Curved Koszul duality of algebras over unital versions of binary operads

We develop a curved Koszul duality theory for algebras presented by quadratic-linear-constant relations over unital versions of binary quadratic operads. As an application, we study Poisson n-algebras given by polynomial functions on a standard shifted symplectic space. We compute explicit resolutions of these algebras using curved Koszul duality. We use these resolutions to compute derived enveloping algebras and factorization homology on parallelized simply connected closed manifolds with coefficients in these Poisson n-algebras.     

Functional calculus and ∗-regularity of a class of Banach algebras II

In this article, we define a natural Banach ∗-algebra for a C^∗-dynamical system (A,G,α) which is slightly bigger than L¹(G;A) (they are the same if A is finite-dimensional). We will show that this algebra is ∗-regular if G has polynomial growth. The main result in this article extends the two main results in [C.W. Leung, C.K. Ng, Functional calculus and ∗-regularity of a class of Banach algebras, Proc. Amer. Math. Soc., in press].     

Batalin-Vilkovisky algebras and the J-homomorphism

Let X be a topological space. The homology of the iterated loop space H_∗ΩnX is an algebra over the homology of the framed n-disks operad H_∗fDn [E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (2) (1994) 265-285; P. Salvatore, N. Wahl, Framed discs operads and Batalin-Vilkovisky algebras, Q. J. Math. 54 (2) (2003) 213-231]. We explicitly determine this H_∗fDn-algebra structure on H_∗(ΩnX;Q). We show that the action of H_∗(SO(n)) on the iterated loop space H_∗ΩnX is related to the J-homomorphism and that the BV-operator on H_∗(Ω²X) vanishes on spherical classes only in characteristic other than 2.     

Involutions and representations for reduced quantum algebras

In the context of deformation quantization, there exist various procedures to deal with the quantization of a reduced space Mred. We shall be concerned here mainly with the classical Marsden-Weinstein reduction, assuming that we have a proper action of a Lie group G on a Poisson manifold M, with a moment map J for which zero is a regular value. For the quantization, we follow Bordemann et al. (2000) [6] (with a simplified approach) and build a star product red_? on Mred from a strongly invariant star product ? on M. The new questions which are addressed in this paper concern the existence of natural ^∗-involutions on the reduced quantum algebra and the representation theory for such a reduced ^∗-algebra.We assume that ? is Hermitian and we show that the choice of a formal series of smooth densities on the embedded coisotropic submanifold C=J−1(0), with some equivariance property, defines a ^∗-involution for red_? on the reduced space. Looking into the question whether the corresponding ^∗-involution is the complex conjugation (which is a ^∗-involution in the Marsden-Weinstein context) yields a new notion of quantized modular class.We introduce a left (C^∞(M)?λ?,?)-submodule and a right (C^∞(Mred)?λ?,red_?)-submodule of C^∞(C)?λ?; we define on it a C^∞(Mred)?λ?-valued inner product and we establish that this gives a strong Morita equivalence bimodule between C^∞(Mred)?λ? and the finite rank operators on . The crucial point is here to show the complete positivity of the inner product. We obtain a Rieffel induction functor from the strongly non-degenerate ^∗-representations of (C^∞(Mred)?λ?,red_?) on pre-Hilbert right D-modules to those of (C^∞(M)?λ?,?), for any auxiliary coefficient ^∗-algebra D over C?λ?.     

A characterization of Leonard pairs using the notion of a tail

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A^∗:V→V that satisfy (i) and (ii) below:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. In this paper, we characterize the Leonard pairs using the notion of a tail. This notion is borrowed from algebraic graph theory.     

Calabi–Yau Algebras Viewed as Deformations of Poisson Algebras

From any algebra A defined by a single non-degenerate homogeneous quadratic relation f, we prove that the quadratic algebra B defined by the potential w?=?fz is 3-Calabi–Yau. The algebra B can be viewed as a 3-Calabi–Yau completion of Keller. The algebras A and B are both Koszul. The classification of the algebras B in three generators, i.e., when A has two generators, leads to three types of algebras. The second type (the most interesting one) is viewed as a deformation of a Poisson algebra S whose Poisson bracket is non-diagonalizable quadratic. Although the potential of S has non-isolated singularities, the homology of S is computed. Next the Hochschild homology of B is obtained.     

Lie theory and the Chern-Weil homomorphism

Let P→B be a principal G-bundle. For any connection θ on P, the Chern-Weil construction of characteristic classes defines an algebra homomorphism from the Weil algebra Wg=Sg^∗⊗∧g^∗ into the algebra of differential forms A=Ω(P). Invariant polynomials inv(Sg^∗)⊂Wg map to cocycles, and the induced map in cohomology inv(Sg^∗)→H(Abasic) is independent of the choice of θ. The algebra Ω(P) is an example of a commutativeg-differential algebra with connection, as introduced by H. Cartan in 1950. As observed by Cartan, the Chern-Weil construction generalizes to all such algebras.In this paper, we introduce a canonical Chern-Weil map Wg→A for possibly non-commutativeg-differential algebras with connection. Our main observation is that the generalized Chern-Weil map is an algebra homomorphism “up to g-homotopy”. Hence, the induced map inv(Sg^∗)→Hbasic(A) is an algebra homomorphism. As in the standard Chern-Weil theory, this map is independent of the choice of connection.Applications of our results include: a conceptually easy proof of the Duflo theorem for quadratic Lie algebras, a short proof of a conjecture of Vogan on Dirac cohomology, generalized Harish-Chandra projections for quadratic Lie algebras, an extension of Rouvière's theorem for symmetric pairs, and a new construction of universal characteristic forms in the Bott-Shulman complex.     

The Category and Operad of Matching Dialgebras

This paper gives a systematic study of matching dialgebras corresponding to the operad As ⁽²⁾ in Zinbiel (2012) as the only Koszul self dual operad there other than the operads of associative algebras and Poisson algebras. The close relationship of matching dialgebras with semi-homomorphisms and matched pairs of associative algebras are established. By anti-symmetrizing, matching dialgerbas are also shown to give compatible Lie algebras, pre-Lie algebras and PostLie algebras. By the rewriting method, the operad of matching dialgebras is shown to be Koszul and the free objects are constructed in terms of tensor algebras. The operadic complex computing the homology of the matching dialgebras is made explicit.     

Lie algebras generated by bounded linear operators on Hilbert spaces

It is proved that the operator Lie algebra ε(T,T^∗) generated by a bounded linear operator T on Hilbert space H is finite-dimensional if and only if T=N+Q, N is a normal operator, [N,Q]=0, and dimA(Q,Q^∗)<+∞, where ε(T,T^∗) denotes the smallest Lie algebra containing T,T^∗, and A(Q,Q^∗) denotes the associative subalgebra of B(H) generated by Q,Q^∗. Moreover, we also give a sufficient and necessary condition for operators to generate finite-dimensional semi-simple Lie algebras. Finally, we prove that if ε(T,T^∗) is an ad-compact E-solvable Lie algebra, then T is a normal operator.     

Quasi-hereditary covers of higher zigzag algebras of type A

In this paper we define and study some quasi-hereditary covers for higher zigzag algebras of type A. We show how these algebras satisfy three different Koszul properties: they are Koszul in the classical sense, standard Koszul and Koszul with respect to the standard module Δ, according to the definition given in [24]. This last property gives rise to a well defined duality and we compute the Δ-Koszul dual as the path algebra of a quiver with relations.     

Piecewise-Koszul algebras

It is a small step toward the Koszul-type algebras.The piecewise-Koszul algebras are, in general,a new class of quadratic algebras but not the classical Koszul ones,simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases.We give a criteria theorem for a graded algebra A to be piecewise-Koszul in terms of its Yoneda-Ext algebra E(A),and show an A_∞-structure on E(A).Relations between Koszul algebras and piecewise-Koszul algebras are discussed.In particular,our results are related to the third question of Green-Marcos.     

Interactions

Given a C^∗-algebra B, a closed ^*-subalgebra A⊆B, and a partial isometry S in B which interacts with A in the sense that S^∗aS=H(a)S^∗S and SaS^∗=V(a)SS^∗, where V and H are positive linear operators on A, we derive a few properties which V and H are forced to satisfy. Removing B and S from the picture we define an interaction as being a pair of maps (V,H) satisfying the derived properties. Starting with an abstract interaction (V,H) over a C^∗-algebra A we construct a C^∗-algebra B containing A and a partial isometry S whose interaction with A follows the above rules. We then discuss the possibility of constructing a covariance algebra from an interaction. This turns out to require a generalization of the notion of correspondences (also known as Pimsner bimodules) which we call a generalized correspondence. Such an object should be seen as an usual correspondence, except that the inner-products need not lie in the coefficient algebra. The covariance algebra is then defined using a natural generalization of Pimsner's construction of the celebrated Cuntz-Pimsner algebras.     

A local duality principle for the Baire classes of functions

A local dual of a Banach space X is a closed subspace of X^∗ that satisfies the properties that the principle of local reflexivity assigns to X as a subspace of X∗∗. We show that, for every ordinal 1?α?ω₁, the spaces Bα[0,1] of bounded Baire functions of class α are local dual spaces of the space M[0,1] of all Borel measures. As a consequence, we derive that each annihilator Bα^⊥[0,1] is the kernel of a norm-one projection.     

On topological centre problems and SIN quantum groups

Let A be a Banach algebra with a faithful multiplication and ^∗〈A^∗A〉 be the quotient Banach algebra of A∗∗ with the left Arens product. We introduce a natural Banach algebra, which is a closed subspace of ^∗〈A^∗A〉 but equipped with a distinct multiplication. With the help of this Banach algebra, new characterizations of the topological centre Zt(^∗〈A^∗A〉) of ^∗〈A^∗A〉 are obtained, and a characterization of Zt(^∗〈A^∗A〉) by Lau and Ülger for A having a bounded approximate identity is extended to all Banach algebras. The study of this Banach algebra motivates us to introduce the notion of SIN locally compact quantum groups and the concept of quotient strong Arens irregularity. We give characterizations of co-amenable SIN quantum groups, which are even new for locally compact groups. Our study shows that the SIN property is intrinsically related to topological centre problems. We also give characterizations of quotient strong Arens irregularity for all quantum group algebras. Within the class of Banach algebras introduced recently by the authors, we characterize the unital ones, generalizing the corresponding result of Lau and Ülger. We study the interrelationships between strong Arens irregularity and quotient strong Arens irregularity, revealing the complex nature of topological centre problems. Some open questions by Lau and Ülger on Zt(^∗〈A^∗A〉) are also answered.     

On transitive Lie bialgebroids and Poisson groupoids

We prove that, for any transitive Lie bialgebroid (A, A^∗), the differential associated to the Lie algebroid structure on A^∗ has the form d_∗=A[Λ,⋅]+Ω, where Λ is a section of ^∧2A and Ω is a Lie algebroid 1-cocycle for the adjoint representation of A. Globally, for any transitive Poisson groupoid (Γ,Π), the Poisson structure has the form , where ΠF is a bivector field on Γ associated to a Lie groupoid 1-cocycle.     

Thin Hessenberg pairs

A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let V denote a nonzero finite-dimensional vector space over a field K. We consider an ordered pair of linear transformations A:V→V and A^∗:V→V which satisfy both (i) and (ii) below.

(i)

There exists a basis for V with respect to which the matrix representing A is Hessenberg and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^∗ is Hessenberg.

We call such a pair a thin Hessenberg pair (or TH pair). This is a special case of a Hessenberg pair which was introduced by the author in an earlier paper. We investigate several bases for V with respect to which the matrices representing A and A^∗ are attractive. We display these matrices along with the transition matrices relating the bases. We introduce an “oriented” version of called a TH system. We classify the TH systems up to isomorphism.     

Koszul-like algebras and modules

In this paper, the notion of Koszul-like algebra is introduced; this notion generalizes the notion of Koszul algebra and includes some Artin-Schelter regular algebras of global dimension 5 as special examples. Basic properties of Koszul-like modules are discussed. In particular, some necessary and sufficient conditions for KL(A) = L(A) are provided, where KL(A) and L(A) denote the categories of Koszul-like modules and modules with linear presentations (see [1]–[3], etc.) respectively, and A is a Koszul-like algebra. We construct new Koszul-like algebras from the known ones by the “one-point extension.” Some criteria for a graded algebra to be Koszul-like are provided. Finally, we construct many classical Koszul objects from the given Koszul-like objects.