Symplectic algebras and poisson algebras

Abstract

A ring R is called left P-injective if for every a ∈ R, aR = r(l(a)) where l( ? ) and r( ? ) denote left and right annihilators respectively. The ring R is called left GP-injective if for any 0 ≠ a ∈ R, there exists n > 0 such that a ⁿ  ≠ 0 and a ⁿ R = r(l(a ⁿ )). As a response to an open question on GP -injective rings, an example of a left GP-injective ring which is not left P-injective is given. It is also proved here that a ring R is left FP -injective if and only if every matrix ring 𝕄 _n (R) is left GP-injective.     

Hopf algebras over hopf algebras

Summary In any category with products and a terminal object one may define the notions of group, module over a group etc. if f: R′→R is a homomorphism of groups, and M an R-module, then one has an induced R′-module f*(M). If one is working in the category of sets, one may define a functor left adjoint to f* by N→R⊗_R′ N, where N is an R′-module. In this paper we show that f* has a left adjoint when one is working in the category of graded connected coalgebras over a field.     

Baxter algebras and Hopf algebras

By applying a recent construction of free Baxter algebras, we obtain a new class of Hopf algebras that generalizes the classical divided power Hopf algebra. We also study conditions under which these Hopf algebras are isomorphic.

     

Coloured peak algebras and Hopf algebras

For G a finite abelian group, we study the properties of general equivalence relations on G _n = G ⁿ ⋊ _n , the wreath product of G with the symmetric group _n , also known as the G-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of G _n as well as graded connected Hopf subalgebras of ⨁ _{n≥ o} G _n . In particular we construct a G-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or G-coloured descent algebra). We show that the direct sum of the G-coloured peak algebras is a Hopf algebra. We also have similar results for a G-colouring of the Loday-Ronco Hopf algebras of planar binary trees. For many of the equivalence relations under study, we obtain a functor from the category of finite abelian groups to the category of graded connected Hopf algebras. We end our investigation by describing a Hopf endomorphism of the G-coloured descent Hopf algebra whose image is the G-coloured peak Hopf algebra. We outline a theory of combinatorial G-coloured Hopf algebra for which the G-coloured quasi-symmetric Hopf algebra and the graded dual to the G-coloured peak Hopf algebra are central objects. 2000 Mathematics Subject Classification Primary: 16S99; Secondary: 05E05, 05E10, 16S34, 16W30, 20B30, 20E22Bergeron is partially supported by NSERC and CRC, CanadaHohlweg is partially supported by CRC     

Leibniz algebras constructed by Witt algebras

ABSTRACT

We describe infinite-dimensional Leibniz algebras whose associated Lie algebra is the Witt algebra and we prove the triviality of low-dimensional Leibniz cohomology groups of the Witt algebra with the coefficients in itself.     

Bordism of algebras over Hopf algebras

An algebraic theory of bordism via characteristic numbers, analogous to topological bordism, is given. The Steenrod algebra is replaced by a fairly general graded Hopf algebra A, topological spaces by algebras over A, vector bundles by Thom modules, and closed manifolds by Poincaré algebras over A.     

Lie algebras and algebras of associative type

In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type A = ⊕ _αεG A _α graded by some group G, over a field of characteristic zero, has a nonzero component A ₁ (where 1 stands for the identity element of G), and A ₁ is a semisimple associative algebra. Let B = ⊕ _αεG B _α be a finite-dimensional semisimple Lie algebra over a prime field F _p , and let B be graded by a commutative group G. If B = F _p ? _? A _L , where A _L is the commutator algebra of a ?-algebra A = ⊕ _αεG A _α ; if ? ? _? A is an algebra of associative type, then the 1-component of the algebra K ? _? B, where K stands for the algebraic closure of the field F _p , is the sum of some algebras of the form gl(n _i ,K).     

Boolean algebras R-generated by MV-effect algebras

We prove that every MV-effect algebra M is, as an effect algebra, a homomorphic image of its R-generated Boolean algebra. We characterize central elements of M in terms of the constructed homomorphism.     

Duality and operator algebras II: operator algebras as Banach algebras

We answer, by counterexample, several questions concerning algebras of operators on a Hilbert space. The answers add further weight to the thesis that, for many purposes, such algebras ought to be studied in the framework of operator spaces, as opposed to that of Banach spaces and Banach algebras. In particular, the ‘nonselfadjoint analogue’ of a w*-algebra resides naturally in the category of dual operator spaces, as opposed to dual Banach spaces. We also show that an automatic w*-continuity result in the preceding paper of the authors, is sharp.     

Lie-admissible algebras and generalized witt algebras ∗

The symmetric group 𝔖_n+1 of degree n + 1 admits an n-dimensional irreducible Q𝔖_n-module V corresponding to the hook partition (2, 1^n?1). By the work of Craig and Plesken, we know that there are σ(n + 1) many isomorphism classes of Z𝔖_n+1-lattices which are rationally equivalent to V, where σ denotes the divisor counting function. In the present article, we explicitly compute the Solomon zeta function of these lattices. As an application we obtain the Solomon zeta function of the Z𝔖_n+1-lattice defined by the Specht basis.     

Localization algebras and deformations of Koszul algebras

We show that the center of a flat graded deformation of a standard Koszul algebra A behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed point set. In particular, the center of A acts by characters on the deformed standard modules, providing a “localization map”. We construct a universal graded deformation of A and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming from the algebra Koszul dual to A. This is an algebraic version of a duality discovered by Goresky and MacPherson between the equivariant cohomology rings of partial flag varieties and Springer fibers; we recover and generalize their result by showing that the center of the universal deformation for the ring governing a block of parabolic category O{\mathcal{O}} for \mathfrakgl_n{\mathfrak{gl}_n} is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version of the “category O{\mathcal{O}}” of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric variety.     

Decompositions of universal algebras by idempotent algebras

An algebra with units is an algebra in which every subalgebra contains a singleton subalgebra. A one-unit-algebra is an algebra in which every subalgebra contains exactly one singleton subalgebra. IfU,V are subclasses of a classK of algebras,U _K V is the class of all K on which there is a congruence such that /V and every -class that is a subalgebra of belonging toK belongs also toU, e.g., ifK is the class of all semigroups,V is the class of all bands andU is the class of all groups,U _K V is the class of all bands of groups. We studyU _K V andU _K U whereU is a class of one-unit-K-algebras andV is a class of idempotentK-algebras. IfK is a class of algebras of type closed under subalgebras and homomorphisms,U is the class of all one-unit-K-algebras andV is the class of all idempotentK-algebras, thenU _K V is the class of allK-algebras that are -reducts of , e-algebras satisfying e(x) is a singleton subalgebra for everyx A belonging to the -subalgebra of generated byx and e(f(– x₁, x₂,..., x_n))=e(fe(x₁), e(x₂),..., e(x_n)) for every n-ary operationf of type . IfK is a variety of algebras with units and of finite type,U andV are finitely based (relative toK) subquasivarieties ofK, thenU _K V is finitely based relative toK. IfK is the variety of all commutative groupoids with an additional unary operatione satisfying e(e(x))=e(x)=e(x)· e(x), e(x · y)=e(x)· e(y),U andV are the subvarities ofK defined by e(x)=e(y) andx=e(x) respectively, thenU _K U is neither a variety nor finitely based. Some applications to semigroups and quasigroups are considered.Presented by G. Grätzer.     

Type A Hecke algebras and related algebras

Let be the Hecke algebra of the symmetric group over a field K of characteristic and a primitive -th root of one in K. We show that an -module is projective if and only if its restrictions to any -parabolic subalgebra of is projective. Moreover, we give a new construction of blocks of -parabolic subalgebras, in terms of skew group algebras over local commutative algebras. Received: 30 June 2003     

Twisted split category algebras as quasi-hereditary algebras

We show that if C is a finite split category, k is a field of characteristic 0, and α is a 2-cocycle of C with values in k  ^× , then the twisted category algebra k _α C is quasi-hereditary.     

Elliptic Lie algebras and tubular algebras

This article is to study relations between tubular algebras of Ringel and elliptic Lie algebras in the sense of Saito-Yoshii. Using the explicit structure of the derived categories of tubular algebras given by Happel-Ringel, we prove that the elliptic Lie algebra of type , , or is isomorphic to the Ringel-Hall Lie algebra of the root category of the tubular algebra with the same type. As a by-product of our proof, we obtain a Chevalley basis of the elliptic Lie algebra following indecomposable objects of the root category of the corresponding tubular algebra. This can be viewed as an analogue of the Frenkel-Malkin-Vybornov theorem in which they described a Chevalley basis for each untwisted affine Kac-Moody Lie algebra by using indecomposable representations of the corresponding affine quiver.