The Yoneda Algebra of a 𝒦2 Algebra Need not be Another 𝒦2 Algebra

The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. 𝒦₂ algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a 𝒦₂ algebra would be another 𝒦₂ algebra. We show that this is not necessarily the case by constructing a monomial 𝒦₂ algebra for which the corresponding Yoneda algebra is not 𝒦₂.     

Generalization of quasi-Koszul algebras

In this paper, we generalize two kinds of graded algebras, δ-Koszul algebras and K p algebras, to the non-graded cases. The trivial modules of δ-Koszul algebras have pure resolutions, while those of K p algebras admit non-pure resolutions. We provide necessary and sufficient conditions for a notherian semiperfect algebra either to be a quasi-δ-Koszul algebra or to be a quasi-K p algebra.     

The Birman–Murakami–Wenzl Algebras of Type D n

The Birman–Murakami–Wenzl algebra (BMW algebra) of type D _n is shown to be semisimple and free of rank (2 ⁿ  + 1)n!! ? (2 ^n?1 + 1)n! over a specified commutative ring R, where n!! =1·3…(2n ? 1). We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type D _n is the image of an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the ring ?[δ^±1]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley–Lieb algebra of type D _n is a subalgebra of the BMW algebra of the same type.     

Poisson color algebras of arbitrary degree

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g₀, where g₀ is some element of the grading group G such that g₀ = 0 or 4g₀≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.     

Presenting Schur Algebras as Quotients of the Universal Enveloping Algebra of gl2

We give a presentation of the Schur algebras S _Q (2,d) by generators and relations, in fact a presentation which is compatible with Serre's presentation of the universal enveloping algebra of a simple Lie algebra. In the process we find a new basis for S _Q (2,d), a truncated form of the usual PBW basis. We also locate the integral Schur algebra within the presented algebra as the analogue of Kostant's Z-form, and show that it has an integral basis which is a truncated version of Kostant's basis.     

Class Partition Algebras as Centralizer Algebras of Wreath Products

In this article, we study an important subalgebra of the tensor product partition algebra P _k (x)? P _k (y), denoted by P _k (x, y) and called “Class Partition Algebra.” We show that the algebra P _k (n, m) is the centralizer algebra of the wreath product S _m ? S _n . Furthermore, the algebra P _k (x, y) and the tensor product partition algebra P _k (x)? P _k (y) are subalgebras of the G-colored partition algebra P _k (x;G) and G-vertex colored partition algebra P _k (x, G) respectively, for every group G with |G|=y ≥ 2k.     

Some Examples of Affine Monoid Algebras

We construct a finitely generated monoid S with a zero element such that for every field K the Jacobson radical of the monoid algebra K[S] is a sum of nilpotent ideals but is not nilpotent. Moreover, the contracted monoid algebra K ₀[S] is a monomial algebra.

If K is a field of characteristic p > 0, then we construct a finitely presented group H _p such that the Jacobson radical J of the group algebra K[H _p ] is a sum of nilpotent ideals, but is not nilpotent. Moreover, K[H _p ]/J is a domain.     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

On Hochschild Cohomology of a Self-injective Special Biserial Algebra Obtained by a Circular Quiver with Double Arrows

We calculate the dimensions of the Hochschild cohomology groups of a self-injective special biserial algebra Λ _s obtained by a circular quiver with double arrows. Moreover, we give a presentation of the Hochschild cohomology ring modulo nilpotence of Λ _s by generators and relations. This result shows that the Hochschild cohomology ring modulo nilpotence of Λ _s is finitely generated as an algebra.     

Gröbner–Shirshov bases for brace algebras

Let A be a brace algebra. This structure implies that A is also a pre-Lie algebra. In this paper, we establish Composition-Diamond lemma for brace algebras. For each pre-Lie algebra L, we find a Gröbner–Shirshov basis for its universal brace algebra U_b(L). As applications, we determine an explicit linear basis for U_b(L) and prove that L is a pre-Lie subalgebra of U_b(L).     

On the McCool group M3 and its associated Lie algebra

We prove that the Lie algebra of the McCool group M₃ is torsion free. As a result, we are able to give a presentation for the Lie algebra of M₃. Furthermore, M₃ is a Magnus group.     

Ideals and simplicity of unitary Lie algebras

Double graded ideals and simplicity of elementary unitary Lie algebra eu _n (R,⁻, γ) and Steinberg unitary Lie algebra stu _n (R,⁻, γ) are characterized, where R is a unital involutory associative algebra over a field F of characteristic zero, n ⩾ 5.     

Graded automorphism group of TKK algebra

The classification of extended affine Lie algebras of type A_1 depends on the Tits-Kantor- Koecher (TKK) algebras constructed from semilattices of Euclidean spaces.One can define a unitary Jordan algebra J(S) from a semilattice S of R~v (v≥1),and then construct an extended affine Lie algebra of type A_1 from the TKK algebra T(J(S)) which is obtained from the Jordan algebra J(S) by the so-called Tits-Kantor-Koecher construction.In R~2 there are only two non-similar semilattices S and S′,where S is a lattice and S′is a non-lattice semilattice.In this paper we study the Z~2-graded automorphisms of the TKK algebra T(J(S)).     

Finite vs affine W-algebras

In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the λ-bracket. In Section 2 we construct, in the most general framework, the Zhu algebra Zhu_ΓV, an associative algebra which “controls” Γ-twisted representations of the vertex algebra V with a given Hamiltonian operator H. An important special case of this construction is the H-twisted Zhu algebra Zhu_H V. In Section 3 we review the theory of non-linear Lie conformal algebras (respectively non-linear Lie algebras). Their universal enveloping vertex algebras (resp. universal enveloping algebras) form an important class of freely generated vertex algebras (resp. PBW generated associative algebras). We also introduce the H-twisted Zhu non-linear Lie algebra Zhu_H R of a non-linear Lie conformal algebra R and we show that its universal enveloping algebra is isomorphic to the H-twisted Zhu algebra of the universal enveloping vertex algebra of R. After a discussion of the necessary cohomological material in Section 4, we review in Section 5 the construction and basic properties of affine and finite W-algebras, obtained by the method of quantum Hamiltonian reduction. Those are some of the most intensively studied examples of freely generated vertex algebras and PBW generated associative algebras. Applying the machinery developed in Sections 3 and 4, we then show that the H-twisted Zhu algebra of an affine W-algebra is isomorphic to the finite W-algebra, attached to the same data. In Section 6 we define the Zhu algebra of a Poisson vertex algebra, and we discuss quasiclassical limits. In the Appendix, the equivalence of three definitions of a finite W-algebra is established. “I am an old man, and I know that a definition cannot be so complicated.” I.M. Gelfand (after a talk on vertex algebras in his Rutgers seminar)     

Twisted product and cohomology

LetH be a Hopf algebra,H ₁ be a sub-Hopf algebra ofH, H ₂ be the quotient Hopt algebra ofH modularH ₁. This paper gives a simplified complex by defining a new base for the cobar complex and proves that the cobar complex ofH has the same cohomology algebra with a twisted product of the cobar complexes ofH ₁ andH ₂. Supported by National Natural Science Foundation of China     

On balanced bases

It is proved that either a given balanced basis of the algebra (n + 1)M₁ M_n or the corresponding complementary basis is of rank n + 1. This result enables us to claim that the algebra (n + 1)M₁ M_n is balanced if and only if the matrix algebra M_n admits a WP-decomposition, i.e., a family of n + 1 subalgebras conjugate to the diagonal algebra and such that any two algebras in this family intersect orthogonally (with respect to the form tr XY) and their intersection is the trivial subalgebra. Thus, the problem of whether or not the algebra (n + 1)M₁ M_n is balanced is equivalent to the well-known Winnie-the-Pooh problem on the existence of an orthogonal decomposition of a simple Lie algebra of type A_n–1 into the sum of Cartan subalgebras.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 213–218.Original Russian Text Copyright © 2005 by D. N. Ivanov.This revised version was published online in April 2005 with a corrected issue number.     

Vertex Poisson Algebras Associated with Courant Algebroids

We study relationships between vertex Poisson algebras and Courant algebroids. For any ?-graded vertex Poisson algebra A = ? _n∈? A _(n), we show that A ₍₁₎ is a Courant A ₍₀₎-algebroid. On the other hand, for any Courant 𝒜-algebroid ?, we construct an ?-graded vertex Poisson algebra A = ? _n∈? A _(n) such that A ₍₀₎ is 𝒜 and the Courant 𝒜-algebroid A ₍₁₎ is isomorphic to ? as a Courant 𝒜-algebroid.     

Compact DG modules and Gorenstein DG algebras

When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras. When A is a regular DG algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul DG algebra. For a general regular DG algebra A, we give some equivalent conditions for the Gorensteiness. For a finite connected DG algebra A, we prove that D^c(A) and D^c(A ^op) admit Auslander-Reiten triangles if and only if A and A ^op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, D^c(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D_lf^b(A) and D_lf^b (A ^op) instead, when A is a regular DG algebra. This work was supported by the National Natural Science Foundation of China (Grant No. 10731070) and the Doctorate Foundation of Ministry of Education of China (Grant No. 20060246003)