Hochschild cohomology of triangular string algebras and its ring structure

We compute the Hochschild cohomology groups HH^?(A)${\mathrm{HH}}^{?}(A)$ in case A is a triangular string algebra, and show that its ring structure is trivial.     

Symmetric (co)homologies of Lie algebras

Cohomologies of Lie algebras are usually calculated using the Chevalley-Eilenberg cochain complex of skew-symmetric forms. We consider two cochain complexes consisting of forms with some symmetric properties. First, cochains C^*(L) are symmetric in the last 2 arguments, skew-symmetric in the others and satify moreover some kind of Jacobi condition in the last 3 arguments. In characteristic 0, its cohomologies are isomorphic to the cohomologies of the factor-complex C^*(L,L’)/C^*+1(L,K). Second, a symmetric version C_λ^*(A) is defined for an associative algebra A. It is a subcomplex of the cyclic cochain complex. These symmetric cochain complexes are used for the calculation of 3-cohomologies of Cartan Type Lie algebras with trivial coefficients.     

Gerstenhaber Algebra Structure on the Hochschild Cohomology of Quadratic String Algebras

We describe the Gerstenhaber algebra structure on the Hochschild cohomology HH^?(A) when A is a quadratic string algebra. First we compute the Hochschild cohomology groups using Barzdell’s resolution and we describe generators of these groups. Then we construct comparison morphisms between the bar resolution and Bardzell’s resolution in order to get formulae for the cup product and the Lie bracket. We find conditions on the bound quiver associated to string algebras in order to get non-trivial structures.     

Depth One Extensions of Semisimple Algebras and Hopf Subalgebras

Depth one extensions of finite dimensional semisimple algebras are completely characterized in terms of their algebra centers. For extensions of semisimple Hopf algebras this characterization translates into a trivial monoidal action of the dual fusion category Rep(A ^*) on Rep(B).     

Hochschild Cohomology of skew group rings and invariants

Let A be a k-algebra and G be a group acting on A. We show that G also acts on the Hochschild cohomology algebra HH ^⊙ (A) and that there is a monomorphism of rings HH ^⊙ (A) ^G →HH ^⊙ (A[G]). That allows us to show the existence of a monomorphism from HH ^⊙ (Ã) ^G into HH ^⊙ (A), where à is a Galois covering with group G.     

On the radical for some class of Banach algebras

Let A be a complex Banach algebra. It is well known that the second dual A** of A can be equipped with a multiplication that extends the original multiplication on A and makes A** a Banach algebra. We show that Rad(A) = ^⊥(A ^* · A) and Rad(A ^**) = (A ^* · A)^⊥ for some classes of Banach algebras A with scattered structure space. Some applications of these results are given.     

When does continuity on the center imply continuity?

LetA be a Banach algebra. We give a condition forA which forces a homomorphism fromA into a Banach algebra to be continuous if the closure of its continuity ideal has finite codimension, and if its restriction to the center ofA is continuous. We apply this result to answer the question in the title for centralC ^*-algebras,AW ^*-algebras, andL ¹ (G) for certain [FIA]^?-groupsG.     

The Coherent Cohomology Ring of an Algebraic Group

Let G be a group scheme of finite type over a field, and consider the cohomology ring H ^*(G) with coefficients in the structure sheaf. We show that H ^*(G) is a free module of finite rank over its component of degree 0, and is the exterior algebra of its component of degree 1. When G is connected, we determine the Hopf algebra structure of H ^*(G).     

Automorphism groups of trivial extensions

Given a split basic finite dimensional algebra A over a field, we study the relationship between the groups of categorical automorphisms of A and its trivial extension A?D(A). Our results cover all triangular algebras and all 2-nilpotent algebras whose quiver has no nontrivial oriented cycle of length ?2. In this latter as well as in the hereditary case, we give structure theorem for CAut(A?D(A)) in terms of CAut(A). As a byproduct, we get the precise relationship between the first Hochschild cohomology groups of A and A?D(A).     

On a conjecture by Ghahramani-Lau and related problems concerning topological centres

Let A be a Banach algebra, and consider A^** equipped with the first Arens product. We establish a general criterion which ensures that A is left strongly Arens irregular, i.e., the first topological centre of A^** is reduced to A itself. Using this criterion, we prove that for a very large class of locally compact groups, Ghahramani-Lau's conjecture (cf. [Lau 94] and [Gha-Lau 95]) stating the left strong Arens irregularity of the measure algebra M(G), holds true. (Our methods obviously yield as well the right strong Arens irregularity in the situation considered.)Furthermore, the same condition used above implies that every linear left A^**-module homomorphism on A^* is automatically bounded and w^*-continuous. We finally show that our criterion also yields a partial answer to a question raised by Lau-Ülger (Trans. Amer. Math. Soc. 348 (3) (1996) 1191) on the topological centre of the algebra (A^*⊙A)^*, for A having a right approximate identity bounded by 1.     

An Operator Theoretical Approach to Enveloping ϕ* – and C* – Algebras of Melrose Algebras of Totally Characteristic Pseudodifferential Operators

Let X be a compact manifold with boundary. It will be shown (Theorem 3.4) that the small Melrose algebra A? ?_b,cl (χ,^bΩ^1/2) (cf. [22], [23]) of classical, totally characteristic pseudodifferential operators carries no topology such that it is a topological algebra with an open group of invertible elements, in particular, the algebra A cannot be spectrally invariant in any C* – algebra. On the other hand, the symbolic structure of A can be extended continuously to the C* – algebra B generated by A as a subalgebra of ζ(σ^bL²(χ, ^bΩ^1/2)) by a generalization of a method of Gohberg and Krupnik. Furthermore, A is densely embedded in a Fréchet algebra A ? B which is a ?* – algebra in the sense of Gramsch [9, Definition 5.1], reflecting also smooth properties of the original algebra A.     

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.     

The cup product of Hilbert schemes for K3 surfaces

To any graded Frobenius algebra A we associate a sequence of graded Frobenius algebras A ^[n] so that there is canonical isomorphism of rings (H ^*(X;ℚ)[2]) ^[n] ≅H ^*(X ^[n] ;ℚ)[2n] for the Hilbert scheme X ^[n] of generalised n-tuples of any smooth projective surface X with numerically trivial canonical bundle. Oblatum 25-I-2001 & 18-IX-2002?Published online: 24 February 2003     

Sufficiency and strong commutants in quantum probability theory

A probability algebra (A, *, ω) consisting of a^*algebraA with a faithful state ω provides a framework for an unbounded noncommutative probability theory. A characterization of symmetric probability algebra is obtained in terms of an unbounded strong commutant of the left regular representation ofA. Existence of coarse-graining is established for states that are absolutely continuous or continuous in the induced topology. Sufficiency of a^*subalgebra relative to a family of states is discussed in terms of noncommutative Radon-Nikodym derivatives (a form of Halmos-Savage theorem), and is applied to couple of examples (including the canonical algebra of one degree of freedom for Heisenberg commutation relation) to obtain unbounded analogues of sufficiency results known in probability theory over a von Neumann algebra.     

The arens closure of a nonassociative algebra

Given a nonassociative algebra A and an Arens pair A₁, A₂, for A, we identify a subalgcbra ? of A₂ with i (A) ? A ? A₂ and show that ? better reflects the algebraic structure ot A, in parti-cular. any multilinear identity satisfied by ? is also satisfied by ? Hence, ? is commutative or Lie when A is and Jordan when A is a Jordan algebra of characteristic not 2 or 3. Also, we list examples (1) where ? = End_D(V) for A a primitive, associative algebra with commuting ring D and irreducible faithful module V,(2) where ? is the norm closure of A in the arens algebra of all bounded functionals of the bounded functionals for a normed algebra A and (3) where ? is the Arens algebra of all bounded functionals of the bounded functionals with A again normed. Note that dif-ferent Arens closures can arise form the same choice of A, A₁, , A₂ since ? is determined by A, A₁, A₂ and subspaces A₃ ? A₂ ^*, A₄,?A₃ ^*.     

Homomorphisms and composition operators on algebras of analytic functions of bounded type

Let U and V be convex and balanced open subsets of the Banach spaces X and Y, respectively. In this paper we study the following question: given two Fréchet algebras of holomorphic functions of bounded type on U and V, respectively, that are algebra isomorphic, can we deduce that X and Y (or X^* and Y^*) are isomorphic? We prove that if X^* or Y^* has the approximation property and H_wu(U) and H_wu(V) are topologically algebra isomorphic, then X^* and Y^* are isomorphic (the converse being true when U and V are the whole space). We get analogous results for H_b(U) and H_b(V), giving conditions under which an algebra isomorphism between H_b(X) and H_b(Y) is equivalent to an isomorphism between X^* and Y^*. We also obtain characterizations of different algebra homomorphisms as composition operators, study the structure of the spectrum of the algebras under consideration and show the existence of homomorphisms on H_b(X) with pathological behaviors.     

A spectral sequence for string cohomology

Let X be a 1-connected space with free-loop space ΛX. We introduce two spectral sequences converging towards H^*(ΛX;Z/p) and H^*((ΛX)_hT;Z/p). The E₂-terms are certain non-Abelian-derived functors applied to H^*(X;Z/p). When H^*(X;Z/p) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p=2.     

More on the Hochschild and Cyclic Homologies of Crossed Modules of Algebras

We investigate the Hochschild and cyclic homologies of crossed modules of algebras in some special cases. We prove that the cotriple cyclic homology of a crossed module of algebras (I, A, ρ) is isomorphic to HC _*(ρ): HC _*(I) → HC _*(A), provided I is H-unital and the ground ring is a field with characteristic zero. We also calculate the Hochschild and cyclic homologies of a crossed module of algebras (R, 0, 0) for each algebra R with trivial multiplication. At the end, we give some applications proving a new five term exact sequence.     

Representation-finite trivial extension algebras

Let A be a finite-dimensional basic connected associative algebra over an algebraically closed field, and $\text{T(A)=A ? DA}$ its trivial extension by its minimal injective cogenerator. We prove that T(A) is representation-finite of Cartan class Δ if and only if A is an iterated tilted algebra of Dynkin class Δ. The proof also yields a construction procedure for iterated tilted algebras of Dynkin type.     

Hochschild Cohomology of an Algebra Associated with a Circular Quiver

In this article we show that an algebra A = K Γ/(f(X ^s )) has a periodic projective bimodule resolution of period 2, where KΓ is the path algebra of the circular quiver Γ with s vertices and s arrows over a commutative ring K, f(x) is a monic polynomial over K and X is the sum of all arrows in KΓ. Moreover, by means of this projective bimodule resolution, we compute the Hochschild cohomology group of A, and we give a presentation of the Hochschild cohomology ring HH?(A) by the generators and the relations in the case K is a field.