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In a recent paper, the author defined an operation of tensor product for a large class of 2-representations of U + , the positive half of the 2-category associated to sl 2. In this paper, we prove that the operation extends to give an operation of tensor product for 2-representations of the full 2-category U : when the inputs are 2-representations of the full U , the 2-product is also a 2-representation of the full U. As in the previous paper, the 2-product is given for a simple 2-representation L (1) and an abelian 2-representation V taken from the 2-category of algebras. This is the first construction of an operation of tensor product for higher representations of a full Lie algebra in the abelian setting. [ABSTRACT FROM AUTHOR]

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]

Abstract: In 1.9 of the paper [A. Hahn, A Hermitian Morita theorem for algebras with anti-structure, J. Algebra 93 (1985) 215–235], should be replaced by . This leads to minor changes in the rest of the paper where the ring should be replaced by its opposite and vice versa. [Copyright &y& Elsevier]

The computational aspect of tropical Gröbner bases for a polynomial ring K [ x ] with respect to tropical term orders studied by Chan and Maclagan in 2019 is extended to the Weyl algebra D n (K) , where K is a field with a valuation. The term order in this paper is not only an extension of the tropical term order on K [ x ] by Chan and Maclagan, but also of the tropical term order on K [ x ] studied by Vaccon et al. (2021). Due to the involvement of the valuations of term coefficients, this term order is not well-ordering. Therefore, a suitable division algorithm with respect to this term order is needed. This algorithm holds only for homogeneous operators, so utilizing the homogenized Weyl algebra is required. A computation example and an implementation in Risa/Asir Computer Algebra System are also presented in this paper. [ABSTRACT FROM AUTHOR]

For an eventually periodic module, we have the degree and the period of its first periodic syzygy. This paper studies the former under the name "periodic dimension". We give a bound for the periodic dimension of an eventually periodic module with finite Gorenstein projective dimension. We also provide a method of computing the Gorenstein projective dimension of an eventually periodic module under certain conditions. Besides, motivated by recent results of Dotsenko, Gélinas and Tamaroff and of the author, we determine the bimodule periodic dimension of an eventually periodic Gorenstein algebra. Another aim of this paper is to obtain some of the basic homological properties of eventually periodic algebras. We show that a lot of homological conjectures hold for this class of algebras. As an application, we characterize eventually periodic Gorenstein algebras in terms of bimodules Gorenstein projective dimensions. [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]

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]

Almost split sequences lie in the heart of Auslander-Reiten theory. This paper deals with the structure of almost split sequences with certain ending terms in the morphism category of an Artin algebra Λ. Firstly we try to interpret the Auslander-Reiten translates of particular objects in the morphism category in terms of the Auslander-Reiten translations within the category of Λ-modules, and then use them to calculate almost split sequences. In classical representation theory of algebras, it is quite important to recognize the middle term of almost split sequences. As such, another part of the paper is devoted to discuss the middle term of certain almost split sequences in the morphism category of Λ. As an application, we restrict in the last part of the paper to self-injective algebras and present a structural theorem that illuminates a link between representation-finite morphism categories and Dynkin diagrams. [ABSTRACT FROM AUTHOR]

This paper gives a necessary and sufficient condition for Gorensteinness in Rees algebras of the d th power of parameter ideals in certain Noetherian local rings of dimension d ≥ 2. The main result of this paper produces many Gorenstein Rees algebras over non-Cohen-Macaulay local rings. For example, the Rees algebra R (q d) = ⊕ i ≥ 0 q d i is Gorenstein for every parameter ideal q that is a reduction of the maximal ideal in a d -dimensional Buchsbaum local ring of depth 1 and multiplicity 2. [ABSTRACT FROM AUTHOR]

Over fields of characteristic 2, Specht modules may decompose and there is no upper bound for the dimension of their endomorphism algebra. A classification of the (in)decomposable Specht modules and a closed formula for the dimension of their endomorphism algebra remain two important open problems in the area. In this paper, we introduce a novel description of the endomorphism algebra of the Specht modules and provide infinite families of Specht modules with one-dimensional endomorphism algebra. [ABSTRACT FROM AUTHOR]

Skew axets were first defined by McInroy and Shpectorov where they used the term of axets to classify shapes of an algebra. When they first submitted their paper, it was not known if skew axial algebras exist and now we will present such examples with axet X ′ (1 + 2). Looking at 2-generated primitive axial algebras of Monster type, we will be able to state and prove the classification of such algebras with axet X ′ (1 + 2). We will conclude by looking at larger skew axets and give a suggestion on how one could extend the classification. [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]

Let F be a non-archimedean local field and G be a connected reductive group over F. For a Bernstein block in the category of smooth complex representations of G (F) , we have two kinds of progenerators: the compactly induced representation ind K G (F) (ρ) of a type (K , ρ) , and the parabolically induced representation I P G (Π M) of a progenerator Π M of a Bernstein block for a Levi subgroup M of G. In this paper, we construct an explicit isomorphism of these two progenerators. Moreover, we compare the description of the endomorphism algebra End G (F) (ind K G (F) (ρ)) for a depth-zero type (K , ρ) in [20] with the description of the endomorphism algebra End G (F) (I P G (Π M)) in [33] , that are described in terms of affine Hecke algebras. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new class of simple smooth modules over the affine algebra A 1 (1). Specifically, we present a family of sl (2) ˆ -modules W φ for functions φ which are not Whittaker functions, and provide a simplicity criterion as well as an isomorphism theorem for such modules. For each simple sl (2) ˆ -module W φ , a class of simple sl (2) ˜ -module W φ , γ is also constructed for any constant γ. The sl (2) ˆ -modules W φ are weight modules, while the sl (2) ˜ -modules W φ , γ are not. [ABSTRACT FROM AUTHOR]

In this paper we study the (Cohen-Macaulay) type of orders over Dedekind domains in étale algebras. We provide a bound for the type, and give formulas to compute it. We relate the type of the overorders of a given order to the size of minimal generating sets of its fractional ideals, generalizing known results for Gorenstein and Bass orders. Finally, we give a classification of the ideal classes with multiplicator ring of type 2, with applications to the computations of the conjugacy classes of integral matrices and the isomorphism classes of abelian varieties over finite fields. [ABSTRACT FROM AUTHOR]

In this paper, we classify the simple Harish-Chandra modules over the superconformal current algebra g ˆ , which is the semi-direct sum of the N = 1 superconformal algebra with the affine Lie superalgebra g ˙ ⊗ A ⊕ C C 1 , where g ˙ is a finite-dimensional simple Lie algebra, and A is the tensor product of the Laurent polynomial algebra and the Grassmann algebra. As an application, we can directly get the classification of the simple Harish-Chandra modules over the N = 1 Heisenberg-Virasoro algebra. [ABSTRACT FROM AUTHOR]

In this paper, we classify 1-cocycles of the Witt algebra with coefficients in the tensor product of two arbitrary tensor density modules. In a special case, we recover a theorem originally established by Ng and Taft in [24]. Furthermore, by these 1-cocycles, we determine Lie bialgebra structures over certain infinite-dimensional Lie algebras containing the Witt algebra. [ABSTRACT FROM AUTHOR]

In [21] , we give a Ringel-Hall algebra approach to the canonical bases in the symmetric affine cases. In this paper, we extend the results to general symmetrizable affine cases by using Ringel-Hall algebras of representations of a valued quiver. We obtain a bar-invariant basis B ′ = { C (c , t λ) | (c , t λ) ∈ G a } in the generic composition algebra C ⁎ and prove that B ′ = B ′ ⊔ (− B ′) coincides with Lusztig's signed canonical basis B. Moreover, in type B ˜ n , C ˜ n , B ′ is the canonical basis B. [ABSTRACT FROM AUTHOR]

A pseudo-Euclidean left-symmetric algebra (A ,. , 〈 , 〉) is a left-symmetric algebra endowed with a non-degenerate symmetric bilinear form 〈 , 〉 such that left multiplications by any element of A are skew-symmetric with respect to 〈 , 〉. We recall that a pseudo-Euclidean Lie algebra (g , [ , ] , 〈 , 〉) is flat if and only if (g ,. , 〈 , 〉) , its underlying vector space endowed with the Levi-Civita product associated with 〈 , 〉 , is a pseudo-Euclidean left-symmetric algebra. In this paper, we study pseudo-Euclidean left-symmetric algebras (A ,. , 〈 , 〉) such that commutators of all elements of A are contained in the left annihilator of (A ,.) , these algebras will be called pseudo-Euclidean left-symmetric L -algebras. Next, we introduce and study pseudo-Euclidean modules of pseudo-Euclidean left-symmetric L -algebras. We develop double extension processes that allow us to have inductive descriptions of all pseudo-Euclidean left-symmetric L -algebras and of all its pseudo-Euclidean modules of any signature. This, in particular, improves some of the results obtained in [11]. [ABSTRACT FROM AUTHOR]

Kauer moves are local moves of an edge in a Brauer graph that yield derived equivalences between Brauer graph algebras [10]. These derived equivalences may be interpreted in terms of silting mutations. In this paper, we generalize the notion of Kauer moves to any finite number of edges. Their construction is based on cutting and pasting actions on the Brauer graph. To define these actions, we use an alternative definition of Brauer graphs coming from combinatorial topology [12]. Using the link between Brauer graph algebras and gentle algebras via the trivial extension [19] , we show that the generalized Kauer moves also yield derived equivalences of Brauer graph algebras and also may be interpreted in terms of silting mutations. [ABSTRACT FROM AUTHOR]

In this paper, we determine a class of modules over the affine-Virasoro algebra of Nappi-Witten type, which are free of rank one when restricted to the subalgebra U (C L 0 ⊕ C d 0). We study when such modules are simple and determine the isomorphism classes. As a by-product, we obtain a class of indecomposable modules for the affine Nappi-Witten algebra. We also consider the tensor product of finitely many rank one free simple modules with an arbitrary simple restricted module. The simplicity and isomorphism classes are investigated. [ABSTRACT FROM AUTHOR]

Multiparametric quantum semigroups M q ˆ , p ˆ (n) are generalization of the one-parameter general linear semigroups M q (n) , where q ˆ = (q i j) and p ˆ = (p i j) are 2 n 2 parameters satisfying certain conditions. In this paper, we study the algebra of multiparametric Manin matrices using the R-matrix method. The systematic approach enables us to obtain several classical identities such as Muir's identities, Newton's identities, Capelli-type identities, Cauchy-Binet's identity both for determinant and permanent as well as a rigorous proof of the MacMahon master equation for the quantum algebra of multiparametric Manin matrices. Some of the generalized identities are also lifted to multiparameter q -Yangians. [ABSTRACT FROM AUTHOR]

In this paper, we study the category of modules over the Smith algebra which are free of finite rank over the unital polynomial subalgebra generated by the Cartan element h and obtain families of such simple modules of arbitrary rank. In the case of rank one we obtain a full description of the isomorphism classes, a simplicity criterion, and an algorithm to produce all composition series. We show that all such modules have finite length and describe the composition factors and their multiplicity. [ABSTRACT FROM AUTHOR]

In 1998, Georgia Benkart and Tom Roby introduced the down-up algebra A. The algebra A is associative, noncommutative, and infinite-dimensional. It is defined by two generators A , B and two relations called the down-up relations. In the present paper, we introduce the Z 3 -symmetric down-up algebra A. We define A by generators and relations. There are three generators A , B , C and any two of these satisfy the down-up relations. We describe how A is related to some familiar algebras in the literature, such as the Weyl algebra, the Lie algebras sl 2 and sl 3 , the sl 3 loop algebra, the Kac-Moody Lie algebra A 2 (1) , the q -Weyl algebra, the quantized enveloping algebra U q (sl 2) , and the quantized enveloping algebra U q (A 2 (1)). We give some open problems and conjectures. [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]

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]

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]

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]

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]

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]

Our investigation in the present paper is based on three important results. (1) In [14] , Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [6] , Green found a homological formula for the representation category of the quiver and equipped Ringel's Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simple representations realizes the nilpotent part of quantum group of any type. (3) In [11] , Lusztig defined induction and restriction functors for the perverse sheaves on the variety of representations of the quiver which occur in the direct images of constant sheaves on flag varieties, and he found a formula between his induction and restriction functors which gives the comultiplication as algebra homomorphism for quantum group. In the present paper, we prove the formula holds for all semisimple complexes with Weil structure. This establishes the categorification of Green's formula. [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]

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]

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]

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]

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 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]

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]