Initial Chain Algebras on Pseudotrees

We investigate a new class of Boolean algebra, called initial chain algebras on pseudotrees. We discuss the relationship between this class and other classes of Boolean algebras. Every interval algebra, and hence every countable Boolean algebra, is an initial chain algebra. Every initial chain algebra on a tree is a superatomic Boolean algebra, and every initial chain algebra on a pseudotree is a minimally-generated Boolean algebra.We show that a free product of two infinite Boolean algebras is an initial chain algebra if and only if both factors are countable.     

Homotopy G-algebra structure on the cochain complex of hom-type algebras

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. We show that the Hochschild type cochain complex of a hom-associative algebra carries a homotopy G-algebra structure. As a consequence, we get a Gerstenhaber algebra structure on the cohomology of a hom-associative algebra. We also find similar results for hom-dialgebras.     

Universal algebra of a Hom-Lie algebra and group-like elements

We construct the universal enveloping algebra of a Hom-Lie algebra and endow it with a Hom-Hopf algebra structure. We discuss group-like elements that we see as a Hom-group integrating the initial Hom-Lie algebra.     

Projections in a Synaptic Algebra

A synaptic algebra is an abstract version of the partially ordered Jordan algebra of all bounded Hermitian operators on a Hilbert space. We review the basic features of a synaptic algebra and then focus on the interaction between a synaptic algebra and its orthomodular lattice of projections. Each element in a synaptic algebra determines and is determined by a one-parameter family of projections—its spectral resolution. We observe that a synaptic algebra is commutative if and only if its projection lattice is boolean, and we prove that any commutative synaptic algebra is isomorphic to a subalgebra of the Banach algebra of all continuous functions on the Stone space of its boolean algebra of projections. We study the so-called range-closed elements of a synaptic algebra, prove that (von Neumann) regular elements are range-closed, relate certain range-closed elements to modular pairs of projections, show that the projections in a synaptic algebra form an M-symmetric orthomodular lattice, and give several sufficient conditions for modularity of the projection lattice.     

(Quasi-)Poisson enveloping algebras

We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.     

Superalgebraic representation of Dirac matrices

We consider a Clifford extension of the Grassmann algebra in which operators are constructed from products of Grassmann variables and derivatives with respect to them. We show that this algebra contains a subalgebra isomorphic to a matrix algebra and that it additionally contains operators of a generalized matrix algebra that mix states with different numbers of Grassmann variables. We show that these operators are extensions of spin-tensors to the case of superspace. We construct a representation of Dirac matrices in the form of operators of a generalized matrix algebra.     

几类特殊的BCI代数

本文给出具有散子代数的ＢＣＩ代数的条件，讨论了Ｌ（Ｘ）为Ｘ的理想的ＢＣＩ代数，完全刻划了每个子代数都是理想的ＢＣＩ代数。     

A compact quantum semigroup generated by an isometry

In this paper we construct a compact quantum semigroup structure on a Toeplitz algebra. We prove the existence of a subalgebra in the dual algebra isomorphic to the algebra of regular Borel measures on a circle with the convolution product. We also prove the existence of Haar functionals in the dual algebra and in the mentioned subalgebra. We show that this compact quantum semigroup contains a dense subalgebra with the structure of a weak Hopf algebra.     

Primitive idempotents in central simple algebras over Fq(t) with an application to coding theory

We consider the algorithmic problem of computing a primitive idempotent of a central simple algebra over the field of rational functions over a finite field. The algebra is given by a set of structure constants. The problem is reduced to the computation of a division algebra Brauer equivalent to the central simple algebra. This division algebra is constructed as a cyclic algebra, once the Hasse invariants have been computed. We give an application to skew constacyclic convolutional codes.     

The Heisenberg envelope for the Hochschild algebra of a finite-dimensional lie algebra

We consider some kind of Hopf algebra assigned to any finite-dimensional Lie algebra. This algebra was pointed out by Hochschild. We prove several statements on its embeddings into an algebra of formal power series. In particular, we obtain similar results for Lie algebras. More precisely, a Lie algebra can be embedded into a Lie algebra of special derivations with coefficients in rational functions in (quasi)polynomials.     

Structure of the Loday–Ronco Hopf algebra of trees

Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of non-commutative symmetric functions in the Malvenuto–Reutenauer Hopf algebra of permutations factors through their Hopf algebra of trees, and these maps correspond to natural maps from the weak order on the symmetric group to the Tamari order on planar binary trees to the boolean algebra.We further study the structure of this Hopf algebra of trees using a new basis for it. We describe the product, coproduct, and antipode in terms of this basis and use these results to elucidate its Hopf-algebraic structure. In the dual basis for the graded dual Hopf algebra, our formula for the coproduct gives an explicit isomorphism with a free associative algebra. We also obtain a transparent proof of its isomorphism with the non-commutative Connes–Kreimer Hopf algebra of Foissy, and show that this algebra is related to non-commutative symmetric functions as the (commutative) Connes–Kreimer Hopf algebra is related to symmetric functions.     

Hom-Hopf代数的对极和Drinfel'd偶

本文证明了带有等价Hom- 范畴的Hom- 双代数是Hom-Hopf 代数, 在基本结构定理成立的前提下, 构造了对极使其成为Hom-Hopf 代数. 研究了拟三角Hom-Hopf 代数的对极, 并给出了Hom-Hopf 代数的Radford 公式. 最后, 对有限维Hom-Hopf 代数, 引入了Drinfel''d 偶的概念, 同时证明了Drinfel''d 偶是拟三角Hom-Hopf 代数.     

The ring of physical states in the <Emphasis Type="Italic">M</Emphasis>(2, 3) minimal Liouville gravity

We consider the M(2, 3) minimal Liouville gravity, whose state space in the gravity sector is realized as irreducible modules of the Virasoro algebra. We present a recursive construction for BRST cohomology classes based on using an explicit form of singular vectors in irreducible modules of the Virasoro algebra. We find a certain algebra acting on the BRST cohomology space and use this algebra to find the operator algebra of physical states.     

On the Semisimplicity of the Outer Derivations of Monomial Algebras

We show that the Hochschild cohomology of a monomial algebra over a field of characteristic zero vanishes from degree two if the first Hochschild cohomology is semisimple as a Lie algebra. We also prove that first Hochschild cohomology of a radical square zero algebra is reductive as a Lie algebra. In the case of the multiple loops quiver, we obtain the Lie algebra of square matrices of size equal to the number of loops.     

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.     

Classical r-Matrices and Novikov Algebras

We study the existence problem for Novikov algebra structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov algebra is necessarily solvable. Conversely we present a 2-step solvable Lie algebra without any Novikov structure. We use extensions and classical r-matrices to construct Novikov structures on certain classes of solvable Lie algebras.     

Spin Hecke algebras of finite and affine types

We introduce the spin Hecke algebra, which is a q-deformation of the spin symmetric group algebra, and its affine generalization. We establish an algebra isomorphism which relates our spin (affine) Hecke algebras to the (affine) Hecke-Clifford algebras of Olshanski and Jones-Nazarov. Relation between the spin (affine) Hecke algebra and a nonstandard presentation of the usual (affine) Hecke algebra is displayed, and the notion of covering (affine) Hecke algebra is introduced to provide a link between these algebras. Various algebraic structures for the spin (affine) Hecke algebra are established.     

Cluster automorphism groups of cluster algebras with coefficients

We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. We introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra (i.e., the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients, cluster algebras with universal geometric coefficients, and cluster algebras from surfaces (except a 4-gon) with coefficients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coefficients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coefficients, its cluster automorphism group is isomorphic to the automorphism group of its initial quiver.     

On integro-differential algebras

The concept of integro-differential algebra has been introduced recently in the study of boundary problems of differential equations. We generalize this concept to that of integro-differential algebra with a weight, in analogy to the differential Rota–Baxter algebra. We construct free commutative integro-differential algebras with weight generated by a differential algebra. This gives in particular an explicit construction of the integro-differential algebra on one generator. Properties of the free objects are studied.     

A solvability criterion for the Lie algebra of derivations of a fat point

We consider the Lie algebra of derivations of a zero-dimensional local complex algebra. We describe an inequality involving the embedding dimension, the order, and the first deviation that forces this Lie algebra to be solvable. Our result was motivated by and generalizes the solvability of the Yau algebra of an isolated hypersurface singularity.