})$. We provide a homeomorphism between the corresponding topological spaces (on the level of points, it gives a bijection between the primitive ideals of $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$). We also show that all primitive ideals of $\mathrm{S}(\mathfrak{n})$ from an open set in a properly chosen topology are generated by their intersections with the Poisson center. Under the assumption that $\mathfrak{n}$ is a nil-Dynkin Lie algebra, we give two criteria for primitive ideals $I(\lambda)\subset\mathrm{S}(\mathfrak{n})$ and $J(\lambda)\subset\mathrm{U}(\mathfrak{n})$, $\lambda\in\mathfrak{n}^*$, to be nonzero. Most of these results generalize the known facts about primitive and Poisson spectrum for finite-dimensional nilpotent Lie algebras (but note that for a finite-dimensional nilpotent Lie algebra all primitive ideals $I(\lambda)$, $J(\lambda)$ are nonzero)., Comment: 43 pages

Let $\mathbb F$ be a finite field. Consider a direct sum $V$ of an infinite number of copies of $\mathbb F$, consider the dual space $V^\diamond$, i.~e., the direct product of an infinite number of copies of $\mathbb F$. Consider the direct sum ${\mathbb V}=V\oplus V^\diamond$. The object of the paper is the group $\mathbf{GL}$ of continuous linear operators in $\mathbb V$. We reduce the theory of unitary representations of $\mathbf{GL}$ to projective representations of a certain category whose morphisms are linear relations in finite-dimensional linear spaces over $\mathbb F$. In fact we consider a certain family $ Q_\alpha$ of subgroups in $\mathbb V$ preserving two-element flags, show that there is a natural multiplication on spaces of double cosets with respect to $ Q_\alpha$, and reduce this multiplication to products of linear relations. We show that this group has type $\mathrm{I}$ and obtain an 'upper estimate' of the set of all irreducible unitary representations of $\mathbf{GL}$., Comment: 48pp, a revised version

For any integers $m,n$ with $m\ne 0$ and $n>0$, let $G_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[z^m,x][z^n,y]\rangle$; for any integers $m,n>0$, let $H_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[x^m,z^n][y,z]\rangle$. By investigating cohomology jump loci of irreducible ${\rm GL}(2,\mathbb{C})$-character varieties, we show: if $m,m'\ne 0$, $n,n'>0$ and $G_{m',n'}\cong G_{m,n}$, then $m=m',n=n'$; if $m,m',n,n'>0$ and $H_{m',n'}\cong H_{m,n}$, then $m'=m, n'=n$., 18 pages

This paper studies local properties of Jacobson-Witt algebras over fields of prime characteristic, i.e., initiates the study on 2-local derivations of Lie algebras of prime characteristic. Let W n be a simple Jacobson-Witt algebra over a field F of prime characteristic p with | F | ≥ p n . In this paper, it is shown that every 2-local derivation on W n is a derivation.

Throughout this paper, all groups are finite and G always denotes a finite group. Let σ = { σ i | i ∈ I } be a partition of the set of all primes P . The group G is said to be: σ-primary if G is a σ i -group for some i = i ( G ) ; σ-nilpotent if G = G 1 × … × G n for some σ-primary groups G 1 , … , G n ; σ-soluble if every chief factor of G is σ-primary; σ-full if G possesses a Hall σ i -subgroup for all i such that σ i ∩ π ( G ) ≠ ∅ . A subgroup A of G is said to be σ-permutable in G provided G is σ-full and A permutes with every Hall σ i -subgroup H of G, that is, A H = H A for all i; G is said to be a PσT-group if σ-permutability is a transitive relation in G, that is, if K is a σ-permutable subgroup of H and H is a σ-permutable subgroup of G, then K is a σ-permutable subgroup of G. Let F be a class of group. Then a set Σ of subgroups of G is called a G-covering subgroup system for the class F if G ∈ F whenever Σ ⊆ F . We prove that: If a set of subgroups Σ of G contains at least one supplement to each maximal subgroup of every Sylow subgroup of G, then Σ is a G-covering subgroup system for the classes of all σ-soluble groups, all σ-nilpotent groups, and all σ-soluble PσT-groups. This result gives positive answers to Questions 19.87 and 19.88 in the Kourovka Notebook and, also, allows us to obtain the following characterization of σ-soluble PσT-groups: G is a σ-soluble PσT-group if and only if each maximal subgroup of every Sylow subgroup of G has a supplement T in G such that T is a σ-soluble PσT-group.

We construct orbifolds of holomorphic lattice vertex operator algebras for non-abelian finite automorphism groups G. To this end, we construct twisted modules for automorphisms g together with the projective representation of the centralizer of g on the twisted module. This allows us to extract the irreducible modules of the fixed-point VOA VG, and to compute their characters and modular transformation properties. We then construct holomorphic VOAs by adjoining such modules to VG. Applying these methods to extremal lattices in d=48 and d=72, we construct more than fifty new holomorphic VOAs of central charge 48 and 72, many of which have a very small number of light states., Journal of Algebra, 585, ISSN:0021-8693, ISSN:1090-266X

In the paper we describe the class of all solvable extensions of an infinite-dimensional filiform Leibniz algebra. The filiform Leibniz algebra is taken as a maximal pro-nilpotent ideal of a residually solvable Leibniz algebra. It is proven that the second cohomology group of the extension is trivial.

We introduce the notions of τ-exceptional and signed τ-exceptional sequences for any finite dimensional algebra. We prove that for a fixed algebra of rank n, and for any positive integer t ≤ n , there is a bijection between the set of signed τ-exceptional sequences of length t, and (basic) ordered support τ-rigid objects with t indecomposable direct summands. If the algebra is hereditary, our notions coincide with exceptional and signed exceptional sequences. The latter were recently introduced by Igusa and Todorov, who constructed a similar bijection in the hereditary setting.

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

We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8., Comment: 19 pages

We introduce and discuss an elementary tool from representation theory of finite groups for constructing linear codes invariant under a given permutation group G. The tool gives theoretical insight as well as a recipe for computations of generator matrices and weight distributions. In some interesting cases a classification of code vectors under the action of G can be obtained. As an explicit example a class of binary codes is studied extensively which is closely related to the class of binary codes associated to triangular graphs. A second explicit application is related to the action of the Mathieu simple group M 24 on the set of octads giving many binary codes of length 759 with interesting properties. We also obtain new alternative proofs for several other theorems and construct several new codes invariant under various subgroups of the Conway simple group Co 1 .

We study multiplication modules. The rings are not assumed to be commutative. Several criteria with some applications are given for a direct sum of modules to be a multiplication module.

This paper investigates algebraic objects equipped with an operator, such as operated monoids, operated algebras etc. Various free object functors in these operated contexts are explicitly constructed. For operated algebras whose operator satisfies a set $\Phi$ of relations (usually called operated polynomial identities (aka. OPIs)), Guo defined free objects, called free $\Phi$-algebras, via universal algebra. Free $\Phi$-algebras over algebras are studied in details. A mild sufficient condition is found such that $\Phi$ together with a Gr\"obner-Shirshov basis of an algebra $A$ form a Gr\"obner-Shirshov basis of the free $\Phi$-algebra over algebra $A$ in the sense of Guo et al.. Ample examples for which this condition holds are provided, such as all Rota-Baxter type OPIs, a class of differential type OPIs, averaging OPIs and Reynolds OPI., Comment: Slightly revised version of the published paper in Journal of Algebra

A pseudo-Riemannian Lie group ( G , 〈 ⋅ , ⋅ 〉 ) is a connected and simply connected Lie group with a left-invariant pseudo-Riemannian metric of type ( p , q ) . This paper is to study pseudo-Riemannian Lie groups with non-Killing conformal vector fields induced by derivations which is an extension from non-Killing left-invariant conformal vector fields. First we prove that a Riemannian (i.e. type ( n , 0 ) ), Lorentzian (i.e. type ( n − 1 , 1 ) ) or trans-Lorentzian (i.e. type ( n − 2 , 2 ) ) Lie group with such a vector field is solvable. Then we construct non-solvable unimodular pseudo-Riemannian Lie groups with such vector fields for any min ⁡ ( p , q ) ≥ 3 . Finally, we give the classification for the Riemannian and Lorentzian cases.

We study Gorenstein flat objects in the category Rep ( Q , R ) of representations of a left rooted quiver Q with values in Mod ( R ) , the category of all left R-modules, where R is an arbitrary associative ring. We show that a representation X in Rep ( Q , R ) is Gorenstein flat if and only if for each vertex i the canonical homomorphism φ i X : ⊕ a : j → i X ( j ) → X ( i ) is injective, and the left R-modules X ( i ) and Coker φ i X are Gorenstein flat. As an application, we obtain a Gorenstein flat model structure on Rep ( Q , R ) in which we give explicit descriptions of the subcategories of trivial, cofibrant and fibrant objects.

In this paper, we define the notion of Hopf crossed square for cocommutative Hopf algebras extending the notions of crossed squares of groups and of Lie algebras. We prove the equivalence between the category of Hopf crossed squares and the category of double internal groupoids in the category of cocommutative Hopf algebras. The Hopf crossed squares turn out to be the internal crossed modules in the category of crossed modules in the category of cocommutative Hopf algebras., Comment: 40 pages

By constructing explicit examples, we show that the core of a group-like element in a cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra over the group ring of a finite abelian group is not always completely trivial.

Let G be a finite group, and T ( G ) be the sum of all complex irreducible character degrees of G. In this paper, we aim to characterize the structure of finite groups in terms of T ( G ) . We show that if | G | / T ( G ) ( p + 1 ) / 2 , then G has a normal Sylow p-subgroup; and that if | G | / T ( G ) 3 p 2 / ( p 2 + 2 ) , then G is p-supersolvable, where p is a prime.

We prove that the isomorphism type of the subrack lattice of a finite group determines the nilpotence class. We analyze the problem of estimating the orders of the group elements corresponding to the atoms of the subrack lattice. As a result, we show that the subrack lattice determines p-nilpotence of the group if a certain condition is met.

We give a short proof of Green's formula on Hall numbers. By using rotation of triangles, we find that Green's formula is just the associativity of derived Hall numbers.

Given a monomial $m$ in a polynomial ring and a subset $L$ of the variables of the polynomial ring, the principal $L$-Borel ideal generated by $m$ is the ideal generated by all monomials which can be obtained from $m$ by successively replacing variables of $m$ by those which are in $L$ and have smaller index. Given a collection $\mathcal{I}=\{I_1,\ldots,I_r\}$ where $I_i$ is $L_i$-Borel for $i=1,\ldots,r$ (where the subsets $L_1,\ldots,L_r$ may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets $L_1,\ldots,L_r$ is chordal bipartite, then the defining equations of the multi-Rees algebra of $\mathcal{I}$ has a Gr\"obner basis of quadrics with squarefree lead terms under lexicographic order. Thus the multi-Rees algebra of such a collection of ideals is Koszul, Cohen-Macaulay, and normal. This significantly generalizes a theorem of Ohsugi and Hibi on Koszul bipartite graphs. As a corollary we obtain that the multi-Rees algebra of a collection of principal Borel ideals is Koszul. To prove our main result we use a fiber-wise Gr\"obner basis criterion for the kernel of a toric map and we introduce a modification of Sturmfels' sorting algorithm., Comment: 29 pages, 4 figures