On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

On Universal Central Extensions of Leibniz Algebras

We construct the endofunctor 𝔲𝔠𝔢 between the category of Leibniz algebras which assigns to a perfect Leibniz algebra its universal central extension, and we obtain the isomorphism 𝔲𝔠𝔢_Lie(𝔮_Lie) ? (𝔲𝔠𝔢_Leib(𝔮))_Lie, where 𝔮 is a perfect Leibniz algebra satisfying the condition [x, [x, y]] + [[x, y], x] = 0, for all x, y ∈ 𝔮. Moreover, we obtain several results concerning the lifting of automorphisms and derivations in a covering. We also study the relationship between the universal central extension of a semidirect product of perfect Leibniz algebras and the semidirect product of the universal central extension of both of them.     

Generalized skew derivations on triangular algebras determined by action on zero products

For a triangular algebra 𝒜 and an automorphism σ of 𝒜, we describe linear maps F,G:𝒜→𝒜 satisfying F(x)y+σ(x)G(y) = 0 whenever x,y∈𝒜 are such that xy = 0. In particular, when 𝒜 is a zero product determined triangular algebra, maps F and G satisfying the above condition are generalized skew derivations of the form F(x) = F(1)x+D(x) and G(x) = σ(x)G(1)+D(x) for all x∈𝒜, where D:𝒜→𝒜 is a skew derivation. When 𝒜 is not zero product determined, we show that there are also nonstandard solutions for maps F and G.     

Projective Summands in Tensor Products of Simple Modules of Finite Dimensional Hopf Algebras

Abstract

Let H be a finite dimensional Hopf algebra over a field k. We show that H contains a unique maximal Hopf ideal J _w (H) contained in J(H), the Jacobson radical of H. We give various characterizations of J _w (H), for example J _w (H) = Ann _H ((H/J(H))^?n ) for all large enough n. The smallest positive integer n with this property is denoted by l _w (H). We prove that l _w (H) equals the smallest number n such that (H/J(H))^?n contains every projective indecomposable H/J _w (H)-module as a direct summand. This also equals the minimal n such that the tensor product of n suitable simple H-modules contains the projective cover of the trivial H/J _w (H)-module as a direct summand. We define projective homomorphisms between H-modules, which are used to obtain various reciprocity laws for tensor products of simple H-modules and their projective indecomposable direct summands. We also discuss some consequences of our general results in case H = kG is a group algebra of a finite group G and k is a field of characteristic p.     

On the Invariant Fields and Rings of Some Groups of Polynomial Automorphisms

Abstract

We consider the group G of C-automorphisms of C(x, y) (resp. C[x, y]) generated by s, t such that t(x) = y, t(y) = x and s(x) = x, s(y) = ? y + u(x) where u ∈ C[x] is of degree k ≥ 2. Using Galois's theory, we show that the invariant field and the invariant algebra of G are equal to C.     

Commuting Graphs of Group Algebras

The commuting graph of a ring R, denoted by Γ(R), is a graph of all whose vertices are noncentral elements of R, and 2 distinct vertices x and y are adjacent if and only if xy = yx. In this article we investigate some graph-theoretic properties of Γ(kG), where G is a finite group, k is a field, and 0 ≠ |G| ∈k. Among other results it is shown that if G is a finite nonabelian group and k is an algebraically closed field, then Γ(kG) is not connected if and only if |G| = 6 or 8. For an arbitrary field k, we prove that Γ(kG) is connected if G is a nonabelian finite simple group or G′ ≠ G″ and G″ ≠ 1.     

Non-abelian Tensor Product of Leibniz Algebras and an Exact Sequence in Leibniz Homology

Abstract

Let 𝔪 and 𝔫 be two-sided ideals of a Leibniz algebra 𝔤 such that 𝔤 = 𝔪 + 𝔫. The goal of the paper is to achieve the exact sequence Ker(𝔪  𝔫 + 𝔫  𝔪 → 𝔤) → HL ₂(𝔤) → HL ₂(𝔤/𝔪) ⊕ HL ₂(𝔤/𝔫) → 𝔪 ∩ 𝔫/ [𝔪,𝔫] → HL ₁(𝔤) → HL ₁(𝔤/𝔪) ⊕ HL ₁(𝔤/𝔫) → 0, where HL denotes the Leibniz homology with trivial coefficients of a Leibniz algebra and denotes a non-abelian tensor product of Leibniz algebras.     

An Equivalence of Categories for Lie Color Algebras

Abstract

Let k be a field with char k = p > 0 and G an abelian group with a bicharacter λ on G. For each p-(G,λ)-Lie color algebra L over k the p-universal enveloping algebra u(L) is a G-graded Hopf algebra,i.e.,a Hopf algebra in the category ^kG ? of kG-comodules. In this paper we describe a subcategory of ^kG ? which is equivalent to the category of the finite dimensional p-(G,λ)-Lie color algebras over k.     

On Derived Equivalences for Selfinjective Algebras

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^? ? K ^b(𝒫 _A ) with Hom _K(Mod?A)(P ^?,P ^?[i]) = 0 for i ≠ 0 and add(P ^?) = add(νP ^?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m  = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.     

Abelian Group Gradings on Full Matrix Rings

For positive integers ? and n, several authors studied ?_?-gradings of the full matrix ring M _n (k) over a field k. In this article, we show that every (G × H)-grading of M _n (k) can be constructed by a pair of compatible G-grading and H-grading of M _n (k), where G and H are any finite groups. When G and H are finite cyclic groups, we characterize all (G × H)-gradings which are isomorphic to a good grading. Moreover, the results can be generalized for any finite abelian group grading of M _n (k).     

Tensor product q-Bernstein polynomials

An affine de Casteljau type algorithm to compute q-Bernstein Bézier curves is introduced and its intermediate points are obtained explicitly in two ways. Furthermore we define a tensor product patch, based on this algorithm, depending on two parameters. Degree elevation procedure is studied. The matrix representation of tensor product patch is given and we find the transformation matrix between a classical tensor product Bézier patch and a tensor product q-Bernstein Bézier patch. Finally, q-Bernstein polynomials B _n,m (f;x,y) for a function f(x,y), (x,y)∈[0,1]×[0,1] are defined and fundamental properties are discussed. AMS subject classification (2000)  65D17     

Modular <Emphasis Type="Italic">W</Emphasis> and <Emphasis Type="Italic">H</Emphasis> Type Lie Algebras

The modular Witt algebra W(p, n) and H(p, 2n) are defined on the polynomial rings Z_p[x₁,...,x_n] and Z_p[X₁,...,x_n, y₁,...,y_n] respectively. We generalize Z_p[x₁,...,xn] and Zp[x₁,...,xn, y₁,...,y_n], so we get the generalized W-type and H-type modular Lie algebras. We find all the derivations of W(p, 1).AMS Subject Classification: Primary 17B40, 17B56.     

Vertex Poisson Algebras Associated with Courant Algebroids

We study relationships between vertex Poisson algebras and Courant algebroids. For any ?-graded vertex Poisson algebra A = ? _n∈? A _(n), we show that A ₍₁₎ is a Courant A ₍₀₎-algebroid. On the other hand, for any Courant 𝒜-algebroid ?, we construct an ?-graded vertex Poisson algebra A = ? _n∈? A _(n) such that A ₍₀₎ is 𝒜 and the Courant 𝒜-algebroid A ₍₁₎ is isomorphic to ? as a Courant 𝒜-algebroid.     

The Relative Brauer Group of an Affine Double Plane

We study algebra classes and divisor classes on a normal affine surface of the form z ² = f(x, y). The affine coordinate ring is T = k[x, y, z]/(z ² ? f), and if R = k[x, y][f ^?1] and S = R[z]/(z ² ? f), then S is a quadratic Galois extension of R. If the Galois group is G, we show that the natural map H¹(G, Cl(T)) → H¹(G, Pic(S)) factors through the relative Brauer group B(S/R) and that all of the maps are onto. Sufficient conditions are given for H¹(G, Cl(T)) to be isomorphic to B(S/R). The groups and maps are computed for several examples.     

On the Hochschild (Co)Homology of Quantum Exterior Algebras

We compute the Hochschild cohomology and homology of the algebra Λ = k 〈x, y〉/(x ², xy + qyx, y ²) with coefficients in ₁ Λ_ψ for every degree preserving k-algebra automorphism ψ : Λ → Λ. As a result we obtain several interesting examples of the homological behavior of Λ as a bimodule.     

Commuting Graphs of Matrix Algebras

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(M _n (F)), where F is a field and n ≥ 2. Also we study the commuting graphs of some classical groups such as GL _n (F) and SL _n (F). We show that Γ(M _n (F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GL _n (F)) and Γ(SL _n (F)). Also we show that for two fields F and E and integers n, m ≥ 2, if Γ(M _n (F))?Γ(M _m (E)), then n = m and |F|=|E|.     

Hopf-Type Algebras

In this article, we provide an alternative approach to the definition of a weak Hopf algebra (WHA). For an associative unital algebra A with a coassociative comultiplication Δ ∈Alg ^u (A, A ? A), the set of homomorphisms from A to A ? A, which do not preserve the units. If the linear maps Ξ₁, Ξ₂ ∈ End(A ? A), defined by Ξ₁(a ? b) = Δ(a)(1 ? b), Ξ₂(a ? b) = (a ? 1)Δ(b), are von Neumann regular elements in the ring End(A ? A) of endomorphisms of A ? A satisfying some appropriate assumptions, we call the A a Hopf-type algebra. We show the existence of a target, a source, a counit, and an antipode of A as in the usual WHA.     

Twisted Automorphisms and Twisted Right Gyrogroups

In this paper we make an attempt to study right loops (S, o) in which, for each y ∈ S, the map σ _y from the inner mapping group G _S of (S, o) to itself given by σ _y (h)(x)o h(y) = h(xoy), x ∈ S, h ∈ G _S is a homomorphism. The concept of twisted automorphisms of a right loop and also the concept of twisted right gyrogroup appears naturally and it turns out that the study is almost equivalent to the study of twisted automorphisms and a twisted right gyrogroup. We also study relationship between twisted gyrotransversals and twisted subgroups.     

Adjunctions of Hom and Tensor as Endofunctors of (Bi-)Module Categories Over Quasi-Hopf Algebras

For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor functors ? ? _k V and V ? _k  ? are known to be left adjoint to some kind of Hom-functors as endofunctors of _H 𝕄. The units and counits of adjunctions, in this case, are formally trivial as in the classical case.

In this paper, we generalize this Hom-tensor adjunction for (bi-)module categories over a quasi-Hopf algebra H and show that these (bi-)module categories are biclosed monoidal. However, the units and counits of adjunctions in these generalized cases are not as trivial as in the Hopf algebra case, and they should be modified in terms of the reassociator and the quasi-antipode. Also, if the H-module V is finitely generated and projective as a k-module, we will obtain a generalized form of adjunction between the tensor functors ? ?V and ? ?V* depending on the reassociator and quasi-antipode of H and describe a natural isomorphism between functors ? ?V* and Hom _k (V, ?) explicitly. Furthermore, we consider the special case V = A being an H-module algebra. In this case, each tensor functor will be a monad and its corresponding right adjoint is a comonad. We describe isomorphisms between the (Eilenberg–Moore) module categories over these monads and the (Eilenberg–Moore) comodule categories over their corresponding comonads explicitly.     

Presentations of the Schützenberger Product of n Groups

ABSTRACT

In this article, we first consider n × n upper-triangular matrices with entries in a given semiring k. Matrices of this form with invertible diagonal entries form a monoid B _n (k). We show that B _n (k) splits as a semidirect product of the monoid of unitriangular matrices U _n (k) by the group of diagonal matrices. When the semiring is a field, B _n (k) is actually a group and we recover a well-known result from the theory of groups and Lie algebras. Pursuing the analogy with the group case, we show that U _n (k) is the ordered set product of n(n ? 1)/2 commutative monoids (the root subgroups in the group case). Finally, we give two different presentations of the Schützenberger product of n groups G ₁,…, G _n , given a monoid presentation ?A _i  | R _i ? of each group G _i . We also obtain as a special case presentations for the monoid of all n × n unitriangular Boolean matrices.