Identities of PI-Algebras Graded by a Finite Abelian Group

We consider associative PI-algebras over an algebraically closed field of zero characteristic graded by a finite abelian group G. It is proved that in this case the ideal of graded identities of a G-graded finitely generated PI-algebra coincides with the ideal of graded identities of some finite dimensional G-graded algebra. This implies that the ideal of G-graded identities of any (not necessary finitely generated) G-graded PI-algebra coincides with the ideal of G-graded identities of the Grassmann envelope of a finite dimensional (G × ?₂)-graded algebra, and is finitely generated as GT-ideal. Similar results take place for ideals of identities with automorphisms.     

Representability and Specht problem for G-graded algebras

Let W be an associative PI-algebra over a field F of characteristic zero, graded by a finite group G. Let idG(W) denote the T-ideal of G-graded identities of W. We prove: 1. [G-graded PI-equivalence] There exists a field extension K of F and a finite-dimensional Z/2Z×G-graded algebra A over K such that idG(W)=idG(A^∗) where A^∗ is the Grassmann envelope of A. 2. [G-graded Specht problem] The T-ideal idG(W) is finitely generated as a T-ideal. 3. [G-graded PI-equivalence for affine algebras] Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite-dimensional algebra A over K such that idG(W)=idG(A).     

Polynomial Identities of Finite Dimensional Simple Algebras

Let F be an algebraically closed field and let A and B be arbitrary finite dimensional simple algebras over F. We prove that A and B are isomorphic if and only if they satisfy the same identities.     

Existence of Gradings on Associative Algebras

In this article, we study the existence of gradings on finite dimensional associative algebras. We prove that a connected algebra A does not have a nontrivial grading if and only if A is basic, its quiver has one vertex, and its group of outer automorphisms is unipotent. We apply this result to prove that up to graded Morita equivalence there do not exist nontrivial gradings on the blocks of group algebras with quaternion defect groups and one isomorphism class of simple modules.     

On the structure of graded <Emphasis Type="Italic">λ</Emphasis>-Hopf algebras

Let G be an abelian group, B the G-graded λ-Hopf algebra with A being a bicharacter on G. By introducing some new twisted algebras （coalgebras）, we investigate the basic properties of the graded antipode and the structure for B. We also prove that a G-graded λ-Hopf algebra can be embedded in a usual Hopf algebra. As an application, it is given that if G is a finite abelian group then the graded antipode of a finite dimensional G-graded A-Hopf algebra is invertible.     

When Is a Graded PI Algebra a PI Algebra?

Abstract

Let G be a monoid with cancellation and let ? be a strongly G-graded algebra with finite G-grading satisfying a graded polynomial identity. We show that ? is a PI algebra.     

A Decomposition Theorem for CK1 of Central Simple Algebras

Let A be a central simple algebra over its center F. Define CK₁ A = Coker(K₁ F → K₁ A). We prove that if A and B are F-central simple algebras of coprime degrees, then CK₁(A? _F B) = CK₁ A × CK₁ B.     

Algebraic properties of isometries between groups of invertible elements in Banach algebras

We prove that an isometry T between open subgroups of the invertible groups of unital Banach algebras A and B is extended to a real-linear isometry up to translation between these Banach algebras. While a unital isometry between unital semisimple commutative Banach algebras need not be multiplicative, we prove in this paper that if A is commutative and A or B are semisimple, then (T(eA))⁻¹T is extended to an isometric real algebra isomorphism from A onto B. In particular, A−1 is isometric as a metric space to B−1 if and only if they are isometrically isomorphic to each other as metrizable groups if and only if A is isometrically isomorphic to B as a real Banach algebra; it is compared by the example of ?elazko concerning on non-isomorphic Banach algebras with the homeomorphically isomorphic invertible groups. Isometries between open subgroups of the invertible groups of unital closed standard operator algebras on Banach spaces are investigated and their general forms are given.     

Identities of Group Algebras

We consider polynomial identities of group algebras over a field F of characteristic zero. We prove that any PI group algebra satisfies the same identities as a matrix algebra M _n (F ), where n is the maximal degree of finite dimensional representations of the group over algebraic extensions of F.     

Extensions and isomorphisms for the generalized Fourier algebras of a locally compact group

It is shown that for amenable groups, all finite-dimensional extensions of A_p(G) algebras split strongly. Furthermore, each extension of A_p(G) which splits algebraically also splits strongly. We also show that if G is an almost connected locally compact group, or a subgroup of GL_n(V) (V being a finite-dimensional vector space), and if for a fixed p∈(1,∞), all finite-dimensional singular extensions of A_p(G) split strongly, then G is amenable. Continuous order isomorphisms for the pointwise order of A_p(G) algebras, are characterized as weighted composition maps. Similarly, order isomorphisms for the pointwise order of B_p(G) algebras, are characterized as ∗-algebra isomorphisms followed by multiplication by an invertible positive multiplier. In addition, it is shown that for amenable groups, an order isomorphism for the pointwise order between A_p(G) algebras that preserve cozero sets is necessarily continuous, and hence induces an algebra isomorphism.     

Infinite Rings with Planar Zero-Divisor Graphs

In this article, we give an extension of the Fundamental Theorem of finite dimensional algebras to the case of ?₂-graded algebras. Essentially, the results are the same as in the classical case, except that the notion of a ?₂-graded division algebra needs to be modified. We classify all finite dimensional ?₂-graded division algebras over ? and ?.     

Graded tensor products

We study the polynomial identities of regular algebras, introduced in [A. Regev, T. Seeman, Z₂-graded tensor products of P.I. algebras, J. Algebra 291 (2005) 274-296]. For example, a finite-dimensional algebra is regular if it has a basis whose multiplication table satisfies some commutation relations. The matrix algebra M_n(F) over the field F is regular, which is closely related to M_n(F) being Z_n-graded. We study the polynomial identities of various types of tensor products of such algebras. In particular, using the theory of Hopf algebras, we prove a far reaching extension of the A⊗B theorem for Z₂-graded PI algebras.     

On Characteristic Modules of Graded Quasi-hereditary Algebras

Abstract

Let G be a group and A a G-graded quasi-hereditary algebra. Then its characteristic module is proved to be G-gradable, i.e., it is isomorphic to a G-graded module as A-modules. This implies that the Ringel dual A′ of A admits a canonical G-grading which extends to the graded situation the typical equivalence between Δ-good and ?-good modules of A and A′, respectively. It follows some consequences: the derived category of finitely generated G-graded A-modules is equivalent to the derived category of finitely generated G-graded A'-modules; if G is finite, then the Ringel dual of the smash product A#G* is isomorphic to the smash product A'#G* of A' with G.     

Higman ideal, stable Hochschild homology and Auslander-Reiten conjecture

Let A and B be two finite dimensional algebras over an algebraically closed field, related to each other by a stable equivalence of Morita type. We prove that A and B have the same number of isomorphism classes of simple modules if and only if their 0-degree Hochschild Homology groups HH ₀(A) and HH ₀(B) have the same dimension. The first of these two equivalent conditions is claimed by the Auslander-Reiten conjecture. For symmetric algebras we will show that the Auslander-Reiten conjecture is equivalent to other dimension equalities, involving the centers and the projective centers of A and B. This motivates our detailed study of the projective center, which now appears to contain the main obstruction to proving the Auslander-Reiten conjecture for symmetric algebras. As a by-product, we get several new invariants of stable equivalences of Morita type.     

Graded Polynomial Identities for Tensor Products by the Grassmann Algebra

Abstract

Let 𝕂 be a field of characteristic zero, and R be a G-graded 𝕂-algebra. We consider the algebra R ? E, then deduce its G × ?₂-graded polynomial identities starting from the G-graded polynomial identities of R. As a consequence, we describe a basis for the ? _n  × ?₂-graded identities of the algebras M _n (E). Moreover we give the graded cocharacter sequence of M ₂(E), and show that M ₂(E) is PI-equivalent to M _1,1(E) ? E. This fact is a particular case of a more general result obtained by Kemer.     

On transitive algebras containing a standard finite von Neumann subalgebra

Let M be a finite von Neumann algebra acting on a Hilbert space H and A be a transitive algebra containing M^′. In this paper we prove that if A is 2-fold transitive, then A is strongly dense in B(H). This implies that if a transitive algebra containing a standard finite von Neumann algebra (in the sense of [U. Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975) 271-283]) is 2-fold transitive, then A is strongly dense in B(H). Non-selfadjoint algebras related to free products of finite von Neumann algebras, e.g., LFn and , are studied. Brown measures of certain operators in are explicitly computed.     

On a Regev-Seeman conjecture about ℤ2-graded tensor products

If A,B are superalgebras then, besides A?B, a ?₂-graded tensor product A $ \bar \otimes $ B arises. Kemer proved that if A,B are T-prime algebras then A? B is multi-linear equivalent to a suitable T-prime algebra C. Regev and Seeman conjectured that this holds for A $ \bar \otimes $ B as well. In this paper we prove their conjecture is true indeed, by means of G-graded polynomial identities. The results obtained are valid over any infinite field of characteristic ≠ 2.     

On the cohomology of joins of operator algebras

By analogy with the join in topology, the join A*B for operator algebras A and B acting on Hilbert spaces H and K, respectively, was defined by Gilfeather and Smith (Amer. J. Math. 116 (1994) 541-561). Assuming that K is finite dimensional, they calculated the Hochschild cohomology groups for A*B with coefficients in L(K⊕H). We assume that A is a maximal abelian von Neumann algebra acting on H, A is a subalgebra of , and B is an ultraweakly closed subalgebra of M_n(A) containing A⊗1_n. We show that B may be decomposed into a finite sum of free modules. In this context, we redefine the join of A and B, generalize the calculations of Gilfeather and Smith, and calculate , for all m?0.     

Tilt-Critical Algebras of Tame Type

Let A be a finite dimensional k-algebra over an algebraically closed field. Assume A = kQ/I where Q is a quiver without oriented cycles. We say that A is tilt-critical if it is not tilted but every proper convex subcategory of A is tilted. We describe the tilt-critical algebras which are strongly simply connected and tame.