Dual Coalgebra Research Articles

The finite dual coalgebra as a quantization of the maximal spectrum

In pursuit of a noncommutative spectrum functor, we argue that the Heyneman-Sweedler finite dual coalgebra can be viewed as a quantization of the maximal spectrum of a commutative affine algebra, integrating prior perspectives of Takeuchi, Batchelor, Kontsevich-Soibelman, and Le Bruyn. We introduce fully residually finite-dimensional algebras A as those with enough finite-dimensional representations to let A∘ act as an appropriate depiction of the noncommutative maximal spectrum of A; importantly, this class includes affine noetherian PI algebras. In the case of prime affine algebras that are module-finite over their center, we describe how the Azumaya locus is represented in the finite dual. This is used to describe the finite dual of quantum planes at roots of unity as an endeavor to visualize the noncommutative space on which these algebras act as functions. Finally, we discuss how a similar analysis can be carried out for other maximal orders over surfaces.

Calabi-Yau algebras and the shifted noncommutative symplectic structure

In this paper we show that for a Koszul Calabi-Yau algebra, there is a shifted bi-symplectic structure in the sense of Crawley-Boevey-Etingof-Ginzburg [15], on the cobar construction of its co-unitalized Koszul dual coalgebra, and hence its DG representation schemes, in the sense of Berest-Khachatryan-Ramadoss [3], have a shifted symplectic structure in the sense of Pantev-Toën-Vaquié-Vezzosi [28].

Dual Lie Bialgebra Structures of Twisted Schrödinger-Virasoro Type

In this paper, the structures of dual Lie bialgebras of twisted Schrödinger-Virasoro type are investigated. By studying the maximal good subspaces, we determine the dual Lie coalgebras of the twisted Schrödinger-Virasoro algebras. Then based on this, we construct the dual Lie bialgebra structures of this type. As by-products, four new infinite dimensional Lie algebras are obtained.

Representation homology, Lie algebra cohomology and the derived Harish-Chandra homomorphism

We study the derived representation scheme DRep _n(A) parametrizing the n -dimensional representations of an associative algebra A over a field of characteristic zero. We show that the homology of DRep _n(A) is isomorphic to the Chevalley–Eilenberg homology of the current Lie coalgebra \mathfrak {gl}_n^*(\bar{C}) defined over a Koszul dual coalgebra of A . This gives a conceptual explanation to some of the main results of [BKR] and [BR], relating them (via Koszul duality) to classical theorems on (co)homology of current Lie algebras \mathfrak {gl}_n(A) . We extend the above isomorphism to representation schemes of Lie algebras: for a finite-dimensional reductive Lie algebra \mathfrak g , we define the derived affine scheme DRep _{\mathfrak g}(\mathfrak a) parametrizing the representations (in \mathfrak g ) of a Lie algebra \mathfrak{a} ; we show that the homology of DRep_ {\mathfrak g}(\mathfrak a) is isomorphic to the Chevalley–Eilenberg homology of the Lie coalgebra \mathfrak g^*(\bar{C}) , where C is a cocommutative DG coalgebra Koszul dual to the Lie algebra \mathfrak a . We construct a canonical DG algebra map \Phi_{\mathfrak g}(\mathfrak a): \mathrm {DRep}_{\mathfrak g}(\mathfrak a)^G \to \mathrm {DRep}_{\mathfrak h}(\mathfrak a)^W , relating the G -invariant part of representation homology of a Lie algebra \mathfrak a in \mathfrak g to the W -invariant part of representation homology of \mathfrak a in a Cartan subalgebra of \mathfrak g . We call this map the derived Harish-Chandra homomorphism as it is a natural homological extension of the classical Harish-Chandra restriction map. We conjecture that, for a two-dimensional abelian Lie algebra \mathfrak{a} , the derived Harish-Chandra homomorphism is a quasi-isomorphism. We provide some evidence for this conjecture, including proofs for \mathfrak {gl}_2 and \mathfrak {sl}_2 as well as for \mathfrak {gl}_n, \mathfrak {sl}_n, \mathfrak{so}_n and \mathfrak{sp}_{2n} in the inductive limit as n \to \infty . For any complex reductive Lie algebra \mathfrak g , we compute the Euler characteristic of DRep _{\mathfrak g}(\mathfrak a)^G in terms of matrix integrals over G and compare it to the Euler characteristic of DRep _{\mathfrak h}(\mathfrak a)^W . This yields an interesting combinatorial identity, which we prove for \mathfrak {gl}_n and \mathfrak{sl}_n (for all n ). Our identity is analogous to the classical Macdonald identity, and our quasi-isomorphism conjecture is analogous to the strong Macdonald conjecture proposed in [Ha1, F] and proved in [FGT]. We explain this analogy by giving a new homological interpretation of Macdonald's conjectures in terms of derived representation schemes, parallel to our Harish-Chandra quasi-isomorphism conjecture.

Dual Hodge decompositions and derived Poisson brackets

We study general properties of Hodge-type decompositions of cyclic and Hochschild homology of universal enveloping algebras of (DG) Lie algebras. Our construction generalizes the operadic construction of cyclic homology of Lie algebras due to Getzler and Kapranov. We give a topological interpretation of such Lie Hodge decompositions in terms of $$S^1$$ -equivariant homology of the free loop space of a simply connected topological space. We prove that the canonical derived Poisson structure on a universal enveloping algebra arising from a cyclic pairing on the Koszul dual coalgebra preserves the Hodge filtration on cyclic homology. As an application, we show that the Chas–Sullivan Lie algebra of any simply connected closed manifold carries a natural Hodge filtration. We conjecture that the Chas–Sullivan Lie algebra is actually graded, i.e. the string topology bracket preserves the Hodge decomposition.

Dualizing Complexes of Noetherian Complete Algebras via Coalgebras

Let A be a noetherian complete basic semiperfect algebra over an algebraically closed field, and C = A°be its dual coalgebra. If A is Artin–Schelter regular, then the local cohomology of A is isomorphic to a shift of twisted bimodule 1 C σ* with σ a coalgebra automorphism. This yields that the balanced dualinzing complex of A is a shift of the twisted bimodule σ* A 1. If σ is an inner automorphism, then A is Calabi–Yau. An appendix is included to prove a duality theorem of the bounded derived category of quasi-finite comodules over an artinian coalgebra.

Enriched P-partitions and peak algebras

We develop a more general view of Stembridge's enriched P-partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group. Initially we focus on commutative peak algebras, spanned by sums of permutations with the same number of peaks, where we consider several variations on the definition of “peak.” Whereas Stembridge's enriched P-partitions are related to quasisymmetric functions (the dual coalgebra of Solomon's type A descent algebra), our generalized enriched P-partitions are related to type B quasisymmetric functions (the dual coalgebra of Solomon's type B descent algebra). Using these functions, we move on to explore (non-commutative) peak algebras spanned by sums of permutations with the same set of peaks. While some of these algebras have been studied before, our approach gives explicit structure constants with a combinatorial description.

Dual Coalgebras of Jordan Bialgebras and Superalgebras

W. Michaelis showed for Lie bialgebras that the dual coalgebra of a Lie algebra is a Lie bialgebra. In the present article we study an analogous question in the case of Jordan bialgebras. We prove that a simple infinite-dimensional Jordan superalgebra of vector type possesses a nonzero dual coalgebra. Thereby, we demonstrate that the hypothesis formulated by W. Michaelis for Lie coalgebras fails in the case of Jordan supercoalgebras.

The locally finite part of the dual coalgebra of quantized irreducible flag manifolds

The notion of locally finite part of the dual coalgebra of certain quantized coordinate rings is introduced. In the case of irreducible flag manifolds this locally finite part is shown to coincide with a natural quotient coalgebra V of U_q(g). On the way the coradical filtration of V is determined. A graded version of the duality between V and the quantized coordinate ring is established. This leads to a natural construction of several examples of quantized vector spaces. As an application covariant first order differential calculi on quantized irreducible flag manifolds are classified. Keywords: quantum groups, quantized flag manifolds

On coreflexive coalgebras and comodules over commutative rings

In this paper we study dual coalgebras of algebras over arbitrary (Noetherian) commutative rings. We present and study a generalized notion of coreflexive comodules and use the results obtained for them to characterize the so called coreflexive coalgebras. Our approach in this note is an algebraically topological one.

Podleś' quantum sphere: dual coalgebra and classification of covariant first-order differential calculus

The dual coalgebra of Podleś' quantum sphere O q( S 2 c) is determined explicitly. This result is used to classify all finite-dimensional covariant first-order differential calculi over O q( S 2 c) for all but exceptional values of the parameter c.

THE DUAL COALGEBRA OF CERTAIN INFINITE-DIMENSIONAL LIE ALGEBRAS

ABSTRACT Using translates, a characterization for the dual coalgebra of any Lie algebra is given. This characterization is analogous to the well known characterization for the dual coalgebra of any associative algebra. For any commutative, associative algebra and finite set of commuting derivations of satisfying a certain additional hypothesis, the structure of the dual coalgebra of the Lie subalgebra of is determined. This generalizes a result Nichols[1] proved for the case . As an application, the family of dual coalgebras which corresponds to the family of infinite-dimensional Lie algebras of derivations of polynomials in several indeterminates is then given.

Dual coalgebras of algebras over commutative rings

In the study of algebraic groups the representative functions related to monoid algebras over fields provide an important tool which also yields the finite dual coalgebra of any algebra over a field. The purpose of this note is to transfer this basic construction to monoid algebras over commutative rings R. As an application we obtain a bialgebra (Hopf algebra) structure on the finite dual of the polynomial ring R[ x] over a noetherian ring R. Moreover, we give a sufficient condition for the finite dual of any R-algebra A to become a coalgebra. In particular, this condition is satisfied provided R is noetherian and hereditary.

On the definition of the dual Lie coalgebra of a Lie algebra

Let $L$ be a Lie algebra over a field $K$. The dual Lie coalgebra $L^\circ$ of $L$ has been defined by W. Michaelis to be the sum of all good subspaces $V$ of the dual space $L^*$ of $L$: $V$ is good if ${}^tm (V) \subset V \otimes V$, where $m$ is the multiplication of $L$. We show that $ L^\circ = {}^t m^{-1} ( L^* \otimes L^* )$ as in the associative case.

Quantum dressing orbits on compact groups

The quantum double is shown to imply the dressing transformation on quantum compact groups and the quantum Iwasawa decompositon in the general case. Quantum dressing orbits are described explicitly as *-algebras. The dual coalgebras consisting of differential operators are related to the quantum Weyl elements. Besides, the differential geometry on a quantum leaf allows a remarkably simple construction of irreducible *-representations of the algebras of quantum functions. Representation spaces then consist of analytic functions on classical phase spaces. These representations are also interpreted in the framework of quantization in the spirit of Berezin applied to symplectic leaves on classical compact groups. Convenient “coherent states” are introduced and a correspondence between classical and quantum observables is given.

On a ?super? version of Lie's third fundamental theorem

Invoking the techniques of nonstandard analysis, we associate a super Lie group in the sense of M. Rothstein to any finite-dimensional super Lie module over an arbitrary graded-commutative Banach algebra, which obeys duality (in the sense that it may be recaptured from its dual coalgebra).

Graded manifolds and vector bundles: A functorial correspondence

A functorial correspondence between the category of graded manifolds and the category of vectors bundles is given. Given a graded manifold (X,A), a vector bundle G over X is given as a subset of the product X×A0 where A0 is the dual coalgebra of A. This bundle has an obvious coalgebra structure on each fiber. The correspondence is achieved by showing that the sheaf A is isomorphic as a sheaf of algebras to the sheaf of sections of the vector bundle G′ dual to G.

Generalized dual coalgebras of algebras, with applications to cofree coalgebras

Generalized dual coalgebras of algebras, with applications to cofree coalgebras