The number of simple modules of a cellular algebra

Let n be a natural number, and let A be an indecomposable cellular algebra such that the spectrum of its Cartan matrix C is of the form {n, 1, …., 1}. In general, not every natural number could be the number of non-isomorphic simple modules over such a cellular algebra. Thus, two natural questions arise: (1) which numbers could be the number of non-isomorphic simple modules over such a cellular algebra A ? (2) Given such a number, is there a cellular algebra such that its Cartan matrix has the desired property ? In this paper, we shall completely answer the first question, and give a partial answer to the second question by constructing cellular algebras with the pre-described Cartan matrix.     

New simple modular Lie superalgebras as generalized prolongs

Over algebraically closed fields of characteristic p > 2, —prolongations of simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. We discover several new simple Lie superalgebras, serial and exceptional, including super versions of Brown and Melikyan algebras, and thus corroborate the super analog of the Kostrikin-Shafarevich conjecture. Simple Lie superalgebras with 2 × 2 Cartan matrices are classified.     

Lie algebras induced by a nonzero field derivation

Given a finite-dimensional associative commutative algebra A over a field F, we define the structure of a Lie algebra using a nonzero derivation D of A. If A is a field and charF > 3; then the corresponding algebra is simple, presenting a nonisomorphic analog of the Zassenhaus algebra W ₁(m).     

On a Variety Related to the Commuting Variety of a Reductive Lie Algebra

For a reductive Lie algbera over an algbraically closed field of charasteristic zero, we consider a Borel subgroup B of its adjoint group, a Cartan subalgebra contained in the Lie algebra of B and the closure X of its orbit under B in the Grassmannian. The variety X plays an important role in the study of the commuting variety. In this note, we prove that X is Gorenstein with rational singularities.     

On Ext-transfer for Reductive Lie Algebras

Let G be a connected reductive algebraic group over an algebraically closed field of prime characteristic p and ?? be the Lie algebra of G. In this paper, we study the representations of ?? when p-character has standard Levi form. An Ext-transfer from the Ext-groups of induced ??-modules to its Levi subalgebras is obtained. Furthermore, we reduce the computation of the multiplicities of simple factors in baby Verma modules over ?? to its Levi subalgebras.     

A class of simple Lie algebras attached to unit forms

Let n ≥ 3. The complex Lie algebra, which is attached to a unit form q(x ₁, x ₂,..., x _n) = \({\sum\nolimits_{i = 1}^n {x_i^2 + \sum\nolimits_{1 \leqslant i \leqslant j \leqslant n} {\left( { - 1} \right)} } ^{j - i}}{x_i}{x_j}\) and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A _n , and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.     

Invariants of the action of a semisimple Hopf algebra on PI-algebra

We extend several classical results in the theory of invariants of finite groups to the case of action of a finite-dimensional Hopf algebra H on an algebra satisfying a polynomial identity. In particular, we prove that an H-module algebra A over an algebraically closed field k is integral over the subalgebra of invariants, if H is a semisimple and cosemisimple Hopf algebra. We show that for char k > 0, the algebra Z\({\left( A \right)^{{H_0}}}\) is integral over the subalgebra of central invariants Z(A)^H, where Z(A) is the center of algebra A, H⁰ is the coradical of H. This result allowed us to prove that the algebra A is integral over the subalgebra Z(A)^H in some special case. We also construct a counterexample to the integrality of the algebra \({A^{{H_0}}}\) over the subalgebra of invariants A^H for a pointed Hopf algebra over a field of non-zero characteristic.     

<Emphasis Type="Italic">δ</Emphasis>-derivations of classical Lie superalgebras

We consider the δ-derivations of classical Lie superalgebras and prove that these superalgebras admit nonzero δ-derivations only when δ = 0, ½, 1. The structure of ½-derivations for classical Lie superalgebras is completely determined.     

Invariants and rings of quotients of <Emphasis Type="Italic">H</Emphasis>-semiprime <Emphasis Type="Italic">H</Emphasis>-module algebras satisfying a polynomial identity

We consider an action of a finite-dimensional Hopf algebra H on a PI-algebra. We prove that an H-semiprime H-module algebra A has a Frobenius artinian classical ring of quotients Q, provided that A has a finite set of H-prime ideals with zero intersection. The ring of quotients Q is an H-semisimple H-module algebra and a finitely generated module over the subalgebra of central invariants. Moreover, if algebra A is a projective module of constant rank over its center, then A is integral over its subalgebra of central invariants.     

Characterization of 2-Local Derivations and Local Lie Derivations on Some Algebras

We prove that each 2-local derivation from the algebra M_n(A ) (n > 2) into its bimodule M_n(M) is a derivation, where A is a unital Banach algebra and M is a unital A -bimodule such that each Jordan derivation from A into M is an inner derivation, and that each 2-local derivation on a C*-algebra with a faithful traceable representation is a derivation. We also characterize local and 2-local Lie derivations on some algebras such as von Neumann algebras, nest algebras, the Jiang–Su algebra, and UHF algebras.     

ABELIAN BALANCED HERMITIAN STRUCTURES ON UNIMODULAR LIE ALGEBRAS

Let g be a 2n-dimensional unimodular Lie algebra equipped with a Hermitian structure (J; F) such that the complex structure J is abelian and the fundamental form F is balanced. We prove that the holonomy group of the associated Bismut connection reduces to a subgroup of SU(n – k), being 2k the dimension of the center of g. We determine conditions that allow a unimodular Lie algebra to admit this particular type of structures. Moreover, we give methods to construct them in arbitrary dimensions and classify them if the Lie algebra is 8-dimensional and nilpotent.     

Irreducible <Emphasis Type="Italic">A</Emphasis><Stack><Subscript>1</Subscript><Superscript>(1)</Superscript></Stack>-modules from modules over two-dimensional non-abelian Lie algebra

For any module V over the two-dimensional non-abelian Lie algebra b and scalar α ∈ C, we define a class of weight modules F ^α (V) with zero central charge over the affine Lie algebra A ₁ ⁽¹⁾ . These weight modules have infinitedimensional weight spaces if and only if V is infinite dimensional. In this paper, we will determine necessary and sufficient conditions for these modules F ^α(V) to be irreducible. In this way, we obtain a lot of irreducible weight A ₁ ⁽¹⁾ -modules with infinite-dimensional weight spaces.     

A Characterization of Stem Algebras in Terms of Central Derivations

Let L be a Lie algebra, and Der _z (L) denote the set of all central derivations of L, that is, the set of all derivations of L mapping L into the center. In this paper, by using the notion of isoclinism, we study the center of Der _z (L) for nilpotent Lie algebras with nilindex 2. We also give a characterization of stem Lie algebras by their central derivations. In fact we show that for non-abelian nilpotent Lie algebras of finite dimension and any nilpotent Lie algebra with nilindex 2 (not finite dimensional in general), Der _z (L) is abelian if and only if L is a stem Lie algebra.     

Local Superderivatio ns o n Lie Superalgebra $$\mathfrak{}$$ ( n)

Let \(\mathfrak{q}\)(n) be a simple strange Lie superalgebra over the complex field ?. In a paper by A.Ayupov, K.Kudaybergenov (2016), the authors studied the local derivations on semi-simple Lie algebras over ? and showed the difference between the properties of local derivations on semi-simple and nilpotent Lie algebras. We know that Lie superalgebras are a generalization of Lie algebras and the properties of some Lie superalgebras are similar to those of semi-simple Lie algebras, but \(\mathfrak{p}\)(n) is an exception. In this paper, we introduce the definition of the local superderivation on \(\mathfrak{q}\)(n), give the structures and properties of the local superderivations of \(\mathfrak{q}\)(n), and prove that every local superderivation on \(\mathfrak{q}\)(n), n > 3, is a superderivation.     

Generalized Wigner-Inönü contractions and Maxwell (super)algebras

We consider a class of generalized Wigner-Inönü contractions for the semidirect product of two particularly related semisimple Lie (super)algebras. A special class of such contractions provides the D = 4 Maxwell algebra and the recently introduced simple D = 4 Maxwell superalgebra. Further we present two types of D = 4 N-extended Maxwell superalgebras, the nonstandard one for any N with ½N(N?1) central charges and the standard one, for even N = 2k, with k(2k ? 1) internal symmetry generators.     

Off-Shell Supersymmetry and Filtered Clifford Supermodules

An off-shell representation of supersymmetry is a representation of the super Poincaré algebra on a dynamically unconstrained space of fields. We describe such representations formally, in terms of the fields and their spacetime derivatives, and we interpret the physical concept of engineering dimension as an integral grading. We prove that formal graded off-shell representations of one-dimensional N-extended supersymmetry, i.e., the super Poincaré algebra \(\mathfrak {p}^{1|N}\), correspond to filtered Clifford supermodules over Cl(N). We also prove that formal graded off-shell representations of two-dimensional (p,q)-supersymmetry, i.e., the super Poincaré algebra \(\mathfrak {p}^{1,1|p,q}\), correspond to bifiltered Clifford supermodules over Cl(p + q). Our primary tools are Rees superalgebras and Rees supermodules, the formal deformations of filtered superalgebras and supermodules, which give a one-to-one correspondence between filtered spaces and graded spaces with even degree-shifting injections. This generalizes the machinery used by Gerstenhaber to prove that every filtered algebra is a deformation of its associated graded algebra. Our treatment extends the notion of Rees algebras and modules to filtrations which are compatible with a supersymmetric structure. We also describe the analogous constructions for bifiltrations and bigradings.     

Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras

To each multiquiver Γ we attach a solution to the consistency equations associated to twisted generalized Weyl (TGW) algebras. This generalizes several previously obtained solutions in the literature. We show that the corresponding algebras \(\mathcal {A}({\Gamma })\) carry a canonical representation by differential operators and that \(\mathcal {A}({\Gamma })\) is universal among all TGW algebras with such a representation. We also find explicit conditions in terms of Γ for when this representation is faithful or locally surjective. By forgetting some of the structure of Γ one obtains a Dynkin diagram, D(Γ). We show that the generalized Cartan matrix of \(\mathcal {A}({\Gamma })\) coincides with the one corresponding to D(Γ) and that \(\mathcal {A}({\Gamma })\) contains graded homomorphic images of the enveloping algebra of the positive and negative part of the corresponding Kac-Moody algebra. Finally, we show that a primitive quotient U/J of the enveloping algebra of a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero is graded isomorphic to a TGW algebra if and only if J is the annihilator of a completely pointed (multiplicity-free) simple weight module. The infinite-dimensional primitive quotients in types A and C are closely related to \(\mathcal {A}({\Gamma })\) for specific Γ. We also prove one result in the affine case.     

Classification of non-degenerate projective varieties with non-zero prolongation and application to target rigidity

The prolongation \(\mathfrak{g}^{(k)}\) of a linear Lie algebra \(\mathfrak{g}\subset \mathfrak{gl}(V)\) plays an important role in the study of symmetries of G-structures. Cartan and Kobayashi-Nagano have given a complete classification of irreducible linear Lie algebras \(\mathfrak{g}\subset \mathfrak{gl}(V)\) with non-zero prolongations.If \(\mathfrak{g}\) is the Lie algebra \(\mathfrak{aut}(\hat{S})\) of infinitesimal linear automorphisms of a projective variety S??V, its prolongation \(\mathfrak{g}^{(k)}\) is related to the symmetries of cone structures, an important example of which is the variety of minimal rational tangents in the study of uniruled projective manifolds. From this perspective, understanding the prolongation \(\mathfrak{aut}(\hat{S})^{(k)}\) is useful in questions related to the automorphism groups of uniruled projective manifolds. Our main result is a complete classification of irreducible non-degenerate nonsingular variety S??V with \(\mathfrak{aut}(\hat {S})^{(k)}\neq0\), which can be viewed as a generalization of the result of Cartan and Kobayashi-Nagano. As an application, we show that when S is linearly normal and Sec?(S)≠?V, the blow-up Bl _S (?V) has the target rigidity property, i.e., any deformation of a surjective morphism f:Y→Bl _S (?V) comes from the automorphisms of Bl _S (?V).     

Chowla’s cosine problem

A continuous linear map T from a Banach algebra A into another B approximately preserves the zero products if ‖T(a)T(b)‖ ≤ α‖a‖‖b‖ (a,b ∈ A, ab = 0) for some small positive α. This paper is mainly concerned with the question of whether any continuous linear surjective map T: A → B that approximately preserves the zero products is close to a continuous homomorphism from A onto B with respect to the operator norm. We show that this is indeed the case for amenable group algebras.     

Lie bialgebra structures on derivation Lie algebra over quantum tori

We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H ¹(W, W ? W) is trivial.