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Let k be a field, let n ≥ 2 be a nonsquarefree integer not divisible by the characteristic of k. Assume that all roots of unity of degree n are contained in k. In the first part of the paper we consider pairs of symbol algebras over k with common slots D 1 ≃ (e , x) n ≃ (r , u) n , D 2 ≃ (e , y) n ≃ (r , v) n , exp D 1 = exp D 2 = n , and show that in general (e , x , y) n ≠ (r , u , v) n. As a consequence we prove that in general it is impossible to connect the pair { (e , x) n ; (e , y) n } and the pair { (r , u) n ; (r , v) n } by a chain of pairs of symbol algebras with a common slot and isomorphic to (D 1 ; D 2) in such a way that any two neighboring pairs in the chain are obtained from one another by a "natural" transformation. In the second part of the paper we prove that in contrast to the case n = 2 for any n divisible by 4 there exist symbol algebras D 1 , D 2 with deg ⁡ D 1 = deg ⁡ D 2 = n and exp D 1 = exp D 2 = n without common slot such that i D 1 + j D 2 is a symbol algebra of degree n for any i , j ∈ Z. [ABSTRACT FROM AUTHOR]

Let X be a finite connected poset, K a field of characteristic zero and I (X , K) the incidence algebra of X over K seen as a Lie algebra under the commutator product. In the first part of the paper we show that any 1 2 -derivation of I (X , K) decomposes into the sum of a central-valued 1 2 -derivation, an inner 1 2 -derivation and a 1 2 -derivation associated with a map σ : X < 2 → K that is constant on chains and cycles in X. In the second part of the paper we use this result to prove that any transposed Poisson structure on I (X , K) is the sum of a structure of Poisson type, a mutational structure and a structure determined by λ : X e 2 → K , where X e 2 is the set of (x , y) ∈ X 2 such that x < y is a maximal chain not contained in a cycle. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic 0. We study some cohomological properties of Lie subalgebras of the Witt algebra W = Der (k [ t , t − 1 ]) and the one-sided Witt algebra W ≥ − 1 = Der (k [ t ]). In the first part of the paper, we consider finite codimension subalgebras of W ≥ − 1. We compute derivations and one-dimensional extensions of such subalgebras. These correspond to Ext U (L) 1 (M , L) , where L is a subalgebra of W ≥ − 1 and M is a one-dimensional representation of L. We find that these subalgebras exhibit a kind of rigidity: their derivations and extensions are controlled by the full one-sided Witt algebra. As an application of these computations, we prove that any isomorphism between finite codimension subalgebras of W ≥ − 1 extends to an automorphism of W ≥ − 1. The second part of the paper is devoted to explaining the observed rigidity. We define a notion of "completely non-split extension" and prove that W ≥ − 1 is the universal completely non-split extension of any of its subalgebras of finite codimension. In some sense, this means that even when studying subalgebras of W ≥ − 1 as abstract Lie algebras, they remember that they are contained in W ≥ − 1. We also consider subalgebras of infinite codimension, explaining the similarities and differences between the finite and infinite codimension situations. Almost all of the results above are also true for subalgebras of the Witt algebra. We summarise results for W at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we classify simple Harish-Chandra modules over simple generalized Witt algebras. [ABSTRACT FROM AUTHOR]

The goal of this paper is to generalize a statement by Drinfeld, asserting that Yangians can be constructed as limit forms of the quantum loop algebras, to the super case. We establish a connection between quantum loop superalgebra and super Yangian of the general linear Lie superalgebra gl M | N in RTT type presentation. In particular, we derive the Poincaré-Birkhoff-Witt(PBW) theorem for the quantum loop superalgebra U q (Lgl M | N). Additionally, we investigate the application of the same argument to twisted super Yangian of the ortho-symplectic Lie superalgebra. For this purpose, we introduce the twisted quantum loop superalgebra as a one-sided coideal of U q (Lgl M | 2 n) with respect to the comultiplication. [ABSTRACT FROM AUTHOR]

For a ∈ C ⁎ there are two natural evaluation maps ev a and ev a from the affine Hecke algebra H ▵ (r) C to the Hecke algebra H (r) C. The maps ev a and ev a induce evaluation maps ev ˜ a and ev ˜ a from the affine quantum Schur algebra S ▵ (n , r) C to the quantum Schur algebra S (n , r) C , respectively. In this paper we prove that the evaluation map ev ˜ a (resp. ev ˜ a) is compatible with the evaluation map Ev a (resp. Ev (− 1) n a q n ) for quantum affine sl n. Furthermore we compute the Drinfeld polynomials associated with the simple S ▵ (n , r) C -modules which come from the simple S (n , r) C -modules via the evaluation maps ev ˜ a. Then we characterize finite-dimensional irreducible S ▵ (n , r) C -modules which are irreducible as S (n , r) C -modules for n > r. As an application, we characterize finite-dimensional irreducible modules for the affine Hecke algebra H ▵ (r) C which are irreducible as modules for the Hecke algebra H (r) C. [ABSTRACT FROM AUTHOR]

In this paper, a symplectic structure on a Leibniz algebra is defined to be a symmetric nondegenerate bilinear form satisfying certain compatibility condition, and a phase space of a Leibniz algebra is defined to be a symplectic Leibniz algebra satisfying certain conditions. We show that a Leibniz algebra has a phase space if and only if there is a compatible Leibniz-dendriform algebra, and phase spaces of Leibniz algebras are one-to-one corresponds to Manin triples of Leibniz-dendriform algebras. Product (paracomplex) structures and complex structures on Leibniz algebras are studied in terms of decompositions of Leibniz algebras. A para-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a paracomplex structure satisfying a compatibility condition. We show that a symplectic Leibniz algebra admits a para-Kähler structure if and only if the Leibniz algebra is the direct sum of two Lagrangian subalgebras as vector spaces. A complex product structure on a Leibniz algebra consists of a complex structure and a product structure satisfying a compatibility condition. A pseudo-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a complex structure satisfying a compatibility condition. Various properties and relations of complex product structures and pseudo-Kähler structures are studied. In particular, Leibniz-dendriform algebras give rise to complex product structures and pseudo-Kähler structures naturally. [ABSTRACT FROM AUTHOR]

In this paper we mainly consider the notion of Drinfeld double for two weak multiplier Hopf (⁎-)algebras which are paired with each other. Then we show that the Drinfeld double is again a weak multiplier Hopf (⁎-)algebra. Furthermore, we study integrals on the Drinfeld double. Finally, we establish the correspondence between modules over a Drinfeld double D (A) and Yetter-Drinfeld modules over a weak algebraic quantum group A. [ABSTRACT FROM AUTHOR]

This paper concerns commutative plenary train algebras of rank m and their idempotents. We obtain the Peirce decomposition of these algebras having an idempotent element and the multiplication table of Peirce components when the plenary train roots are mutually different. We show that a backcrossing algebra is a plenary train algebra of rank m if and only if, it is a principal train one of rank m. For the backcrossing train algebras, we confirm a first conjecture of Juan Carlos Gutiérrez Fernández on the relation between plenary train roots and principal train roots. A second conjecture on the existence of idempotent in train algebras also obtains a positive answer in the class of backcrossing algebras. [ABSTRACT FROM AUTHOR]

We deal with Perazzo hypersurfaces X = V (f) in P n + 2 defined by a homogeneous polynomial f (x 0 , x 1 , ... , x n , u , v) = p 0 (u , v) x 0 + p 1 (u , v) x 1 + ⋯ + p n (u , v) x n + g (u , v) , where p 0 , p 1 , ... , p n are algebraically dependent but linearly independent forms of degree d − 1 in K [ u , v ] and g is a form in K [ u , v ] of degree d. Perazzo hypersurfaces have vanishing hessian and, hence, the associated graded artinian Gorenstein algebra A f fails the strong Lefschetz property. In this paper, we first determine the maximum and minimum Hilbert function of A f , we prove that the Hilbert function of A f is always unimodal and we determine when A f satisfies the weak Lefschetz property. We illustrate our results with many examples and we show that our results do not generalize to Perazzo hypersurfaces X = V (f) in P n + 3 defined by a homogeneous polynomial f (x 0 , x 1 , ... , x n , u , v , w) = p 0 (u , v , w) x 0 + p 1 (u , v , w) x 1 + ⋯ + p n (u , v , w) x n + g (u , v , w) , where p 0 , p 1 , ... , p n are algebraically dependent but linearly independent forms of degree d − 1 in K [ u , v , w ] and g is a form in K [ u , v , w ] of degree d. [ABSTRACT FROM AUTHOR]

For a smooth affine algebra R of dimension d ≥ 3 over a field k and an invertible alternating matrix χ of rank 2 n , the group S p (χ) of invertible matrices of rank 2 n over R which are symplectic with respect to χ acts on the right on the set U m 2 n (R) of unimodular rows of length 2 n over R. In this paper, we prove that S p (χ) acts transitively on U m 2 n (R) if k is algebraically closed, d ! ∈ k × and 2 n ≥ d. [ABSTRACT FROM AUTHOR]

The extension dimensions of an Artin algebra give a reasonable way of measuring how far an algebra is from being representation-finite. In this paper we mainly study the behavior of the extensions dimensions of algebras under different equivalences. We show that the difference of the extension dimensions of two derived equivalent algebras is bounded above by the length of the tilting complex associated with the derived equivalence, and that the extension dimension is an invariant under the stable equivalence. In addition, we provide two sufficient conditions such that the extension dimension is an invariant under particular derived equivalences. [ABSTRACT FROM AUTHOR]

We present a formula for the class number of a multinorm one torus T L / k associated to any étale algebra L over a global field k. This is deduced from a formula for analogues of invariants introduced by T. Ono, which are interpreted as a generalization of Gauss genus theory. This paper includes the variants of Ono's invariant for arbitrary S -ideal class numbers and the narrow version, generalizing results of Katayama, Morishita, Ono and Sasaki. [ABSTRACT FROM AUTHOR]

Let A be a finite-dimensional algebra, and M be a d -cluster tilting subcategory of mod A. From the viewpoint of higher homological algebra, a natural question to ask is when M induces a d -cluster tilting subcategory in Mod A. In this paper, we investigate this question in a more general form. We consider M as an essentially small d -abelian category, known to be equivalent to a d -cluster tilting subcategory of an abelian category A. The completion of M , denoted by Ind (M) , is defined as the universal completion of M with respect to filtered colimits. We explore Ind (M) and demonstrate its equivalence to the full subcategory L d (M) of Mod M , comprising left d -exact functors. Notably, Ind (M) as a subcategory of Mod M Eff (M) falls short of being a d -cluster tilting subcategory since it satisfies all properties of a d -cluster tilting subcategory except d -rigidity. For a d -cluster tilting subcategory M of mod A , M → consists of all filtered colimits of objects from M , is a generating-cogenerating, functorially finite subcategory of Mod A. The question of whether M → is a d -rigid subcategory remains unanswered. However, if it is indeed d -rigid, it qualifies as a d -cluster tilting subcategory. In the case d = 2 , employing cotorsion theory, we establish that M → is a 2-cluster tilting subcategory if and only if M is of finite type. Thus, the question regarding whether M → is a d -cluster tilting subcategory of Mod A appears to be equivalent to Iyama's question about the finiteness of M. Furthermore, for general d , we address the problem and present several equivalent conditions for Iyama's question. [ABSTRACT FROM AUTHOR]

A pseudo-Euclidean non-associative algebra (g , •) is a finite dimensional algebra over a field K that has a metric, i.e., a bilinear, symmetric, and non-degenerate form 〈 , 〉. The metric is considered L-invariant (resp. R-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if 〈 u • v , w 〉 = 〈 u , v • w 〉 for all u , v , w ∈ g. These three notions coincide when g is a Lie algebra and in this case g endowed with the metric is known as a quadratic Lie algebra. This paper provides a complete description of L-invariant, R-invariant, or associative pseudo-Euclidean metrics on left Leibniz algebras over a commutative field of characteristic zero. It shows that a left Leibniz algebra with an associative metric is also right Leibniz and can be obtained easily from its underlying Lie algebra, which is a quadratic Lie algebra. Additionally, it shows that at the core of a left Leibniz algebra endowed with a L-invariant or R-invariant metric, there are two Lie algebras with one quadratic and the left Leibniz algebra can be built from these Lie algebras. We derive many important results from this complete description. Finally, the paper provides a list of left Leibniz algebras with an associative metric up to dimension 6, as well as a list of left Leibniz algebras with an L-invariant metric, up to dimension 4, and R-invariant metric up to dimension 5. [ABSTRACT FROM AUTHOR]

BiHom-associative algebras are generalized associative algebras with two multiplicative linear maps. In this paper, we give the Hochschild homology and cyclic homology structure of BiHom-associative algebras. Then, generalize the dual bimodule action to define the cyclic cohomology. Finally, we introduce the crossed modules of BiHom-associative algebras and show that the Hochschild cohomology of BiHom-associative algebra classifies crossed modules. [ABSTRACT FROM AUTHOR]

Let g be a complex simple Lie algebra and Z (g) be the center of the universal enveloping algebra U (g). Denote by V λ the finite-dimensional irreducible g -module with highest weight λ. Lehrer and Zhang defined the notion of strongly multiplicity free representations for simple Lie algebras motivated by studying the structure of the endomorphism algebra End U (g) (V λ ⊗ r) in terms of the quotients of the Kohno's infinitesimal braid algebra. Kostant introduced the g -invariant endomorphism algebras R λ (g) = (End V λ ⊗ U (g)) g and R λ , π (g) = (End V λ ⊗ π (U (g))) g. In this paper, we give some other criteria for a multiplicity free representation to be strongly multiplicity free by classifying the pairs (g , V λ) , which are multiplicity free and for such pairs, R λ (g) and R λ , π (g) are generated by generalizations of the quadratic Casimir elements of Z (g). [ABSTRACT FROM AUTHOR]

Hopf monads generalise Hopf algebras. They clarify several aspects of the theory of Hopf algebras and capture several related structures such as weak Hopf algebras and Hopf algebroids. However, important parts of Hopf algebra theory are not reached by Hopf monads, most noticeably Drinfeld's quasi-Hopf algebras. In this paper we introduce a generalisation of Hopf monads, that we call slack Hopf monads. This generalisation retains a clean theory and is flexible enough to encompass quasi-Hopf algebras as examples. A slack Hopf monad is a colax magma monad T on a magma category C such that the forgetful functor U T : C T → C 'slackly' preserves internal Homs. We give a number of different descriptions of slack Hopf monads, and study special cases such as slack Hopf monads on cartesian categories and k -linear exact slack Hopf monads on Vect k , that is comagma algebras such that a modified fusion operator is invertible. In particular, we characterise quasi-Hopf algebras in terms of slackness. [ABSTRACT FROM AUTHOR]

Previously we showed that the tensor product of a weight module over a generalized Weyl algebra (GWA) with a weight module over another GWA is a weight module over a third GWA. In this paper we compute tensor products of simple and indecomposable weight modules over generalized Weyl algebras supported on a finite orbit. This allows us to give a complete presentation by generators and relations of the Grothendieck ring of the categories of weight modules over a tower of generalized Weyl algebras in this setting. We also obtain partial results about the split Grothendieck ring. We described the case of infinite orbits in previous work. [ABSTRACT FROM AUTHOR]

In a categorification of skew-symmetric cluster algebras, each cluster variable corresponds with an indecomposable module over the associated Jacobian algebra. Buan, Marsh and Reiten studied when the denominator vector of each cluster variable in an acyclic cluster algebra coincides with the dimension vector of the corresponding module. In this paper, we give analogues of their results for cluster algebras from triangulated surfaces by comparing two kinds of intersection numbers of tagged arcs. [ABSTRACT FROM AUTHOR]

Dlab–Ringel's standardization method gives a realization of a standardly stratified algebra. In this paper, we construct mixed stratified algebras, which are a generalization of standardly stratified algebras, following Dlab–Ringel's standardization method. Moreover, we study a Ringel duality of mixed stratified algebras from the viewpoint of stratifying systems. [ABSTRACT FROM AUTHOR]

Lie-Yamaguti algebras generalize both the notions of Lie algebras and Lie triple systems. In this paper, we consider the inducibility problem for automorphisms of Lie-Yamaguti algebra extensions. More precisely, given an abelian extension [Display omitted] of a Lie-Yamaguti algebra L , we are interested in finding the pairs (ϕ , ψ) ∈ Aut (V) × Aut (L) , which are inducible by an automorphism in Aut (L ˜). We connect the inducibility problem to the (2 , 3) -cohomology of Lie-Yamaguti algebra. In particular, we show that the obstruction for a pair of automorphisms in Aut (V) × Aut (L) to be inducible lies in the (2 , 3) -cohomology group H (2 , 3) (L , V). We develop the Wells exact sequence for Lie-Yamaguti algebra extensions, which relates the space of derivations, automorphism groups, and (2 , 3) -cohomology groups of Lie-Yamaguti algebras. As an application, we describe certain automorphism groups of semi-direct product Lie-Yamaguti algebras. In a sequel, we apply our results to discuss inducibility problem for nilpotent Lie-Yamaguti algebras of index 2. We give examples of infinite families of such nilpotent Lie-Yamaguti algebras and characterize the inducible pairs of automorphisms for extensions arising from these examples. Finally, we write an algorithm to find out all the inducible pairs of automorphisms for extensions arising from nilpotent Lie-Yamaguti algebras of index 2. [ABSTRACT FROM AUTHOR]

Primitive axial algebras of Monster type are a class of non-associative algebras with a strong link to finite (especially simple) groups. The motivating example is the Griess algebra, with the Monster as its automorphism group. A crucial step towards the understanding of such algebras is the explicit description of the 2-generated symmetric objects. Recent work of Yabe, and Franchi and Mainardis shows that any such algebra is either explicitly known, or is a quotient of the infinite-dimensional Highwater algebra H , or its characteristic 5 cover H ˆ. In this paper, we complete the classification of symmetric axial algebras of Monster type by determining the quotients of H and H ˆ. We proceed in a unified way, by defining a cover of H in all characteristics. This cover has a previously unseen fusion law and provides an insight into why the Highwater algebra has a cover which is of Monster type only in characteristic 5. [ABSTRACT FROM AUTHOR]

In this paper, we associate the TKK algebra G ˆ (J) with vertex algebras through twisted modules. Firstly, we prove that for any complex number ℓ , the category of restricted G ˆ (J) -modules of level ℓ is canonically isomorphic to the category of σ -twisted V C g ˆ (ℓ , 0) -modules, where V C g ˆ (ℓ , 0) is a vertex algebra arising from the toroidal Lie algebra of type C 2 and σ is an isomorphism of V C g ˆ (ℓ , 0) induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer ℓ , the integrable restricted G ˆ (J) -modules of level ℓ are exactly the σ -twisted modules for the quotient vertex algebra L C g ˆ (ℓ , 0) of V C g ˆ (ℓ , 0). Finally, we classify the irreducible 1 2 N -graded σ -twisted L C g ˆ (ℓ , 0) -modules. [ABSTRACT FROM AUTHOR]

The bricks over preprojective algebras of type A are known to be in bijection with certain combinatorial objects called "arcs". In this paper, we show how one can use arcs to compute bases for the Hom-spaces and first extension spaces between bricks. We then use this description to classify the "weak exceptional sequences" over these algebras. Finally, we explain how our result relates to a similar combinatorial model for the exceptional sequences over hereditary algebras of type A. [ABSTRACT FROM AUTHOR]

Let F be an infinite field and M 1 , 1 (F) be the algebra of 2 × 2 matrices over F endowed with non-trivial Z 2 -grading. We consider the involutions ⁎ defined on M 1 , 1 (F) which preserve the homogeneous components of the grading. In this paper, we deal with the ⁎-superalgebra (M 1 , 1 (F) , ⁎) and determine the generators of its ideal of (Z 2 , ⁎) -identities, considering that F has characteristic zero and also, we explicitly construct the generators of its space of central (Z 2 , ⁎) -polynomials, when the characteristic of F is different from 2. [ABSTRACT FROM AUTHOR]

In this paper, we classify irreducible modules for map extended Witt algebras with finite dimensional weight spaces. They turn out to be either modules with uniformly bounded weight spaces or highest weight modules. We further prove that all these modules are single point evaluation modules (n ≥ 2). So they are actually irreducible modules for extended Witt algebras. [ABSTRACT FROM AUTHOR]

Let U q − be the negative half of the quantum group associated to the Kac-Moody algebra g , and U _ q − the quantum group obtained by a folding of g. Let A = Z [ q , q − 1 ]. McNamara showed that U _ q − is categorified over a certain extension ring A ˜ of A , by using the folding theory of KLR algebras. He posed a question whether A ˜ coincides with A or not. In this paper, we give an affirmative answer for this problem. [ABSTRACT FROM AUTHOR]

In this paper, we constructed and studied two classes of modules over the Heisenberg-Virasoro algebra HVir from Weyl algebra modules, parallel to the similar construction for Virasoro algebra in [13,9]. More precisely, given any module P over the degree-1 Weyl algebra, we define the HVir-modules M (P , V) , where V is a restricted module over the nonnegative part a of HVir, and L (P , W , ξ , b , c) , where W is a restricted module over another subalgebra b ⊆ HVir and ξ , b , c ∈ C with ξ ≠ 0. We determined the irreducibility of these modules as well as the isomorphisms within each class respectively. Moreover, any two such modules from different classes are never isomorphic although they might be closely related, as we showed in the last section. [ABSTRACT FROM AUTHOR]

Let D be an alternative division ring with characteristic different from two. The purpose of this paper is to characterize additive mappings f , g : D → D satisfying certain identities studied by Vukman, Brešar, and Catalano previously on an associative division ring. We also present some new identities concerning Lie and Jordan products, and provide a complete description of commuting maps on octonion algebras. [ABSTRACT FROM AUTHOR]

In this paper we study the discrete-time dynamical systems associated with gonosomal algebras used as algebraic model in the sex-linked genes inheritance. We show that the class of gonosomal algebras is disjoint from the other non-associative algebras usually studied (Lie, alternative, Jordan, associative power). To each gonosomal algebra, with the mapping x ↦ 1 2 x 2 , an evolution operator W is associated that gives the state of the offspring population at the birth stage, then from W we define the operator V which gives the frequency distribution of genetic types. We study discrete-time dynamical systems generated by these two operators, in particular we show that the various stability notions of the equilibrium points are preserved by passing from W to V. Moreover, for the evolution operators associated with genetic disorders in the case of a diallelic gonosomal lethal gene we give complete analysis of fixed and limit points of the dynamical systems. [ABSTRACT FROM AUTHOR]

Let K be a field and let J n , k be the Jordan algebra of a degenerate symmetric bilinear form b of rank n − k over K. Then one can consider the decomposition J n , k = B n − k ⊕ D k , where B n − k represents the corresponding Jordan algebra, denoted as B n − k = K ⊕ V. In this algebra, the restriction of b on the (n − k) -dimensional subspace V is non-degenerate, while D k accounts for the degenerate part of J n , k. This paper aims to provide necessary and sufficient conditions to check if a given multilinear polynomial is an identity for J n , k. As a consequence of this result and under certain hypothesis on the base field, we exhibit a finite basis for the T -ideal of polynomial identities of J n , k. Over a field of characteristic zero, we also prove that the ideal of identities of J n , k satisfies the Specht property. Moreover, similar results are obtained for weak identities, trace identities and graded identities with a suitable Z 2 -grading as well. In all of these cases, we employ methods and results from Invariant Theory. Finally, as a consequence from the trace case, we provide a counterexample to the embedding problem given in [8] in case of infinite dimensional Jordan algebras with trace. [ABSTRACT FROM AUTHOR]

A fundamental ingredient in the noncommutative geometry program is the notion of KK-duality, often called K-theoretic Poincaré duality, that generalises Spanier-Whitehead duality. In this paper we construct a θ -summable Fredholm module that represents the fundamental class in KK-duality between the stable and unstable Ruelle algebras of a Smale space. To find such a representative, we construct dynamical partitions of unity on the Smale space with highly controlled Lipschitz constants. This requires a generalisation of Bowen's Markov partitions. Along with an aperiodic point-sampling technique we produce a noncommutative analogue of Whitney's embedding theorem, leading to the Fredholm module. [ABSTRACT FROM AUTHOR]

We resolve the long-standing problem of constructing the action of the operad of framed (stable) genus-0 curves on Hamiltonian Floer theory; this operad is equivalent to the framed E 2 operad. We formulate the construction in the following general context: we associate to each compact subset of a closed symplectic manifold a new chain-level model for symplectic cohomology with support, which we show carries an action of a model for the chains on the moduli space of framed genus 0 curves. This construction turns out to be strictly functorial with respect to inclusions of subsets, and the action of the symplectomorphism group. In the general context, we appeal to virtual fundamental chain methods to construct the operations over fields of characteristic 0, and we give a separate account, over arbitrary rings, in the special settings where Floer's classical transversality approach can be applied. We perform all constructions over the Novikov ring, so that the algebraic structures we produce are compatible with the quantitative information that is contained in Floer theory. Over fields of characteristic 0, our construction can be combined with results in the theory of operads to produce explicit operations encoding the structure of a homotopy BV algebra. In an appendix, we explain how to extend the results of the paper from the class of closed symplectic manifolds to geometrically bounded ones. [ABSTRACT FROM AUTHOR]

We study linear PDE with constant coefficients. The constant rank condition on a system of linear PDEs with constant coefficients is often used in the theory of compensated compactness. While this is a purely linear algebraic condition, the nonlinear algebra concept of primary decomposition is another important tool for studying such system of PDEs. In this paper we investigate the connection between these two concepts. From the nonlinear analysis point of view, we make some progress in the study of weak lower semicontinuity of integral functionals defined on sequences of PDE constrained fields, when the PDEs do not have constant rank. [ABSTRACT FROM AUTHOR]

The aim of this paper is to unify classification theories of torsion classes of finite dimensional algebras and commutative Noetherian rings. For a commutative Noetherian ring R and a module-finite R -algebra Λ, we study the set tors Λ (respectively, torf Λ) of torsion (respectively, torsionfree) classes of the category of finitely generated Λ-modules. We construct a bijection from torf Λ to ∏ p torf (κ (p) ⊗ R Λ) , and an embedding Φ t from tors Λ to T R (Λ) : = ∏ p tors (κ (p) ⊗ R Λ) , where p runs over all prime ideals of R. When Λ = R , these give classifications of torsionfree classes, torsion classes and Serre subcategories of mod R due to Takahashi, Stanley-Wang and Gabriel. To give a description of Im Φ t , we introduce the notion of compatible elements in T R (Λ) , and prove that all elements in Im Φ t are compatible. We give a sufficient condition on (R , Λ) such that all compatible elements belong to Im Φ t (we call (R , Λ) compatible in this case). For example, if R is semi-local and dim ⁡ R ≤ 1 , then (R , Λ) is compatible. We also give a sufficient condition in terms of silting Λ-modules. As an application, for a Dynkin quiver Q , (R , R Q) is compatible and we have a poset isomorphism tors R Q ≃ Hom poset (Spec R , C Q) for the Cambrian lattice C Q of Q. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new approach that allows to identify the Gelfand spectrum of weighted Fourier algebras as a subset of an abstract complexification of the corresponding group for a wide class of groups and weights. This generalizes recent related results of Ghandehari-Lee-Ludwig-Spronk-Turowska [11] about the spectrum of Beurling-Fourier algebras on some Lie groups. In the case of discrete groups we show that the spectrum of Beurling-Fourier algebra is homeomorphic to the group itself. [ABSTRACT FROM AUTHOR]

Cluster algebra structures for Grassmannians and their (open) positroid strata are controlled by a Postnikov diagram D or, equivalently, a dimer model on the disc, as encoded by either a bipartite graph or the dual quiver (with faces). The associated dimer algebra A , determined directly by the quiver with a certain potential, can also be realised as the endomorphism algebra of a cluster-tilting object in an associated Frobenius cluster category. In this paper, we introduce a class of A -modules corresponding to perfect matchings of the dimer model of D and show that, when D is connected, the indecomposable projective A -modules are in this class. Surprisingly, this allows us to deduce that the cluster category associated to D embeds into the cluster category for the appropriate Grassmannian. We show that the indecomposable projectives correspond to certain matchings which have appeared previously in work of Muller–Speyer. This allows us to identify the cluster-tilting object associated to D , by showing that it is determined by one of the standard labelling rules constructing a cluster of Plücker coordinates from D. By computing a projective resolution of every perfect matching module, we show that Marsh–Scott's formula for twisted Plücker coordinates, expressed as a dimer partition function, is a special case of the general cluster character formula, and thus observe that the Marsh–Scott twist can be categorified by a particular syzygy operation in the Grassmannian cluster category. [ABSTRACT FROM AUTHOR]

Consider a Borcherds-Kac-Moody Lie superalgebra, denoted as g , associated with the graph G. This Lie superalgebra is constructed from a free Lie superalgebra by introducing three sets of relations on its generators: (1) Chevalley relations, (2) Serre relations, and (3) The commutation relations derived from the graph G. The Chevalley relations lead to a triangular decomposition of g as g = n + ⊕ h ⊕ n − , where each root space g α is contained in either n + or n −. Importantly, each g α is determined solely by relations (2) and (3). This paper focuses on the root spaces of g that are unaffected by the Serre relations. We refer to these root spaces as "free roots" of g (these root spaces are free from the Serre relations and can be associated with certain grade spaces of freely partially commutative Lie superalgebras, as detailed in Lemma 3.10. Consequently, we refer to them as "free roots," and the corresponding root spaces in g as "free root spaces" [cf. Definition 2.6]). Since these root spaces only involve commutation relations derived from the graph G , we can examine them purely from a combinatorial perspective. We employ heaps of pieces to analyze these root spaces and establish various combinatorial properties. We develop two distinct bases for these root spaces of g : We extend Lalonde's Lyndon heap basis, originally designed for free partially commutative Lie algebras, to accommodate free partially commutative Lie superalgebras. We expand upon the basis introduced in the reference [1] , designed for the free root spaces of Borcherds algebras, to encompass BKM superalgebras. This extension is achieved by investigating the combinatorial properties of super Lyndon heaps. Additionally, we also explore several other combinatorial properties related to free roots. [ABSTRACT FROM AUTHOR]

In this paper, we provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs in two different ways: (1) We represent spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper-Hafnians (orders and signs are not considered); (2) We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam (2004) [1] and Pfaffian matrix tree theorem by Masbaum and Vaintrob (2002) [15]. • We provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper- Hafnians when orders and signs are not considered. • We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam [Advances in Applied Mathematics (2004) Vol. 33: 51-70] and Pfaffian matrix tree theorem by Masbaum and Vaintrob [Internat. Math. Res. Notices (2002) Vol. 27: 1397-1426]. [ABSTRACT FROM AUTHOR]

The isotropic Grassmannian parametrizes isotropic subspaces of a vector space equipped with a quadratic form. In this paper, we show that any maximal isotropic Grassmannian in its Plücker embedding can be defined by pulling back the equations of G r iso (3 , 7) or G r iso (4 , 8). [ABSTRACT FROM AUTHOR]

In this paper we develop a new groupoid-based structure theory for the class of regular ⁎-semigroups. This class occupies something of a 'sweet spot' between the important classes of inverse and regular semigroups, and contains many natural examples. Some of the most significant families include the partition, Brauer and Temperley-Lieb monoids, among other diagram monoids. Our main result is that the category of regular ⁎-semigroups is isomorphic to the category of so-called 'chained projection groupoids'. Such a groupoid is in fact a triple (P , G , ε) , where: • P is a projection algebra (in the sense of Imaoka and Jones), • G is an ordered groupoid with object set P , and • ε : C → G is a special functor, where C is a certain natural 'chain groupoid' constructed from P. Roughly speaking: the groupoid G = G (S) remembers only the 'easy' products in a regular ⁎-semigroup S ; the projection algebra P = P (S) remembers only the 'conjugation action' of the projections of S ; and the functor ε = ε (S) tells us how G and P 'fit together' in order to recover the entire structure of S. In this way, we obtain the first completely general structure theorem for regular ⁎-semigroups. As a consequence of our main result, we give a new proof of the celebrated Ehresmann–Schein–Nambooripad Theorem, which establishes an isomorphism between the categories of inverse semigroups and inductive groupoids. Other applications will be given in future works. We consider several examples along the way, and pose a number of problems that we believe are worthy of further attention. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic different from 2. Small-dimensional Jordan algebras over F have been extensively studied and classified. In the present paper we show that any two-dimensional Jordan algebras over a finite field are isomorphic if and only if they satisfy the same polynomial identities (the opposite happens in the case F is infinite, even if algebraically closed). We determine a finite generating set for the T-ideal of the polynomial identities of every two-dimensional Jordan algebra when F is finite, and linear bases for the corresponding relatively free algebras are also determined. [ABSTRACT FROM AUTHOR]

We relate the group structure of van der Kallen on orbit sets of unimodular rows ([26]) with values in a smooth algebra A over a field k with the motivic cohomotopy groups of X = Spec A with coefficients in A n ∖ 0 in the sense of [5]. In the last section, we compare the motivic cohomotopy theory studied in this paper and defined by A n + 1 ∖ 0 or, equivalently, by an A 1 -weakly equivalent quadric Q 2 n + 1 to that considered in [5] , defined by a quadric Q 2 n , by means of explicit morphisms Q 2 n + 1 → Q 2 n , Q 2 n × G m → Q 2 n + 1 of quadrics. [ABSTRACT FROM AUTHOR]