Quotient Algebra Research Articles

Simple modules for quiver Hecke algebras and the Robinson–Schensted–Knuth correspondence

We formalize some known categorical equivalences to give a rigorous treatment of smooth representations of p-adic general linear groups, as ungraded modules over quiver Hecke algebras of type A. Graded variants of RSK-standard modules are constructed for quiver Hecke algebras. Exporting recent results from the p-adic setting, we describe an effective method for construction and classification of all simple modules as quotients of modules induced from maximal homogenous data. It is established that the products involved in the RSK construction fit the Kashiwara-Kim notion of normal sequences of real modules. We deduce that RSK-standard modules have simple heads, devise a formula for the shift of grading between RSK-standard and simple self-dual modules, and establish properties of their decomposition matrix, thus confirming expectations for p-adic groups raised in a work of the author with Lapid. Subsequent work will exhibit how the presently introduced RSK construction generalizes the much-studied Specht construction, when inflated from cyclotomic quotient algebras.

Symmetric approximation sequences, Beilinson-Green algebras and derived equivalences

In this paper, we will consider a class of locally $\Phi$-Beilinson-Green algebras, where $\Phi$ is an infinite admissible set of the integers, and show that symmetric approximation sequences in $n$-exangulated categories give rise to derived equivalences between quotient algebras of locally $\Phi$-Beilinson-Green algebras in the principal diagonals modulo some factorizable ghost and coghost ideals by the locally finite tilting family. Then we get a class of derived equivalent algebras that have not been obtained by using previous techniques. From higher exact sequences, we obtain derived equivalences between subalgebras of endomorphism algebras by constructing tilting complexes, which generalizes Chen and Xi's result for exact sequences. From a given derived equivalence, we get derived equivalences between locally $\Phi$-Beilinson-Green algebras of semi-Gorenstein modules. Finally, from given graded derived equivalences of group graded algebras, we get derived equivalences between associated Beilinson-Green algebras of group graded algebras.

A bi-Hamiltonian nature of the Gaudin algebras

Let q be a Lie algebra over a field k and p,p˜∈k[t] two different normalised polynomials of degree n⩾2. As vector spaces, both quotient Lie algebras q[t]/(p) and q[t]/(p˜) can be identified with W=q⋅1⊕qt¯⊕…⊕qt¯n−1. If deg⁡(p−p˜)⩽1, then the Lie brackets [,]p, [,]p˜ induced on W by p and p˜, respectively, are compatible. Making use of the Lenard–Magri scheme, we construct a subalgebra Z=Z(p,p˜)⊂S(W)q⋅1 such that {Z,Z}p={Z,Z}p˜=0. If tr.degS(q)q=indq and q has the codim–2 property, then tr.degZ takes the maximal possible value, which is n−12dim⁡q+n+12indq. If q=g is semisimple, then Z contains the Hamiltonians of a suitably chosen Gaudin model. Furthermore, if p and p˜ do not have common roots, then there is a Gaudin subalgebra C⊂U(g⊕n) such that Z=gr(C), up to a certain identification. In a non-reductive case, we obtain a completely integrable generalisation of Gaudin models.For a wide class of Lie algebras, which extends the reductive setting, Z(p,p+t) coincides with the image of the Poisson-commutative algebra Z(qˆ,t)=S(tq[t])q[t−1] under the quotient map ψp:S(q[t])→S(W), providing p(0)≠0.

Graded systems of quotients and Martindale-like quotients of graded Lie triple systems

The paper is devoted to introducing graded systems of quotients of graded Lie triple systems, graded weak systems of quotients of graded Lie triple systems and graded systems of Martindale-like quotients of graded Lie triple systems with respect to power graded filters of sturdy graded ideals. And then we investigate certain vital properties of graded Lie triple systems, which can be lifted from a graded Lie triple system to its graded systems of quotients. Furthermore, the connections among graded systems of quotients, graded weak systems of quotients and graded systems of Martindale-like quotients are surveyed. What’s more, the relationship between graded systems of quotients of graded Lie triple systems and systems of quotients of Lie triple systems is highlighted. Besides, the relationship between graded systems of quotients of a graded Lie triple system and graded algebras of quotients of the graded Lie algebra L(T) (the standard embedding of a graded Lie triple system T) is considered, too.

Arnold’s Piecewise Linear Filtrations, Analogues of Stanley–Reisner Rings and Simplicial Newton Polyhedra

In 1974, the author proved that the codimension of the ideal (g1,g2,…,gd) generated in the group algebra K[Zd] over a field K of characteristic 0 by generic Laurent polynomials having the same Newton polytope Γ is equal to d!×Volume(Γ). Assuming that Newtons polytope is simplicial and super-convenient (that is, containing some neighborhood of the origin), the author strengthens the 1974 result by explicitly specifying the set Bsh of monomials of cardinality d!×Volume(Γ), whose equivalence classes form a basis of the quotient algebra K[Zd]/(g1,g2,…,gd). The set Bsh is constructed inductively from any shelling sh of the polytope Γ. Using the Bsh structure, we prove that the associated graded K -algebra grΓ(K[Zd]) constructed from the Arnold–Newton filtration of K -algebra K[Zd] has the Cohen–Macaulay property. This proof is a generalization of B. Kind and P. Kleinschmitt’s 1979 proof that Stanley–Reisner rings of simplicial complexes admitting shelling are Cohen–Macaulay. Finally, we prove that for generic Laurent polynomials (f1,f2,…,fd) with the same Newton polytope Γ, the set Bsh defines a monomial basis of the quotient algebra K[Zd]/(g1,g2,…,gd).

Bernstein algebras that are algebraic and the Kurosh problem

We study the class of Bernstein algebras that are algebraic, in the sense that each element generates a finite-dimensional subalgebra. Every Bernstein algebra has a maximal algebraic ideal, and the quotient algebra is a zero-multiplication algebra. Several equivalent conditions for a Bernstein algebra to be algebraic are given. In particular, known characterizations of Bernstein train algebras in terms of nilpotency are generalized to the case of locally train algebras. Along the way, we show that if a Banach Bernstein algebra is algebraic (respectively, locally train), then it is of bounded degree (respectively, train).Then we investigate the Kurosh problem for Bernstein algebras: whether a finitely generated Bernstein algebra which is algebraic of bounded degree is finite-dimensional. This problem turns out to have a closed link with a question about associative algebras. In particular, when the bar-ideal is nil, the Kurosh problem asks whether a finitely generated Bernstein train algebra is finite-dimensional. We prove that the answer is positive for some specific cases and for low degrees, and construct counter-examples in the general case.On the other hand, by results of Yagzhev the Jacobian conjecture is equivalent to a certain statement about Engel and nilpotence identities of multioperator algebras. We show that the generalized Jacobian conjecture for quadratic mappings holds for Bernstein algebras.

Central extensions of 2-filiform Leibniz algebras

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.

\(n\)-Fold Filters of EQ-Algebras

In this paper, we apply the notion of \(n\)-fold filters to the \(EQ\)-algebras and introduce the concepts of \(n\)-fold pseudo implicative, \(n\)-fold implicative, \(n\)-fold obstinate, \(n\)-fold fantastic prefilters and filters on an \(EQ\)-algebra \(\mathcal{E}\). Then we investigate some properties and relations among them. We prove that the quotient algebra \(\mathcal{E}/F\) modulo an 1-fold pseudo implicative filter of an \(EQ\)-algebra \(\mathcal{E}\) is a good \(EQ\)-algebra and the quotient algebra \(\mathcal{E}/F\) modulo an 1-fold fantastic filter of a good \(EQ\)-algebra \(\mathcal{E}\) is an \(IEQ\)-algebra.

Properties of Homomorphism and Quotient Implication Algebra on Implication Algebras

The concept of homomorphisms on implication algebra is introduced. The notion of sub algebras, normal subalgebras in an implication algebra are investigated. Quotient implication algebras and kernels in an implication algebra ,and Homomorphisms and isomorphism theorems are elaborated.

On Parafree Leibniz Algebras

The parafree Leibniz algebras are a special class of Leibniz algebras which have many properties with a free Leibniz algebra. In this note, we introduce the structure of parafree Leibniz algebras. We survey the
 important results in parafree Leibniz algebras which are analogs of corresponding results in parafree Lie algebras. We first investigate some properties of subalgebras and quotient algebras of parafree Leibniz algebras. Then, we describe the direct sum of parafree Leibniz algebras. We show that the direct sum of two parafree Leibniz algebras is a Leibniz algebra. Furthermore, we prove that the direct sum of two parafree Leibniz algebras is again parafree.

Picard–Borel ideals in Fréchet algebras and Michael's problem

A Picard–Borel algebra is a commutative unital complex algebra A $A$ such that every family of pairwise linearly independent invertible elements of A $A$ is linearly independent, and a Picard–Borel ideal I $I$ in a commutative complex unital algebra A $A$ is an ideal of I $I$ of A $A$ such that the quotient algebra A / I $A/I$ is a Picard–Borel algebra. In a preliminary paper, the author proved that every commutative unital Fréchet algebra which is a Picard–Borel algebra is an integral domain. The main result of the present paper is that all Picard–Borel ideals in commutative unital Fréchet algebras are prime. This result seems to be relevant for Michael's problem, since dense Picard–Borel ideals could play a role in the construction of discontinuous characters on some commutative unital Fréchet algebras.

Fuzzy Ideals in Pseudo-Hoop Algebras

In this study, fuzzy ideals in pseudo-hoop algebras are presented. Also, we investigate fuzzy congruences relations on pseudo-hoop algebras induced by fuzzy ideals. By using fuzzy ideals, we create the fuzzy quotient pseudo-hoop algebras and identify and demonstrate the one-to-one relationship between the set of all normal fuzzy ideals of a pseudo-hoop algebra H with conditions p D N and the set of all fuzzy congruences relation on H . Additionally, we investigate the relationship between fuzzy ideals and fuzzy filters in pseudo-hoop algebras by considering the set of complement elements of pseudo-hoop algebras. Lastly, we study fuzzy prime ideals and fuzzy ⊙ -prime ideals and their relation.

Weak Inflationary BL-Algebras and Filters of Inflationary (Pseudo) General Residuated Lattices

After the research on naBL-algebras gained by the non-associative t-norms and overlap functions, inflationary BL-algebras were also studied as a recent kind of non-associative generalization of BL-algebras, which can be obtained by general overlap functions. In this paper, we show that not every inflationary general overlap function can induce an inflationary BL-algebra by a counterexample and thus propose the new concept of weak inflationary BL-algebras. We prove that each inflationary general overlap function corresponds to a weak inflationary BL-algebra; therefore, two mistaken results in the previous paper are revised. In addition, some properties satisfied by weak inflationary BL-algebras are discussed, and the relationships among some non-classical logic algebras are analyzed. Finally, we establish the theory of filters and quotient algebras of inflationary general residuated lattice (IGRL) and inflationary pseudo-general residuated lattice (IPGRL), and characterize the properties of some kinds of IGRLs and IPGRLs by naBL-filters, (weak) inflationary BL-filters, and weak inflationary pseudo-BL-filters.

Quotient algebras of Banach operator ideals related to non-classical approximation properties

We investigate the quotient algebra AXI:=I(X)/F(X)‾||⋅||I for Banach operator ideals I contained in the ideal of the compact operators, where X is a Banach space that fails the I-approximation property. The main results concern the nilpotent quotient algebras AXQNp and AXSKp for the quasi p-nuclear operators QNp and the Sinha-Karn p-compact operators SKp. The results include the following: (i) if X has cotype 2, then AXQNp={0} for every p≥1; (ii) if X⁎ has cotype 2, then AXSKp={0} for every p≥1; (iii) the exact upper bound of the index of nilpotency of AXQNp and AXSKp for p≠2 is max⁡{2,⌈p/2⌉}, where ⌈p/2⌉ denotes the smallest n∈N such that n≥p/2; (iv) for every p>2 there is a closed subspace X⊂c0 such that both AXQNp and AXSKp contain a countably infinite decreasing chain of closed ideals. In addition, our methods yield a closed subspace X⊂c0 such that the compact-by-approximable algebra AX=K(X)/A(X) contains two incomparable countably infinite chains of nilpotent closed ideals.

On the quotients of coarse structures and Roe algebras

We study the quotients of the Roe algebras of uniformly locally finite coarse spaces which are not necessarily metrizable. We introduce the coarse quotient structure of a coarse structure associated to an ideal and prove that in certain cases, the quotient of the Roe algebra by the ideal induced by the ideal coarse structure is isomorphic to the Roe corona of the coarse quotient structure. We establish that the functor which maps a coarse subspace to the quotient of Roe algebra associated to any ideal actually forms a cosheaf with values in the category of C⁎-algebras. Moreover, we prove a coarse Mayer-Vietories theorem for the relative Roe algebra associated to a coarse subspace.

Study of Quotient Property Topologically

In this article, our focus on the notion of quotient related to topology of some structure algebra several results and generalizations of topological quotient algebra have been given. We investigate some facts about quotient ring and quotient module topological especially when we have product of quotient ring and modules. One pf the results R/I. is a topological ring (TR), as we established. Also some results about topological submodule and topological module was introduced. Finally; some definitions and examples have been presented.

On the annihilator ideal in the bt-algebra of tensor space

We study the representation theory of the braids and ties algebra, or the bt -algebra, E n ( q ) . Using the cellular basis { m s t } for E n ( q ) obtained in previous joint work with J. Espinoza we introduce two kinds of permutation modules M ( λ ) and M ( Λ ) for E n ( q ) . We show that the tensor product module V ⊗ n for E n ( q ) is a direct sum of M ( λ ) 's. We introduce the dual cellular basis { n s t } for E n ( q ) and study its action on M ( λ ) and M ( Λ ) . We show that the annihilator ideal I in E n ( q ) of V ⊗ n enjoys a nice compatibility property with respect to { n s t } . We finally study the quotient algebra E n ( q ) / I , showing in particular that it is a simultaneous generalization of Härterich's ‘generalized Temperley-Lieb algebra’ and Juyumaya's ‘partition Temperley-Lieb algebra’.

New results on topological effect algebras

In this paper, by considering the notion of effect algebra and by using of a new ideal in an effect algebra E, we construct a topology τ on E, and we show that (E,τ) is a topological effect algebra. Then we obtain some conditions under which that (E,τ) is a Hausdorff space. Also, we obtain some results about connected components of this topological space, and we construct a quotient topological effect algebra.

Results on Topological Lattice Effect Algebras

In this paper, we define the notion of topological lattice effect algebras and investigate some of their properties. By using Sasaki arrows and F-balls, we construct two topology on lattice effect algebras. Then we study separation axioms on lattice effect algebras. Specifically, we find some conditions under which a topological lattice algebra is a $T_{0}$, $T_{1}$, and Hausdorff space. Finally, by using a strong filter and a quotient lattice effect algebra constructed by it, we investigate under what conditions this quotient lattice effect algebra will be a topological lattice effect algebra.

Automorphisms of simple quotients of the Poisson and universal enveloping algebras of sl2

Let [Formula: see text] be the Poisson enveloping algebra of the Lie algebra [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic zero. The quotient algebras [Formula: see text] [Formula: see text], where [Formula: see text] is the standard Casimir element of [Formula: see text] in [Formula: see text] and [Formula: see text], are proven to be simple in [U. Umirbaev and V. Zhelyabin, A Dixmier theorem for Poisson enveloping algebras, J. Algebra 568 (2021) 576–600]. Using a result by Makar–Limanov [22], we describe generators of the automorphism group of [Formula: see text] and represent this group as an amalgamated product of its subgroups. Moreover, using similar results by Dixmier [Quotients simples de l’algebre enveloppante de [Formula: see text], J. Algebra 24 (1973) 551–564] and O. Fleury [Sur les sous-groupes finis de [Formula: see text] et [Formula: see text], J. Algebra 200 (1998) 404–427] for the quotient algebras [Formula: see text], where [Formula: see text] is the standard Casimir element of [Formula: see text] in the universal enveloping algebra [Formula: see text], we prove that the automorphism groups of [Formula: see text] and [Formula: see text] are isomorphic.