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Köthe's classical problem posed by G. Köthe in 1935 asks to describe the rings R such that every left R-module is a direct sum of cyclic modules (these rings are known as left Köthe rings). Köthe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved Köthe's problem for basic finite-dimensional algebras. But, so far, Köthe's problem was open in the non-commutative setting. Recently, in the paper [Several characterizations of left Köthe rings, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023) (to appear)]], we classified left Köthe rings into three classes one contained in the other: left Köthe rings, strongly left Köthe rings and very strongly left Köthe rings, and then, we solved Köthe's problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duals of these notions as left co-Köthe rings, strongly left co-Köthe rings and very strongly left co-Köthe rings, and then, we give several structural characterizations for each of them. [ABSTRACT FROM AUTHOR]

This work is devoted to the complete classification of solvable Leibniz algebras with the naturally graded quasi-filiform non-Lie nilradical. It should be noted that the class of naturally graded quasi-filiform Leibniz algebras is sufficiently large, and solvable Leibniz algebras with the nilradical which is isomorphic to several of these algebras are classified in a series of works. This paper considered all omitted cases and completed the problem of classifying solvable Leibniz algebras with naturally graded quasi-filiform non-Lie nilradical. At the end of the paper we combine all results obtained to date, and provide a complete list of solvable Leibniz algebras, whose nilradical is a naturally graded quasi-filiform Leibniz algebra. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the description of local derivations on solvable Lie algebras of maximal rank. Namely, we consider a solvable Lie algebra of the form R = Q ⊕ N , where Q is the maximal torus subalgebra of R , N is the nilradical of R and dim Q = dim N / N 2. We prove that any local derivation of such solvable Lie algebra R is a derivation. Further, we present two examples of solvable Lie algebras which satisfy the condition dim Q < dim N / N 2 and the first algebra admit a local derivation which is not a derivation, while for the second algebra we prove that any local derivation is a derivation. We also apply the main result of the paper to the description of local derivations on so-called standard Borel subalgebras of complex simple Lie algebras. [ABSTRACT FROM AUTHOR]

AbstractLeibniz algebras are certain generalizations of Lie algebras. Motivated by the concept of subinvariance and ascendancy in group theory, Schenkman studied properties of subinvariant subalgebras of a Lie algebra. Kawamoto studied ascendency in Lie algebras. In this paper we define subinvariance and asendency in Leibniz algebras and study their properties. It is shown that the signature results on subinvariance and ascendency in Lie algebras have analogs for Leibniz algebras. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we introduce a combinatorial path model of representation of the quantum affine algebra of type Dn, inspired by Mukhin and Young’s combinatorial path models of representations of the quantum affine algebras of types An and Bn. In particular, we give a combinatorial formula for q-characters of fundamental modules of type Dn by assigning each path to a monomial or binomial. By counting our paths, a new expression on dimensions of fundamental modules of type Dn is obtained. [ABSTRACT FROM AUTHOR]

In this paper, first we give the controlling algebra of Lie triple systems. In particular, the cohomology of Lie triple systems can be characterized by the controlling algebra. Then using controlling algebras, we introduce the notions of homotopy Nambu algebras and homotopy Lie triple systems. We show that 2-term homotopy Lie triple systems is equivalent to Lie triple 2-systems, and the latter is the categorification of a Lie triple system. Finally we study skeletal and strict Lie triple 2-systems. We show that skeletal Lie triple 2-systems can be classified the third cohomology group, and strict Lie triple 2-systems are equivalent to crossed modules of Lie triple systems. [ABSTRACT FROM AUTHOR]

The notion of pre-Leibniz algebras was recently introduced in the study of Rota-Baxter operators on Leibniz algebras. In this paper, we first construct a graded Lie algebra whose Maurer-Cartan elements are pre-Leibniz algebras. Using this characterization, we define the cohomology of a pre-Leibniz algebra with coefficients in a representation. This cohomology is shown to split the Loday-Pirashvili cohomology of Leibniz algebras. As applications of our cohomology, we study formal and finite order deformations of a pre-Leibniz algebra. Finally, we define homotopy pre-Leibniz algebras and classify some special types of homotopy pre-Leibniz algebras. [ABSTRACT FROM AUTHOR]

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra Kn, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K3. This includes a family of 12-dimensional Nichols algebras { B ξ } depending on 3rd roots of unity. Here, B 1 is isomorphic to the well-known Fomin-Kirillov algebra, and B ξ ≃ B ξ 2 as graded algebras but B 1 is not isomorphic to B ξ as algebra for ξ ≠ 1 . As a byproduct we obtain new Hopf algebras of dimension 216. [ABSTRACT FROM AUTHOR]

AbstractPre-Novikov algebras and quasi-Frobenius Novikov algebras naturally appear in the theory of Novikov bialgebras. In this paper, we show that there is a natural pre-Novikov algebra structure associated to a quasi-Frobenius Novikov algebra. Then we introduce the definition of double constructions of quasi-Frobenius Novikov algebras associated to two pre-Novikov algebras and show that it is characterized by a pre-Novikov bialgebra. We also introduce the notion of pre-Novikov Yang-Baxter equation, whose symmetric solutions can produce pre-Novikov bialgebras. Moreover, the operator forms of pre-Novikov Yang-Baxter equation are also investigated. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we study certain combinatorial attributes of Ind-schemes of polynomial automorphisms in positive characteristic. In particular, we prove that over an algebraically closed field K of positive characteristic ≠2 every automorphism of the group of origin-preserving automorphisms of the polynomial algebra K[x1,…,xn] (n≥3), which fixes every diagonal matrix, preserves, up to composition with a linear inner automorphism, every tame automorphism. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a basis for the center of the infinitesimal q-Schur algebra s q (2 , r) 1 in terms of its certain basis. [ABSTRACT FROM AUTHOR]

AbstractFor a simple graph G with edge ideal I(G), we study the N -graded Betti numbers in the linear strand of the minimal free resolution of I(Γ(Zn)), where Γ(Zn) is the zero divisor graph of the ring Zn . We present sharp bounds for the Betti numbers of Γ(Zn) and characterize the graphs attaining these bounds, thereby establishing the correct equality case for one of the results of the earlier published paper (Theorem 4.5, S. Pirzada and S. Ahmad, On the linear strand of edge ideals of some zero divisor graphs, Commun. Algebra 51(2) (2023) 620–632). Also, we present homological invariants of the edge rings of Γ(Zn) for n=p2q and pqr, with primes p

AbstractIn this paper, all super-skewsymmetric super-biderivations of twisted N = 1 Schrödinger-Neveu-Schwarz algebra are determined. It is shown that all the super-biderivations are inner. Furthermore, the linear super-commuting maps of twisted N = 1 Schrödinger-Neveu-Schwarz algebra are standard. [ABSTRACT FROM AUTHOR]

AbstractThe aim of this paper is to examine the structure of genetic algebras associated with permuted Lotka-Volterra operators on a finite-dimensional simplex. Furthermore, three dimensional permuted Lotka-Voltarra algebras are considered and the description of their derivations are given as well. [ABSTRACT FROM AUTHOR]

This paper aims to study the local derivations, 2-local automorphisms and local automorphisms on the super-Virasoro algebras. The primary focus is to establish that every local derivation of the super-Virasoro algebras is indeed a derivation, and to demonstrate that every local or 2-local automorphism of the super-Virasoro algebras is an automorphism. [ABSTRACT FROM AUTHOR]

For a commutative ring R and a Hopf algebra H which is finitely generated projective as an R-module, it is established that there is an (anti)-isomorphism of groups between the Brauer group BQ(R, H) of Hopf Yetter-Drinfel'd H-module algebras and the Brauer group BQ (R , H *) of Hopf Yetter-Drinfel'd H * -module algebras, where H * is the linear dual of H. In this paper, we generalize this result by establishing an anti-isomorphism of groups between BQ(S, H), the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S, H)-module algebras and BQ (S o p , H *) , the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S o p , H *) -module algebras, where S is an H-commutative Hopf Yetter-Drinfel'd H-module algebra and Sop is the opposite algebra of S. Communicated by Alberto Elduque [ABSTRACT FROM AUTHOR]

The present paper is devoted to study local derivations on the conformal Galilei algebra. We prove that every local derivation on the conformal Galilei algebra is a derivation. [ABSTRACT FROM AUTHOR]

In the paper, we show that a uniformly bounded and nonnegative triangle functor between Gorenstein derived categories of two CM-algebras induces a Gorenstein-projectively stable functor between the corresponding Gorenstein-projectively stable categories. Furthermore, we show that a ladder of Gorenstein derived categories of CM-finite algebras induces a recollement of Gorenstein-projectively stable categories under certain nice conditions. As a byproduct, a Gorenstein derived equivalence induces a Gorenstein-projectively stable equivalence for CM-finite Gorenstein algebras. [ABSTRACT FROM AUTHOR]

Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher. Recently, the subject has attracted more attention since it is associated with many areas in mathematics, such as integro-differential algebras. This paper considers differential Rota-Baxter algebras in the quasi-idempotent operator context. We establish a Gröbner-Shirshov basis for free commutative quasi-idempotent differential algebras (resp. Rota-Baxter algebras, resp. differential Rota-Baxter algebras). This provides a linear basis of a free object in each of the three corresponding categories by the Composition-Diamond lemma. Communicated by P. Kolesnikov [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the universal PBW basis and HN system for the derived Hall algebra of a triangulated category satisfying some finiteness conditions. Furthermore, we study the related stability conditions on the module category A of the path algebra A of an acyclic quiver and on the bounded derived category of A. Moreover, we investigate the relation between the Ringel-Hall algebra and the HN system of A defined by Reineke. [ABSTRACT FROM AUTHOR]

Let k be an arbitrary field, Λ be a k-algebra and V be a Λ -module. When it exists, the universal deformation ring R (Λ , V) of V is a k-algebra whose local homomorphisms to R parametrize the lifts of V up to R ⊗ k Λ , where R is any complete, local commutative Noetherian k-algebra with residue field k. Symmetric special biserial algebras, which coincide with Brauer graph algebras, can be viewed as generalizing the blocks of finite type p-modular group algebras. Bleher and Wackwitz classified the universal deformation rings for all modules for symmetric special biserial algebras with finite representation type. In this paper, we begin to address the tame case. Specifically, let Λ be any 1-domestic, symmetric special biserial algebra. By viewing Λ as generalized Brauer tree algebras and making use of a derived equivalence, we classify the universal deformation rings for those Λ -modules V with stable endomorphism ring isomorphic to k. The latter is a natural condition, since it guarantees the existence of the universal deformation ring R (Λ , V) . [ABSTRACT FROM AUTHOR]

In this paper, we define terms generated by transformations preserving a partition on a finite set and then construct their superassociative structures. A generating system of such algebra is determined and the freeness in a variety of all superassociative algebras is investigated. The connection between a semigroup of all mappings whose ranges are terms induced by transformations preserving a partition and substitutions is discussed. In views of applications, we apply these mappings to examine identities of a variety in a higher step. Additionally, we generalize our study to algebraic systems and establish a superassociative algebra of a new type of formulas induced by terms defined by transformations preserving a partition. [ABSTRACT FROM AUTHOR]

In this paper, we study the deformation theory of associative conformal algebras and conformal bimodules. Using the O -operator on bimodules over associative conformal algebras, we introduce the notion of O N -structure on bimodules over associative conformal algebras and consider the relationship between it and the compatibility of O -operators, the PN-structure, the Ω N -structure and the P Ω -structure on associative conformal algebra. We show that a solution of the strong Maurer-Cartan equation of an associative conformal twilled algebra gives rise to an O N -structure, and an O N -structure satisfying certain conditions gives a solution of the strong Maurer-Cartan equation of an associative conformal twilled algebra. [ABSTRACT FROM AUTHOR]

A restricted dth power of an ideal I is obtained by restricting the exponent vectors allowed to appear on the "natural" generating set of Id, for some integer d. In this paper, we study homological properties of restricted powers of complete intersections. We construct a generalization of the L-complex construction of Buchsbaum and Eisenbud. We use this resolution to compute an explicit basis for the Koszul homology which allows us to deduce that the quotient defined by any restricted dth power of a complete intersection is Golod, and construct an algebra structure on this complex. [ABSTRACT FROM AUTHOR]

In this paper, we first propose the concept of DC-nets, where its domain is a DCPO, in convexity spaces and discuss some related properties. And then, we obtain that f : (X , C 1) → (Y , C 2) is convexity-preserving if and only if (f (a α)) α ∈ Λ converges to f(a) with respect to C 2 when (a α) α ∈ Λ converges to a with respect to C 1. Hence, we could discuss convexity-preserving properties of partial binary operations + and – of effect algebras by convergence of DC-nets in convexity spaces. Concretely, we prove that if (E , + , 0 , 1) is a scale effect algebra and C is an interval convexity on E, then + and – are separately convexity-preserving with respect to C. Finally, we provide an example to show that + and – are not jointly convexity-preserving with respect to C when (E , + , 0 , 1) is a lattice effect algebra. Communicated by Ángel del Río Mateos [ABSTRACT FROM AUTHOR]

In this paper, we are devoted to constructing a large number of Novikov H-pseudoalgebras from commutative associative (H-pseudo)algebras with (H-)derivations, infinitesimal H-pseudobialgebras with H-derivations and Novikov (H-pseudo)algebras with Rota-Baxter (H-)operators. Meanwhile, the relationship between Novikov H-pseudoalgebras and Lie H-pseudoalgebras is established. Furthermore, we develop the 1-parameter formal deformation theory of Novikov H-pseudoalgebras and characterize it in terms of cohomology. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Nichols algebra B (W X , r) associated to any non-degenerate involutive solution (X, r) of the Yang-Baxter equation. Infinite examples of finite dimensional Nichols algebras are obtained, including those of dimension nm with m, n ∈ Z ≥ 2 . It turns out that the Nichols algebra B (W X , r) has interesting relations with multinomial expansion. This is a generalization of the work in arXiv:2103.06489, which built a connection between the Nichols algebras of squared dimension and Pascal's triangle. [ABSTRACT FROM AUTHOR]

In this paper we use the cohomological approach to study abelian extensions and formal deformations of Hom-pre-Lie algebras. We show that diagonal abelian extensions of Hom-pre-Lie algebras can be classified by stable points in the second cohomology group under the action of an isomorphism. The infinitesimals of two equivalent formal deformations of a Hom-pre-Lie algebra A are in the same cohomological class in H 2 (A ; A) , and if H 2 (A ; A) is trivial, then the Hom-pre-Lie algebra A is rigid. [ABSTRACT FROM AUTHOR]

In this paper, for any prime p, we propose the notion of a p-transitive association scheme. This notion aims to generalize the fact that the regular module of a group algebra of a finite group has a unique principal submodule to the case of the regular modules of modular adjacency algebras of association schemes. We characterize the p-transitive quasi-thin association schemes and the p-transitive association schemes with thin thin residue by their structure theory properties. We also get some results with independent interests. [ABSTRACT FROM AUTHOR]

For a fixed integer n≥2, let Mn be the algebra of all n × n matrices over the complex field C. Let x1,x2∈Cn be two fixed nonzero vectors, and fix also y1,y2∈Cn. In this paper, we characterize linear bijective maps φ:Mn→Mn having the property that φ(A)φ(B)x2=y2 whenever ABx1=y1. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give some constructions results and examples of n-BiHom-Lie color algebras. Next, we introduce the definition of BiHom-modules over n-BiHom-Lie color algebras and we provide some properties. Moreover we investigate generalized derivations of n-BiHom-Lie color algebras and their BiHom-subalgebras. [ABSTRACT FROM AUTHOR]

In this paper, we provide a new approach to construct monoidal Hom-Hopf algebras. We investigate monoidal Hom-Hopf algebra structure on a left (n, l)-Hom-crossed coproduct structure with a left n-Hom-smash product structure, obtaining Radford [ n , (n , l) ] -biproduct structure theorem. Then, we study a Hom-coaction admissible mapping system to characterize this Radford [ n , (n , l) ] -biproduct structure. Finally, we study the cosemisimplicity of a special Hom-smash coproduct and prove the related Maschke theorem. [ABSTRACT FROM AUTHOR]

An O -operator has been used to extend a Leibniz algebra by its representation. In this paper, we investigate several structures related to O -operators on Leibniz algebras and introduce (dual) O N-structures on Leibniz algebras associated to their representations. It is proved that O -operators and dual O N-structures generate each other under certain conditions. It is also shown that a solution of the strong Maurer-Cartan equation on the twilled Leibniz algebra gives rise to a dual O N-structure. Finally, r – n structures, RBN-structures and B N -structures on Leibniz algebras are thoroughly studied and their interdependent relations are also studied. [ABSTRACT FROM AUTHOR]

In the paper, we consider n − dimensional naturally graded nilpotent associative algebras with the characteristic sequence C (A) = (n − m , m) over an algebraically closed field of characteristic zero. There are two types of such algebras. We give the algebraic classification up to isomorphism of both types of algebras. Moreover, we prove that there are no such algebras of the second type for n > 2 m + 1 . [ABSTRACT FROM AUTHOR]

In this paper we describe the central extensions of some nilpotent Leibniz algebras and one-dimensional central extensions (up to isomorphism) of all naturally graded 2-filiform Leibniz algebras over the field of complex numbers. It is well-know that 2-filiform Leibniz algebras which are not naturally graded have not yet studied. There are such kind of algebras in the algebras obtained along this article. Also in this work we describe some subclasses of 6-dimensional nilpotent Leibniz algebras with condition quotient algebra by two-dimensional center of the algebra is isomorphic to the four-dimensional naturally graded 2-filiform nonsplit Leibniz algebra. [ABSTRACT FROM AUTHOR]

In this paper, we use Gröbner–Shirshov bases method for Leibniz (super)algebras to present a complete characterization of extensions of a Leibniz (super)algebra by another Leibniz (super)algebra, where the former is presented by generators and relations. [ABSTRACT FROM AUTHOR]

Let I (≠ A) be an ideal of a d-dimensional Noetherian local ring A with ht A I ≥ 2 , containing a non-zerodivisor. The problem of when the ring I : I = End A I is Gorenstein is studied, in connection with the problem of the Gorensteinness in Rees algebras R A (Q d) for certain parameter ideals Q of A, that was closely explored by the preceding paper [8] of the authors. Examples are given. Communicated by Scott Chapman [ABSTRACT FROM AUTHOR]

In this paper, we prove that every skew-symmetric biderivation of the conformal Galilei algebra is inner. Based on the result of biderivations, we show that all the linear commuting maps on the conformal Galilei algebra which have the form ψ (x) = λ x + ω (x) z are not standard. [ABSTRACT FROM AUTHOR]

In this paper, we mainly present some new solutions of the Hom-Yang-Baxter equation from Hom-algebras, Hom-coalgebras and Hom-Lie algebras, respectively. Also, we prove that these solutions are all invertible and give some examples. Finally, we introduce the notion of Hom-Yang-Baxter systems and obtain two kinds of Hom-Yang-Baxter systems. [ABSTRACT FROM AUTHOR]

In this paper, we study non-abelian extensions of pre-Lie algebras. First we introduce the non-abelian cohomology for pre-Lie algebras by which we classify non-abelian extensions of pre-Lie algebras. Then we construct the bimultipliers for pre-Lie algebras, which naturally gives rise to a strict Lie 2-algebra. We show that there is a one-to-one correspondence between isomorphic classes of non-abelian extensions of pre-Lie algebras and homotopy classes of homomorphisms from the subadjacent Lie algebra to the strict Lie 2-algebra obtained from the bimultipliers. Finally we classify non-abelian extensions of pre-Lie algebras by Maurer-Cartan elements of a differential graded Lie algebra. [ABSTRACT FROM AUTHOR]

Let A be a unital *-algebra containing a non-trivial projection. In this paper, it is shown that a map Γ: A → A such that Γ (U ° V • W) = Γ (U) ° V • W + U ° Γ (V) • W + U ° V • Γ (W) for all U , V , W ∈ A . Then Γ is an additive *-derivation. [ABSTRACT FROM AUTHOR]

In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Zinbiel algebras. We study Zinbiel algebras containing maximal abelian subalgebras of codimension 1 and supersolvable Zinbiel algebras in which such subalgebras have codimension 2, and we also analyze the case of filiform Zinbiel algebras. We give examples to clarify some results, including listing the values for α and β for the low dimensional Zinbiel algebras over the complex field that have been classified. Communicated by Alberto Elduque [ABSTRACT FROM AUTHOR]

Let P be a partially ordered set, R a 2-torsion free commutative ring with unity and F I (P , R) the finitary incidence algebra of P over R. In this paper, we give an explicit description for the structure of nonlinear Jordan derivations of F I (P , R) . We show that if P has no trivial component, then every nonlinear Jordan derivation of F I (P , R) is proper, and can be presented as a sum of a generalized inner derivation, a transitive induced derivation and an additive induced derivation. If P has a trivial connected component, we prove that every nonlinear Jordan derivation of F I (P , R) is proper if and only if every nonlinear Jordan derivation of R is also proper. [ABSTRACT FROM AUTHOR]

The n-th Schrödinger algebra s c h n : = s l 2 ⋉ h n is the semi-direct of the Lie algebra s l 2 with the n-th Heisenberg Lie algebra h n . In this paper, all derivations and biderivations of the n-th Schrödinger algebra are determined. As applications, all linear commuting maps and commutative post-Lie algebra structures on s c h n are obtained. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study quasitriangular structures on a class of semisimple Hopf algebras k G # σ , τ k Z 2 constructed through abelian extensions of k Z 2 by k G for an abelian group G. We prove that there are only two forms of them and we get a complete list of all universal R -matrices of the generalized Kac-Paljutkin algebra H 2 n 2 (see Section 2 for the definition). [ABSTRACT FROM AUTHOR]

The present paper is devoted to study 2-Local derivations on the Schrödinger algebra in (2 + 1)-dimensional space-time of Schrödinger Lie groups. We prove that every 2-local derivation on this algebra is a derivation. [ABSTRACT FROM AUTHOR]

Hernandez and Leclerc proved that the Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional modules of any untwisted quantum affine algebra U q (g ̂) has a cluster algebra structure. This makes it possible to compute q -characters of real prime simple modules associated to cluster variables as a result of a sequence of cluster mutations. In this paper, for minimal affinizations of type G2, we describe a cluster algebra algorithm to compute their q -characters and prove the geometric q -character formula conjectured by Hernandez and Leclerc. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of transposed BiHom-Poisson (abbr. TBP) algebras which can be constructed by BiHom-Novikov-Poisson algebras. Several useful identities for TBP algebras are provided. We also prove that the tensor products of two (T)BP algebras are closed. The notions of BP 3-Lie algebras and TBP 3-Lie algebras are presented and TBP algebras can induce TBP 3-Lie algebras by two approaches. Finally, we give some examples for the TBP algebras of dimension 2. [ABSTRACT FROM AUTHOR]

This paper shows that the Dong's conjecture in Dong (J. Algebra 591:342–359, 2022) on the principal representation category of infinite reductive group is not true. Thus, it needs to be modified. [ABSTRACT FROM AUTHOR]

The goal of this article is to study the structure and representation theory of affine wreath product algebras W r n aff (A) and its cyclotomic quotients W r n C (A). These algebras appear naturally in Heisenberg categorification and generalize many important algebras (degenerate affine Hecke algebras, affine Sergeev algebras and wreath Hecke algebras). The whole class was introduced by D. Rosso and A. Savage in [16]. In [19], Savage studied both structure and representations under the condition that the trace map of A is even. In this paper we extend the definition for the case of odd trace. Since our approach is analogous to Savage's, we consider the trace map being of arbitrary parity and unify statements and proofs. We also use an approach based on string diagrams, in the spirit of [5]. For simplicity of exposition, we assume A to be symmetric. [ABSTRACT FROM AUTHOR]