Derived brackets and sh Leibniz algebras

We develop a general framework for the construction of various derived brackets. We show that suitably deforming the differential of a graded Leibniz algebra extends the derived bracket construction and leads to the notion of strong homotopy (sh) Leibniz algebra. We discuss the connections among homotopy algebra theory, deformation theory and derived brackets. We prove that the derived bracket construction induces a map from suitably defined deformation theory equivalence classes to the isomorphism classes of sh Leibniz algebras.     

On the Leibniz (co)homology of the Lie algebra of the Euclidean group

The Lie algebra of the Euclidean group is an abelian extension of the orthogonal Lie algebra. We compute its Leibniz (co)homology. It is computed via the identification of certain orthogonal invariants and shown to be an algebra generated by a n−1-fold tensor and an n-fold tensor.     

Rational homotopy of Leibniz algebras

We construct a non-commutative rational homotopy theory by replacing the pair (Lie algebras, commutative algebras) by the pair (Leibniz algebras, Leibniz-dual algebras). Both pairs are Koszul dual in the sense of operads (Ginzburg–Kapranov). We prove the existence of minimal models and the Hurewicz theorem in this framework. We define Leibniz spheres and prove that their homotopy is periodic. Received: 19 September 1997 / Revised version: 23 February 1998     

Quantum Schur algebras revisited

A geometric construction of quantum Schur algebras was given by Beilinson, Lusztig and MacPherson in terms of pairs of flags in a vector space. By viewing such pairs of flags as representations of a poset, we give a recursive formula for the structure constants of quantum Schur algebras which is related to certain Hall polynomials. As an application, we provide a direct proof of the fundamental multiplication formulas which play a key role in the Beilinson-Lusztig-MacPherson realization of quantum gl_n. In the appendix we show how to groupoidify quantum Schur algebras in the sense of Baez, Hoffnung and Walker.     

Representation type of Schur algebras

We give a complete classification of the classical Schur algebras and the infinitesimal Schur algebras which have tame representation type. In combination with earlier work of some of the authors on semisimplicity and finiteness, this completes the classification of representation type of all classical and infinitesimal Schur algebras in all characteristics. Received October 17, 1997; in final form March 5, 1998     

Lie algebras and recurrence relations III:q-analogs and quantized algebras

q-Analogs of the basic structures discussed in Lie Algebras and Recurrence Relations I are presented. Theq-Heisenberg-Weyl (qHW) and qsl(2) algebras are discussed in detail. Presently it is known that such structures are very closely tied in with the theory of quantum groups. Among other topics, coherent state representations and their interpretations asq-identities forq-Hermite and Al-Salam-Chihara (q-Meixner) polynomials are discussed. A discussion of Clebsch-Gordan coefficients for a qsu(2)-type algebra is presented.     

Leibniz cohomology and the calculus of variations

A geometric interpretation of the Leibniz coboundary is given in terms of the calculus of variations. For a differentiable manifold M, Leibniz cohomology generalizes de Rham cohomology by including all tensors as cochains. When applied to two-tensors, the conditions for the vanishing of a Leibniz cochain are related to the necessary conditions to achieve an extreme value of the integral of the tensor over an immersed surface. A local formula for the coboundary of any tensor is given in terms of a coordinate chart, and the Leibniz coboundary of the Riemann curvature tensor is computed in terms of the derivative of sectional curvature.     

Some properties of c-covers of a pair of Lie algebras

In [H. Safa and H. Arabyani, On c-nilpotent multiplier and c-covers of a pair of Lie algebras, Commun. Algebra 45(10) (2017), 4429–4434], we characterized the structure of the c-nilpotent multiplier of a pair of Lie algebras in terms of its c-covering pairs and discussed some results on the existence of c-covers of a pair of Lie algebras. In the present paper, it is shown under some conditions that a relative c-central extension of a pair of Lie algebras is a homomorphic image of a c-covering pair. Moreover, we prove that a c-cover of a pair of finite dimensional Lie algebras, under some assumptions, has a unique domain up to isomorphism and also that every perfect pair of Lie algebras admits at least one c-cover. Finally, we discuss a result concerning the isoclinism of c-covering pairs.     

The fundamental duality of partially ordered sets

The representation of partially ordered sets by subsets of some set such that specified joins (meets) are taken to unions (intersections) suggests two categories, that of partially ordered sets with specified joins and meets, and that of sets equipped with suitable collections of subsets, and adjoint contravariant functors between them. This, in turn, induces a duality including, among several others, the two Stone Dualities and that between spatial locales and sober spaces.     

Higher level affine Schur and Hecke algebras

We define a higher level version of the affine Hecke algebra and prove that, after completion, this algebra is isomorphic to a completion of Webster's tensor product algebra of type A. We then introduce a higher level version of the affine Schur algebra and establish, again after completion, an isomorphism with the quiver Schur algebra. An important observation is that the higher level affine Schur algebra surjects to the Dipper-James-Mathas cyclotomic q-Schur algebra. Moreover, we give nice diagrammatic presentations for all the algebras introduced in this paper.     

Rota-Baxter algebras and dendriform algebras

In this paper we study the adjoint functors between the category of Rota-Baxter algebras and the categories of dendriform dialgebras and trialgebras. In analogy to the well-known theory of the adjoint functor between the category of associative algebras and Lie algebras, we first give an explicit construction of free Rota-Baxter algebras and then apply it to obtain universal enveloping Rota-Baxter algebras of dendriform dialgebras and trialgebras. We further show that free dendriform dialgebras and trialgebras, as represented by binary planar trees and planar trees, are canonical subalgebras of free Rota-Baxter algebras.     

A note on bideterminants for Schur superalgebras

Let S(m|n,r)_Z be a Z-form of a Schur superalgebra S(m|n,r) generated by elements ξ_i,j. We solve a problem of Muir and describe a Z-form of a simple S(m|n,r)-module D_λ,Q over the field Q of rational numbers, under the action of S(m|n,r)_Z. This Z-form is the Z-span of modified bideterminants [T_?:T_i] defined in this work. We also prove that each [T_?:T_i] is a Z-linear combination of modified bideterminants corresponding to (m|n)-semistandard tableaux T_i.     

The Isbell monad

In 1966 [7], John Isbell introduced a construction on categories which he termed the “couple category” but which has since come to be known as the Isbell envelope. The Isbell envelope, which combines the ideas of contravariant and covariant presheaves, has found applications in category theory, logic, and differential geometry. We clarify its meaning by exhibiting the assignation sending a locally small category to its Isbell envelope as the action on objects of a pseudomonad on the 2-category of locally small categories; this is the Isbell monad of the title. We characterise the pseudoalgebras of the Isbell monad as categories equipped with a cylinder factorisation system; this notion, which appears to be new, is an extension of Freyd and Kelly's notion of factorisation system [5] from orthogonal classes of arrows to orthogonal classes of cocones and cones.     

The centralizer algebras of lie color algebras

In his 1977 paper, V.G. Kac classified the finite-dimensional simple complex Lie superalgebras. After Kac’s paper, M. Scheunert initiated the study of a generalization of Lie superalgebras - the Lie color algebras. We construct some new families of simple Lie color algebras. Following the work of A. Berele and A. Regev and A.N. Sergeev, who studied the general linear and sq(n)-series superalgebra cases, and the work of G. Benkart, C. Lee Shader, and A. Ram, who studied the orthosymplectic cases, we examine the centralizer algebras of some classical Lie superalgebras and their Lie color algebra counterparts acting on tensor space and derive Schur-Weyl duality results for these algebras and their centralizers.     

Homological characterization of lean algebras

Certain classes of lean quasi-hereditary algebras play a central role in the representation theory of semisimple complex Lie algebras and algebraic groups. The concept of a lean semiprimary ring, introduced recently in [1] is given here a homological characterization in terms of the surjectivity of certain induced maps between Ext¹-groups. A stronger condition requiring the surjectivity of the induced maps between Ext ^k -groups for allk≥1, which appears in the recent work of Cline, Parshall and Scott on Kazhdan-Lusztig theory, is shown to hold for a large class of lean quasi-hereditary algebras. Research partially supported by NSERC of Canada and by Hungarian National Foundation for Scientific Research grant no. 1903 Research partially supported by NSERC of Canada