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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 ) .

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 .

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.

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

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.

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] .

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

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.

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.

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

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.

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.

Dendriform algebras are certain splitting of associative algebras and arise naturally from Rota-Baxter operators, shuffle algebras and planar binary trees. In this paper, we first consider involutive dendriform algebras, their cohomology and homotopy analogs. The cohomology of an involutive dendriform algebra splits the Hochschild cohomology of an involutive associative algebra. In the next, we introduce a more general notion of oriented dendriform algebras. We develop a cohomology theory for oriented dendriform algebras that closely related to extensions and governs the simultaneous deformations of dendriform structures and the orientation., Comment: 22 pages; Subsection 3.4 is newly added; comments are welcome

We prove that the Diophantine problem for orientable quadratic equations in free metabelian groups is decidable and furthermore, NP-complete. In the case when the number of variables in the equation is bounded, the problem is decidable in polynomial time.

Let $S$ be an unramified regular local ring of mixed characteristic two and $R$ the integral closure of $S$ in a biquadratic extension of its quotient field obtained by adjoining roots of sufficiently general square free elements $f,g\in S$. Let $S^2$ denote the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/2S$. When at least one of $f,g\in S^2$, we characterize the Cohen-Macaulayness of $R$ and show that $R$ admits a birational small Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay in case $f,g\in S^2$ or if $f,g\notin S^2$., Comment: Final version, to appear in Journal of Algebra; minor changes, unabbreviated title, updated references

In this paper we give full classification of rank 3 line arrangements in P 2 (over a field of characteristic 0) that have a minimal logarithmic derivation of degree 3. The classification presents their defining polynomials, up to a change of variables, with their corresponding affine pictures. We also analyze the shape of such a logarithmic derivation, towards obtaining criteria for a line arrangement to possess a cubic minimal logarithmic derivation.

A Lie algebra L is said to be ( Θ n , sl n ) -graded if it contains a simple subalgebra g isomorphic to s l n such that the g -module L decomposes into copies of the adjoint module, the trivial module, the natural module V, its symmetric and exterior squares S 2 V and ∧ 2 V and their duals. We describe the multiplicative structures and the coordinate algebras of ( Θ n , sl n ) -graded Lie algebras for n ≥ 5 , classify these Lie algebras and determine their central extensions.

The quantum plane is the non-commutative polynomial algebra in variables $x$ and $y$ with $xy=qyx$. In this paper, we study the module variety of $n$-dimensional modules over the quantum plane, and provide an explicit description of its irreducible components and their dimensions. We also describe the irreducible components and their dimensions of the GIT quotient of the module variety with respect to the conjugation action of ${\rm GL}_n$., Comment: 31 pages, comments are welcome

Let G be a simple, simply connected linear algebraic group of exceptional type defined over F q with Frobenius endomorphism F : G → G . In this work we give upper bounds for the number of irreducible Brauer characters in the quasi-isolated l-blocks of G F and G F / Z ( G F ) when the prime l is bad for G.

A classical theorem of Veldkamp describes the center of an enveloping algebra of a Lie algebra of a semi-simple algebraic group in characteristic $p.$ We generalize this result to a class of Lie algebras with a property that they arise as the reduction modulo $p\gg 0$ from an algebraic Lie algebra $\mathfrak{g},$ such that $\mathfrak{g}$ has no nontrivial semi-invariants in $Sym(\mathfrak{g})$ and $Sym(\mathfrak{g})^{\mathfrak{g}}$ is a polynomial algebra. As an application, we solve the derived isomorphism problem of enveloping algebras for the above class of Lie algebras., Comment: Significantly revised, final version. To appear in Journal of Algebra

In this paper we study the relations between numerical characteristics of finite dimensional algebras and such classical combinatorial objects as additive chains. We study the behavior of the length function via so-called characteristic sequences of quadratic algebras. As one of our main results, we prove the sharp upper bound for the length of quadratic algebras in terms of the Fibonacci numbers depending on the dimension of the algebra. Moreover, we show that quadratic algebras have the extremal behavior with respect to this bound. In addition, we obtain the description of the set of characteristic sequences for quadratic algebras. Namely, we completely determine the set of combinatorial properties which are satisfied for characteristic sequences of quadratic algebras and show that they belong to the family of additive chains known in combinatorics. Conversely, for a given integer sequence being an additive chain and satisfying these combinatorial properties, we construct a quadratic algebra with a characteristic sequence equal to this sequence. The obtained information on characteristic sequences is then applied to investigate the problem of realizability for the length function. In particular, we determine certain subsets of integers which are not realizable as values of the length function on quadratic algebras.

Let Λ be an Artin algebra. In this paper, the notion of n Z -Gorenstein cluster tilting subcategories will be introduced. It is shown that every n Z -cluster tilting subcategory of mod-Λ is n Z -Gorenstein if and only if Λ is an Iwanaga-Gorenstein algebra. Moreover, it will be shown that an n Z -Gorenstein cluster tilting subcategory of mod-Λ is an n Z -cluster tilting subcategory of the exact category Gprj - Λ , the subcategory of all Gorenstein projective objects of mod-Λ. Some basic properties of n Z -Gorenstein cluster tilting subcategories will be studied. In particular, we show that they are n-resolving, a higher version of resolving subcategories.

We study fibers of the commutator map on general linear maps using representation theory. Especially, we give detailed information on dimension and geometric connectedness of those fibers over regular semisimple and central elements.

Let $A$ be a Koszul Artin-Schelter regular algebra, $\sigma$ a graded automorphism of $A$ and $\delta$ a degree-one $\sigma$-derivation of $A$. We introduce an invariant for $\delta$ called the $\sigma$-divergence of $\delta$. We describe the Nakayama automorphism of the graded Ore extension $B=A[z;\sigma,\delta]$ explicitly using the $\sigma$-divergence of $\delta$, and construct a twisted superpotential $\hat{\omega}$ for $B$ so that it is a derivation quotient algebra defined by $\hat{\omega}$. We also determine all graded Ore extensions of noetherian Artin-Schelter regular algebras of dimension 2 and compute their Nakayama automorphisms.

For a semisimple complex Lie algebra g , the BGG category O is of particular interest in representation theory. It is known that Irving's shuffling functors Sh w , indexed by elements w ∈ W of the Weyl group, induce an action of the braid group B W associated to W on the derived categories D b ( O λ ) of blocks of O . We show that for maximal parabolic subalgebras p of sl n corresponding to the parabolic subgroup W p = S n − 1 × S 1 of S n , the derived shuffling functors L Sh s i are instances of Seidel and Thomas' spherical twist functors. Namely, we show that certain parabolic indecomposable projectives P p ( w ) are spherical objects, and the associated twist functors are naturally isomorphic to L Sh w [ 1 ] as auto-equivalences of D b ( O p ) . We give an overview of the main properties of the BGG category O , the construction of shuffling and spherical twist functors, and give some examples how to determine images of both. To this end, we employ the equivalence of blocks of O and the module categories of certain path algebras.

We compute integral models of real and cohomological induction for finite covering groups of PU(1,1)., Comment: Corollary 5.12 -- Remark 5.15 and their variants are added. To appear in Journal of Algebra 579

In this paper, we prove that the inductive blockwise Alperin weight condition holds for 2-blocks with abelian defect groups, which are not inertial and have a Klein four hyperfocal subgroup.

In this paper we study a generalisation of the Igusa-Todorov functions which gives rise to a vast class of algebras satisfying the finitistic dimension conjecture. This class of algebras is called Lat-Igusa-Todorov and includes, among others, the Igusa-Todorov algebras (defined by J. Wei) and the self-injective algebras which in general are not Igusa-Todorov algebras. Finally, some applications of the developed theory are given in order to relate the different homological dimensions which have been discussed through the paper., 18 pages, submitted to a peer-reviewed journal

An extremal element in a Lie algebra g over a field of characteristic not 2 is an element x ∈ g such that [ x , [ x , g ] ] is contained in the linear span of x. The linear span of an extremal element, called an extremal point, is an inner ideal of g , i.e. a subspace I satisfying [ I , [ I , g ] ] ≤ I . We show that in characteristic different from 2 , 3 the geometry with point set the set of extremal points and as lines the minimal inner ideals containing at least two extremal points is a Moufang spherical building, or in case there are no lines a Moufang set. This last result on the Moufang sets is obtained by connecting Lie algebras to structurable algebras, a class of non-associative algebras with involution generalizing Jordan algebras. It is shown that in characteristic different from 2 , 3 each finite-dimensional simple Lie algebra generated by extremal elements is either a symplectic Lie algebra or can be obtained by applying the Tits-Kantor-Koecher construction to a skew-dimension one structurable algebra. Various relations between the Lie algebra g and its extremal geometry on the one hand and the associated structurable algebra on the other hand are investigated.

We consider the problem of simultaneously finding lower-dimensional subspace structures in a given m-tuple ( X 1 , … , X m ) of possibly corrupted, high-dimensional data sets all of the same size. We refer to this problem as simultaneous robust subspace recovery (SRSR) and provide a quiver invariant theoretic approach to it. We show that SRSR is a particular case of the more general problem of effectively deciding whether a quiver representation is semi-stable (in the sense of Geometric Invariant Theory) and, in case it is not, finding a subrepresentation certifying in an optimal way that the representation is not semi-stable. In this paper, we show that SRSR and the more general quiver semi-stability problem can be solved effectively.

In this paper, we first construct the twisted full toroidal Lie algebra by an extension of a centreless Lie torus LT which is a multiloop algebra twisted by several automorphisms of finite order and equipped with a particular grading. We then provide a complete classification of all the irreducible integrable modules with finite-dimensional weight spaces for this twisted full toroidal Lie algebra having a non-trivial LT-action and where the centre of the underlying Lie algebra acts trivially.

We study the relation between two uncountable groups with remarkable properties (cf. [15] ): the topological free product of infinite cyclic groups G (the fundamental group of the Hawaiian Earring), and the inverse limit of finitely generated free groups F ˆ . The former has a canonical embedding as a proper subgroup of the latter and we examine when G , together with certain naturally defined normal subgroups of F ˆ generate the entire group F ˆ . We are interested in particular in normal subgroups Ker T ( F ˆ ) = ⋂ { Ker φ | φ ∈ hom ⁡ ( F ˆ , T ) } , where T is some finitely-presented n-slender group. Our main results state that if T is the infinite cyclic group or the free nilpotent class 2 group on 2 generators, then G and Ker T ( F ˆ ) generate F ˆ . On the other hand, if T is the free nilpotent class 3 group or a Baumslag-Solitar group, then the product of subgroups G ⋅ Ker T F ˆ is a proper subgroup of F ˆ . In the last section, we provide an interesting geometric interpretation of the above results in terms of path-connectedness of certain fibrations arising as inverse limits of covering spaces over the Hawaiian earring space.

In this paper, we prove that there is an equivariant basic Morita equivalence between an inertial block and its Brauer correspondent in the normalizer of a defect group of the block.

Axial algebras are non-associative algebras generated by semisimple idempotents, known as axes, that all obey a fusion rule. Axial algebras were introduced by Hall, Rehren and Shpectorov as a generalisation of the axioms of Majorana theory, which was in turn introduced by Ivanov as an axiomatisation of certain properties of the 2A-axes of the Griess algebra. Axial algebras of Monster type are axial algebras whose axes obey the Monster, or Majorana, fusion rule. We construct an axial algebra of Monster type $M_{4A}$ over the polynomial ring $\mathbb{R}[t]$ that is generated by six axes whose Miyamoto involutions generate an elementary abelian group of order $4$. This construction automatically provides an infinite-parameter family $\{M(t)\}_{t \in \mathbb{R}}$ of axial algebras of Monster type each of which admit a unique Frobenius form. Moreover, we show that this form on $M(t)$ is positive definite if and only if $0 < t < \frac{1}{6}$ and also satisfies Norton's inequality if and only if $0 \leq t \leq \frac{1}{6}$. Finally, we show that the $4A$ axes of $M_{4A}$ obey a $C_2 \times C_2$-graded fusion rule giving a new infinite family of fusion rules.

We give a full description of the Poisson structures on the finitary incidence algebra F I ( P , R ) of an arbitrary poset P over a commutative unital ring R.