Symmetric Pairs for Quantized Enveloping Algebras

Let θ be an involution of a semisimple Lie algebra g, let g^θ denote the fixed Lie subalgebra, and assume the Cartan subalgebra of g has been chosen in a suitable way. We construct a quantum analog of U(g^θ) which can be characterized as the unique subalgebra of the quantized enveloping algebra of g which is a maximal right coideal that specializes to U(g^θ).     

On Semisimple Hopf Algebras of Dimension 2 m

In this paper we classify all nontrivial semisimple Hopf algebras of dimension 2 ^{n +1} with the group of grouplikes isomorphic to _{2 n–1}×₂. Moreover, we extend some results on irreducible representations from groups to semisimple Hopf algebras and prove that certain semisimple Hopf algebras, including the ones classified in this paper, satisfy the generalized power map property.     

On Auslander-Reiten Quivers of Enveloping Algebras of Restricted Lie Algebras

Let (L, [p]) be a finite dimensional restricted Lie algebra over an algebraically closed field F of characteristic p ≥ 3, X ∈ L* a linear form. In this article we study the Auslander-Reiten quivers of certain blocks of the reduced enveloping algebra u(L,x). In particular, it is shown that the enveloping algebras of supersolvable Lie algebras do not possess AR-components of Euclidean type.     

Bipartite Graphs Whose Edge Algebras Are Complete Intersections

Let R be a monomial subalgebra of k[x₁,…,x_N] generated by square free monomials of degree two. This paper addresses the following question: when is R a complete intersection? For such a k-algebra we can associate a graph G whose vertices are x₁,…,x_N and whose edges are {(x_i, x_j)|x_ix_j  R}. Conversely, for any graph G with vertices {x₁,…,x_N} we define the edge algebra associated with G as the subalgebra of k[x₁,…,x_N] generated by the monomials {x_ix_j|(x_i, x_j) is an edge of G}. We denote this monomial algebra by k[G]. This paper describes all bipartite graphs whose edge algebras are complete intersections.     

Actions of Commutative Hopf Algebras

We show that actions of finite-dimensional semisimple commutativeHopf algebras H on H-module algebras A are essentially group-gradings.Moreover we show that the centralizer of H in the smash productA # H equals A^H H. Using these we invoke results about groupgraded algebras and results about centralizers of separablesubalgebras to give connections between the ideal structureof A, A^H and A # H. Examples of the above occur naturally when one considers: (1) finite abelian groups G of automorphisms of an algebra Awith | G |^–1 A; (2) G-graded algebras, for finite groups G; (3) finite-dimensional restricted Lie algebras L, with semisimplerestricted enveloping algebra u(L), acting as derivations onan algebra A.     

Effacement des dérivations et spectres premiers des algèbres quantiques

Given any (commutative) field k and any iterated Ore extension R=k[X₁][X₂;σ₂,δ₂][X_N;σ_N,δ_N] satisfying some suitable assumptions, we construct the so-called “Derivative-Elimination Algorithm.” It consists of a sequence of changes of variables inside the division ring F=Fract(R), starting with the indeterminates (X₁,…,X_N) and terminating with new variables (T₁,…,T_N). These new variables generate some quantum-affine space such that . This algorithm induces a natural embedding which satisfies the following property:

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. We study both the derivative-elimination algorithm and natural embedding and use them to produce, for the general case, a (common) proof of the “quantum Gelfand–Kirillov” property for the prime homomorphic images of the following quantum algebras: , (wW), R_q[G] (where G denotes any complex, semi-simple, connected, simply connected Lie group with associated Lie algebra and Weyl group W), quantum matrices algebras, quantum Weyl algebras and quantum Euclidean (respectively symplectic) spaces. Another application will be given in [G. Cauchon, J. Algebra, to appear]: In the general case, the prime spectrum of any quantum matrices algebra satisfies the normal separation property.     

Restricted Enveloping Algebras with Minimal Lie Derived Length

Let L be a non-abelian restricted Lie algebra over a field of characteristic p > 0 and let u(L) denote its restricted enveloping algebra. In Siciliano (Publ Math (Debr) 68:503–513, 2006) it was proved that if u(L) is Lie solvable then the Lie derived length of u(L) is at least ⌈log₂(p + 1)⌉. In the present paper we characterize the restricted enveloping algebras whose Lie derived length coincides with this lower bound.     

Algebras of p-Groups of Class 2 Over the Rationals

In this work we analyze p-groups of class 2 G and H, with same rational group algebras. We prove that if QG = QH, then their commutators are equal and the centers, 𝒵(G) and 𝒵(H), have their orders preserved. We apply our results to Frattini Central p-groups, and we present an example of two groups of order p ⁷, with no isomorphic centers and different central cyclic components intersecting the cyclic components of the respective commutators groups.     

PBW-Pairs of Varieties of Linear Algebras

The notion of a Poincaré–Birkhoff–Witt (PBW)-pair of varieties of linear algebras over a field is under consideration. Examples of PBW-pairs are given. We prove that if (𝒱, 𝒲) is a PBW-pair and the variety 𝒱 is homogeneous and Schreier, then so is 𝒲; the results similar to the Schreier property for PBW-pairs are also true for the Freiheitssatz and Word problem. In particular, it follows that the Freiheitssatz is true for the varieties of Akivis and Sabinin algebras. We give also examples of varieties that do not satisfy the Freiheitssatz. It is shown that an element u of a free algebra 𝒲[X] in a homogeneous Schreier variety of algebras 𝒲 satisfying the Freiheitssatz is a primitive element (a coordinate polynomial) if and only if the factor algebra of 𝒲[X] by the ideal generated by the element u is a free algebra in 𝒲. We consider also properties of primitive elements.     

Majorants and Minorants for Elliptic Measures on

Let μ(· ; Σ, G₁) and μ(· ; Ω, G₂) be elliptically contoured measures on ^k centered at 0, having scale parameters (Σ, Ω) and radial cdf′s (G₁, G₂). Elliptical measures v_m(·) and v_M(·), depending on (Σ, Ω, G₁, G₂), are constructed such that V_m(C) ≤ {μ(C; Σ, G₁), μ(C; Ω, G₂)} for every symmetric convex set C ^k with equality for certain sets. These in turn rely on the construction of spectral lower and upper matrix bounds for (Σ, Ω). Extensions include bounds for certain ensembles and mixtures, including versions having star-shaped contours. The lindings specialize to give envelopes for some nonstandard distributions of quadratic forms, with applications to stochastic characteristics of ballistic systems.     

Semiprimitivity of Orthodox Semigroup Algebras

Let S be a finite orthodox semigroup or an orthodox semigroup where the idempotent band E(S) is locally pseudofinite. In this paper, by using principal factors and Rukolaǐne idempotents, we show that the contracted semigroup algebra R₀[S] is semiprimitive if and only if S is an inverse semigroup and R[G] is semiprimitive for each maximal subgroup G of S. This theorem strengthens previous results about the semiprimitivity of inverse semigroup algebras.     

Hopf Algebras and the Penrose Polynomial

Let λ be a positive integer and let G be a plane graph. LetP (G, λ) be the Penrose polynomial of G. We will present an interpretation ofP (G, − λ) in terms of colourings of G. In order to prove our main theorem we construct a Hopf algebra A of graphs and a homomorphism of Hopf algebras Ψ fromA onto a Hopf algebra of polynomials in one indeterminate. If G is a plane graph, thenΨ (G) coincides with the Penrose polynomial of G.     

Degrees of Quantum Function Algebras at Roots of 1

In this paper we will deal with quantum function algebrasF_q[G] in the special case when the parameterqspecializes to a root of 1. Using a combinatorial technique, we will give general formulas for the degree of such algebras and of a particular family of quotients which are fundamental objects in representation theory.     

Orderings and *-Orderings on Cocommutative Hopf Algebras

Our aim is to construct new examples of totally ordered and ∗-ordered noncommutative integral domains. We will discuss the following classes of rings: enveloping algebras U(L), group rings G and smash products U(L) G. All of them are examples of Hopf algebras. Characterizations of orderability for enveloping algebras and group rings and of ∗-orderability for enveloping algebras have been found before and will be recalled in the article. Our main results are: for and L finite–dimensional, we characterize the orderability of U(L) G; for , we give a necessary and a sufficient condition for ∗-orderability of G (G orderable, respectively, G residually ‘torsion-free nilpotent’). Moreover, for and L finite-dimensional, we reduce the problem of characterizing the ∗-orderability of U(L) G to the problem of characterizing the ∗-orderability of G. The latter remains open. The research of the first author was supported by the Ministry of Education, Science and Sport of the Republic of Slovenia under grant P1-0222 (Algebraic methods in operator theory). The research of the second and third author was supported by the Natural Sciences and Engineering Research Council of Canada.     

On Semisimple Hopf Algebras of Dimension pq r

Let p and q be distinct prime numbers. We prove a result on the existence of nontrivial group-like elements in a certain class of semisimple Hopf algebras of dimension pq ^r . We conclude the classification of semisimple Hopf algebras A of dimension pq ² over an algebraically closed field k of characteristic zero, such that both A and A ^* are of Frobenius type. We also complete the classification of semisimple Hopf algebras of dimension pq ²<100.     

Split-By-Nilpotent Extensions Algebras and Stratifying Systems

Let Γ and Λ be artin algebras such that Γ is a split-by-nilpotent extension of Λ by a two sided ideal I of Γ. Consider the change of rings functors G: =_ΓΓ_Λ ?_Λ ? and F: =_ΛΛ_Γ ?_Γ ?. In this article, by assuming that I _Λ is projective, we find the necessary and sufficient conditions under which a stratifying system (Θ, ≤) in modΛ can be lifted to a stratifying system (GΘ, ≤) in mod(Γ). Furthermore, by using the functors F and G, we study the relationship between their filtered categories of modules; and some connections with their corresponding standardly stratified algebras are stated (see Theorem 5.12, Theorem 5.15 and Theorem 5.18). Finally, a sufficient condition is given for stratifying systems in mod(Γ) in such a way that they can be restricted, through the functor F, to stratifying systems in mod(Λ).     

Projective Modules Over Frobenius Algebras and Hopf Comodule Algebras

This note presents some results on projective modules and the Grothendieck groups K ₀ and G ₀ for Frobenius algebras and for certain Hopf Galois extensions. Our principal technical tools are the Higman trace for Frobenius algebras and a product formula for Hattori-Stallings ranks of projectives over Hopf Galois extensions.     

Graded Associative Conformal Algebras of Finite Type

In this paper, we consider graded associative conformal algebras. The class of these objects includes pseudo-algebras over non-cocommutative Hopf algebras of regular functions on some linear algebraic groups. In particular, an associative conformal algebra which is graded by a finite group Γ is a pseudo-algebra over the coordinate Hopf algebra of a linear algebraic group G such that the identity component G ⁰ is the affine line and G/G ⁰???Γ. A classification of simple and semisimple graded associative conformal algebras of finite type is obtained.     

On Strongly Associative Group Algebras

Given a partial action α of a group G on the group algebra FH, where H is a finite group and F is a field whose characteristic p divides the order of H, we investigate the associativity question of the partial crossed product FH*_α G. If FH*_α G is associative for any G and any α, then FH is called “strongly associative.” Using a result of Dokuchaev and Exel (2005 Dokuchaev , M. , Exel , R. ( 2005 ). Associativity of crossed products by partial actions, enveloping actions and partial representations . Trans. Amer. Math. Soc. 357 : 1931 – 1952 .[Crossref], [Web of Science ®] , [Google Scholar]) we characterize the strongly associative modular group algebras FH for several classes of groups H.     

Finite Left-Distributive Algebras and Embedding Algebras

We consider algebras with one binary operation · and one generator (monogenic) and satisfying the left distributive lawa·(b·c)=(a·b)·(a·c). One can define a sequence of finite left-distributive algebrasA_n, and then take a limit to get an infinite monogenic left-distributive algebraA_∞. Results of Laver and Steel assuming a strong large cardinal axiom imply thatA_∞is free; it is open whether the freeness ofA_∞can be proved without the large cardinal assumption, or even in Peano arithmetic. The main result of this paper is the equivalence of this problem with the existence of a certain algebra of increasing functions on natural numbers, called anembedding algebra. Using this and results of the first author, we conclude that the freeness ofA_∞is unprovable in primitive recursive arithmetic.