The Simple Connectedness of a Tame Weakly Shod Algebra

Abstract

We prove that a tame weakly shod algebra A which is not quasi-tilted is simply connected if and only if the orbit graph of its pip-bounded component is a tree, or if and only if its first Hochschild cohomology group H¹(A) with coefficients in _A A _A vanishes. We also show that it is strongly simply connected if and only if the orbit graph of each of its directed components is a tree, or if and only if H¹(A) = 0 and it contains no full convex subcategory which is hereditary of type 𝔸?, or if and only if it is separated and contains no full convex subcategory which is hereditary of type 𝔸?.     

The Structure Theorem of Weak Comodule Algebras

This article mainly gives the structure theorem of weak comodule algebras, that is, assume that H is a weak Hopf algebra, and B a weak right H-comodule algebra, if there exists a morphism φ: H → B of a weak right H-comodule algebras, then there exists an algebra isomorphism: B ? B ^coH #H, where B ^coH denotes the coinvariant subalgebra of B, and B ^coH #H denotes the weak smash product.     

Generalized Group Algebras of Locally Compact Groups

This article studies the homological properties of generalized group algebra L ¹(G, A) of a locally compact group G over a Banach algebra A with an identity of norm 1. It is shown that if L ¹(G, A) is right continuous then G is finite and A is right continuous. It is also shown that L ¹(G, A) is right self-injective if and only if G is finite and A is right self-injective.     

Table algebras with multiple P-polynomial structures

Using covering numbers we prove that a standard real integral table algebra (A, B) with |B| ≥ 6 has a P-polynomial structure with respect to every b ≠ 1 in B if and only if 2|B|-1 is prime and (A, B) is exactly isomorphic to the Bose-Mesner algebra of the association scheme of the ordinary (2|B|-1)-gon. Then we present an example showing that this result is not true if |B| ≤ 5.     

Co-Point Modules Over Frobenius Koszul Algebras

Let A be a Frobenius Koszul algebra such that its Koszul dual A ^! is a quantum polynomial algebra. Co-point modules over A were defined as dual notion of point modules over A ^! with respect to the Koszul duality. In this article, we will see that various important functors between module categories over A used in representation theory of finite dimensional algebras send co-point modules to co-point modules. As a consequence, we will show that if (E, σ) is a geometric pair associated to A ^!, then the map σ:E → E is an automorphism of the point scheme E of A ^!, so that there is a bijection between isomorphism classes of left point modules over A ^! and those of right point modules over A ^!.     

A DESCRIPTION OF NORMAL SEMIGROUPS OF ENDOMORPHISMS OF PROPER INDEPENDENCE ALGEBRAS

Let A be a strong independence algebra of finite rank with at most one constant, and let G he the group of automorphisms of A. Let α be a singular endomorphism of α^G = 〈{α} U G〉. We describe the elements of α^G and give additional characterisations when A is a proper independence algebra and G is a periodic group.     

On a Theorem of V. Dlab

A new proof is given of Dlab's theorem asserting that the left regular representation of an algebra is filtered by the standard modules if and only if the right regular representation of it is filtered by the proper standard modules.     

Flatness Properties of S-Posets

We investigate the Galois coverings of weakly shod algebras. For a weakly shod algebra not quasi-tilted of canonical type, we establish a correspondence between its Galois coverings and the Galois coverings of its connecting component. As a consequence we show that a weakly shod algebra which is not quasi-tilted of canonical type is simply connected if and only if its first Hochschild cohomology group vanishes.     

Derivation Simple Color Algebras and Semisimple Lie Color Algebras

Let K be an algebraically closed field of arbitrary characteristic and Γ an abelian multiplicative group equipped with a bicharacter ε: Γ × Γ → K*. It is proved that, for any finite-dimensional derivation simple color algebra A over K, there exists a simple color algebra S and a color vector space V such that A? S? S^ε(V), where S^ε(V) is the ε-symmetric algebra of V. As an application of this result, a necessary and sufficient condition such that a Lie color algebra is semisimple is obtained.     

The Socle and finite dimensionality of some Banach algebras

The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. One of the main results in Theorem 2 which states that for a locally compact groupG, G is compact if there exists a measure μ in Soc(L ¹(G)) such that μ(G) ≠ 0. We also prove thatG is finite if Soc(M(G)) is closed and every nonzero left ideal inM(G) contains a minimal left ideal.     

F q [M 2], F q [GL 2], and F q [SL 2] as Quantized Hyperalgebras

A ring R is called right Johns if R is right noetherian and every right ideal of R is a right annihilator. R is called strongly right Johns if the matrix ring M _n (R) is right Johns for each integer n ≥ 1. The Faith–Menal conjecture is an open conjecture on QF rings. It says that every strongly right Johns ring is QF. It is proved that the conjecture is true if every closed left ideal of the ring R is finitely generated. This result improves the known result that the conjecture is true if R is a left CS ring.     

Relations Between Algebras and Their Subalgebras of Invariants

We prove that, if H is a finite-dimensional semisimple Hopf algebra, and A is an FCR H-module algebra over an algebraically closed field, then A is a PI-algebra, provided the subalgebra of invariants is a PI-algebra. We also show that if A is an affine algebra with an action of a finite group G by automorphisms, the subalgebra of the fixed points A^G is in the center of A, and the characteristic of the ground field is either zero or relatively prime to the order of G, then A^G is affine. Analogous results are proved for graded algebras and H-module algebras over a semisimple triangular Hopf algebra over a field of characteristic zero. We prove also that, if A is an H-module algebra with an identity element, and H is either a semisimple group algebra or its dual, then, if A is semiprimitive (semiprime), then so is A^H.     

Abelian crossed products and scalar extensions of central simple algebras

Let be a central simple algebra over a field k and G be a commutative group with . It is proved that there exists a regular field extension E/k preserving indices of k-algebras such that is a crossed product with the group G. Bibliography: 11 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 356, 2008, pp. 179-188.     

The Bar-Radical of a Class of Almost Alternative Baric Algebras

In this article we prove that, if (U, ω) is a finite dimensional baric algebra of (γ, δ) type over a field F of characteristic ≠ 2,3,5 such that γ² ? δ² + δ = 1 and δ ≠ 0,1, then rad(U) = R(U) ∩ (bar(U))², where R(U) is the nilradical (maximal nil ideal) of U.     

Automorphisms of Free Left Nilpotent Leibniz Algebras and Fixed Points

ABSTRACT

Let F _m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F _m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F _m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) ^G is not finitely generated.     

The Double Ringel-Hall Algebras of Valued Quivers

This paper is devoted to the study of the structure of the double Ringel-Hall algebra D(Λ) for an infinite dimensional hereditary algebra A, which is given by a valued quiverΓover a finite field, and also to the study of the relations of D(Λ)-modules with representations of valued quiverΓ.     

A superalgebraic interpretation of the quantization maps of Weil algebras

Let G be a Lie group whose Lie algebra g is quadratic. In the paper ＂the non-commutative Weil algebra＂, Alekseev and Meinrenken constructed an explicit G-differential space homomorphism ￡, called the quantization map, between the Well algebra Wg = S（g^＊） χ∧A（g^＊） and Wg= U（g） χ Cl（g） （which they call the noncommutative Weil algebra） for g. They showed that ￡ induces an algebra isomorphism between the basic cohomology rings Hbas^＊（Wg） and Hbas^＊（Wg）. In this paper, we will interpret the quantization map .~ as the super Duflo map between the symmetric algebra S（Tg[1]） and the universal enveloping algebra U（Tg[1]） of a super Lie algebra T9[1] which is canonically associated with the quadratic Lie algebra g. The basic cohomology rings Hbas^＊（Wg） and Hbas^＊（Wg） correspond exactly to S（Tg[1]）^inv and U（Tg[1]）^inv, respectively. So what they proved is equivalent to the fact that the super Duflo map commutes with the adjoint action of the super Lie algebra, and that the super Duflo map is an algebra homomorphism when restricted to the space of invariants.     

Solvability of universal bol 2-loops

ABSTRACT

Let (A, ^?) be a structurable algebra. Then the opposite algebra (A ^op , ^?) is structurable, and we show that the triple system B ^op _A(x, y, z):=V^opx,y(z)=x(y¯z)+z(y¯x)?y(x¯z), x, y, z ∈ A, is a Kantor triple system (or generalized Jordan triple system of the second order) satisfying the condition (A). Furthermore, if A=𝔸₁?𝔸₂ denotes tensor products of composition algebras, (^?) is the standard conjugation, and (^∧) denotes a certain pseudoconjugation on A, we show that the triple systems B ^op _𝔸1?𝔸2 ( x , y¯^∧, z) are models of compact Kantor triple systems. Moreover these triple systems are simple if (dim𝔸₁, dim𝔸₂) ≠ (2, 2). In addition, we obtain an explicit formula for the canonical trace form for compact Kantor triple systems defined on tensor products of composition algebras.     

Approximate Amenability of Certain Semigroup Algebras

In the present paper, it is shown that a left cancellative semigroup S (not necessarily with identity) is left amenable whenever the Banach algebra ℓ¹(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group G with an index set I, then ℓ¹(S) is approximately amenable if and only if G is amenable. Moreover ℓ¹(S) is amenable if and only if G is amenable and I is finite. For a left cancellative foundation semigroup S with an identity such that for every M_a(S)-measurable subset B of S and s ∈ S the set sB is M_a(S)-measurable, it is proved that if the measure algebra M_a(S) is approximately amenable, then S is left amenable. Concrete examples are given to show that the converse is negative.