A Criterion for the Triviality of G(D) and Its Applications to the Multiplicative Structure of D

Let D be an F-central division algebra of index n. Here we present a criterion for the triviality of the group G(D) = D*/Nrd _D/F (D*)D′ and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK ₁(D) = 1 and F ^*2 = F ^*2n . Using this, we investigate the role of some particular subgroups of D* in the algebraic structure of D. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup. More applications of this criterion including the computation of G(D) and the structure of maximal subgroups of D* are also investigated     

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.     

Some Skew Linear Groups Satisfying Generalized Group Identities

In this article, we consider a type of generalized group identity and extend some earlier results. For example, we show that, if D is a division ring with infinite center, then every subnormal subgroup of GL_n(D) satisfying a generalized group identity over GL_n(D) is central.     

Grothendieck groups of unbounded complexes of finitely generated modules

Let Λ be a left Artinian ring, D⁺(mod Λ) (resp., D⁻(mod Λ), D(mod Λ)) the derived category of bounded below complexes (resp., bounded above complexes, unbounded complexes) of finitely generated left Λ-modules. We show that the Grothendieck groups K₀(D⁺(mod Λ)), K₀(D⁻(mod Λ)) and K₀(D(mod Λ)) are trivial. Received: 7 April 2005     

On the structure and representations of non-balanced quantum doubles

For a field k and two finite groups G and X, when G acts on X from the right by group automorphisms, there is a Hopf algebra structure on k-space (kX ^op )* ? kG (see Theorem 2.1), called a non-balanced quantum double and denoted by D _X (G). In this paper, some Hopf algebra properties of D _X (G) are given, the representation types of D _X (G) viewed as a k-algebra are discussed, the algebra structure and module category over D _X (G) are studied. Since the Hopf algebra structure of non-balanced quantum double D _X (G) generalizes the usual quantum double D(G) for a finite group G, all results about D _X (G) in this paper can also be used to describe D(G) as a special case and the universal R-matrix of D _X (G) provides more solutions of Yang-Baxter equation.     

Adjacency preserving bijective maps on triangular matrices over any division ring

Let 𝕋 _n (D) be the set of n × n upper triangular matrices over a division ring D. We characterize the adjacency preserving bijective maps in both directions on 𝕋 _n (D) (n ≥ 3). As applications, we describe the ring semi-automorphisms and the Jordan automorphisms on upper triangular matrices over a simple Artinian ring.     

Generic Modules Over a Class of Drinfeld's Quantum Doubles

Let k be a field and A _n (ω) be the Taft's n ²-dimensional Hopf algebras. When n is odd, the Drinfeld quantum double D(A _n (ω)) of A _n (ω) is a Ribbon Hopf algebra. In the previous articles, we constructed an n ⁴-dimensional Hopf algebra H _n (p, q) which is isomorphic to D(A _n (ω)) if p ≠ 0 and q = ω^?1, and studied the finite dimensional representations of H _n (1, q). We showed that the basic algebra of any nonsimple block of H _n (1, q) is independent of n. In this article, we examine the infinite representations of H ₂(1, ? 1), or equivalently of H _n (1, q)?D(A _n (ω)) for any n ≥ 2. We investigate the indecomposable and algebraically compact modules over H ₂(1, ? 1), describe the structures of these modules and classify them under the elementary equivalence.     

Augmentation quotients for complex representation rings of dihedral groups

Denote by D _m the dihedral group of order 2m. Let ℛ(D _m ) be its complex representation ring, and let Δ(D _m ) be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient Δ ⁿ (D _m )/Δ ⁿ⁺¹(D _m ) for each positive integer n.     

Division algebras with a projective basis

Letk be any field andG a finite group. Given a cohomology class α∈H ²(G,k ^*), whereG acts trivially onk ^*, one constructs the twisted group algebrak ^αG. Unlike the group algebrakG, the twisted group algebra may be a division algebra (e.g. symbol algebras, whereG⋞Z _n×Z_n). This paper has two main results: First we prove that ifD=k ^α G is a division algebra central overk (equivalentyD has a projectivek-basis) thenG is nilpotent andG’ the commutator subgroup ofG, is cyclic. Next we show that unless char(k)=0 and , the division algebraD=k ^α G is a product of cyclic algebras. Furthermore, ifD _p is ap-primary factor ofD, thenD _p is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k)=0 and , the same result holds forD _p, p odd. Ifp=2 we show thatD ₂ is a product of quaternion algebras with (possibly) a crossed product algebra (L/k,β), Gal(L/k)⋞Z ₂×Z₂n.     

Derivations nilpotent on subsets of prime rings

ABSTRACT

Let D be a division algebra with center F. Consider the group CK ₁(D) = D*/F*D′ where D* is the group of invertible elements of D and D′ is its commutator subgroup. In this note we shall show that, assuming a division algebra D is a product of cyclic algebras, the group CK ₁(D) is trivial if and only if D is an ordinary quaternion algebra over a real Pythagorean field F. We also characterize the cyclic central simple algebras with trivial CK ₁ and show that CK ₁ is not trivial for division algebras of index 4. Using valuation theory, the group CK ₁(D) is computed for some valued division algebras.

     

Some Results for Decomposition Numbers for the Hecke Algebra of Type A with q = −1

Let ? = ?_{?, ?1}(𝔖 _n ) be the Hecke algebra of the symmetric group 𝔖 _n . For partitions λ and ν with ν 2 ? regular, define the Specht module S(λ) and the irreducible module D(ν). Define d _λν = [S(λ): D(ν)] to be the composition multiplicity of D(ν) in S(λ). In this paper we compute the decomposition numbers d _λν for all partitions of the form λ = (a, c, 1 ^b ) and ν 2 ? regular.     

SK1 of Azumaya algebras over Hensel Pairs

Let A be an Azumaya algebra of constant rank n ² over a Hensel pair (R, I) where R is a semilocal ring with n invertible in R. Then the reduced Whitehead group SK₁(A) coincides with its reduction SK₁(A/I A). This generalizes a result of Hazrat (J Algebra 305:687–703, 2006) to non-local Henselian rings.     

Identifications in modular group algebras

Let G be a group, S a subgroup of G, and F a field of characteristic p. We denote the augmentation ideal of the group algebra FG by ω(G). The Zassenhaus-Jennings-Lazard series of G is defined by D_n(G)=G∩(1+ωⁿ(G)). We give a constructive proof of a theorem of Quillen stating that the graded algebra associated with FG is isomorphic as an algebra to the enveloping algebra of the restricted Lie algebra associated with the D_n(G). We then extend a theorem of Jennings that provides a basis for the quotient ωⁿ(G)/ωⁿ⁺¹(G) in terms of a basis of the restricted Lie algebra associated with the D_n(G). We shall use these theorems to prove the main results of this paper. For G a finite p-group and n a positive integer, we prove that G∩(1+ω(G)ωⁿ(S))=D_n+1(S) and G∩(1+ω²(G)ωⁿ(S))=D_n+2(S)D_n+1(S∩D₂(G)). The analogous results for integral group rings of free groups have been previously obtained by Gruenberg, Hurley, and Sehgal.     

Endomorphism algebras of transitive permutation modules for p-groups

Let G be a finite p-group with subgroup H and k a field of characteristic p. We study the endomorphism algebra E = End_kG(k_H ↑^G), showing that it is a split extension of a nilpotent ideal by the group algebra kN_G(H)/H. We identify the space of endomorphisms that factor through a projective kG-module and hence the endomorphism ring of k_H ↑^G in the stable module category, and determine the Loewy structure of E when G has nilpotency class 2 and [G, H] is cyclic. Received: 3 November 2008     

On Defining Ideals or Subrings of Hall Algebras

Let A be a finitary algebra over a finite field k, and A- \textmod\text{mod} the category of finite dimensional left A-modules. Let H(A)\mathcal{H}(A) be the corresponding Hall algebra, and for a positive integer r let D _r (A) be the subspace of H(A)\mathcal{H}(A) which has a basis consisting of isomorphism classes of modules in A- \textmod\text{mod} with at least r + 1 indecomposable direct summands. If A is the path algebra of the quiver of type A _n with linear orientation, then D _r (A) is known to be the kernel of the map from the twisted Hall algebra to the quantized Schur algebra indexed by n + 1 and r. For any A, we determine necessary and sufficient conditions for D _r (A) to be an ideal and some conditions for D _r (A) to be a subring of H(A)\mathcal{H}(A). For A the path algebra of a quiver, we also determine necessary and sufficient conditions for D _r (A) to be a subring of H(A)\mathcal{H}(A).     

Augmentation quotients for burnside rings of generalized dihedral groups

Let H be a finite abelian group of odd order, D be its generalized dihedral group, i.e., the semidirect product of C₂ acting on H by inverting elements, where C₂ is the cyclic group of order two. Let Ω (D) be the Burnside ring of D, Δ(D) be the augmentation ideal of Ω (D). Denote by Δⁿ(D) and Q_n(D) the nth power of Δ(D) and the nth consecutive quotient group Δⁿ(D)/Δⁿ⁺¹(D), respectively. This paper provides an explicit Z-basis for Δⁿ(D) and determines the isomorphism class of Q_n(D) for each positive integer n.     

Projective Summands in Tensor Products of Simple Modules of Finite Dimensional Hopf Algebras

Abstract

Let H be a finite dimensional Hopf algebra over a field k. We show that H contains a unique maximal Hopf ideal J _w (H) contained in J(H), the Jacobson radical of H. We give various characterizations of J _w (H), for example J _w (H) = Ann _H ((H/J(H))^?n ) for all large enough n. The smallest positive integer n with this property is denoted by l _w (H). We prove that l _w (H) equals the smallest number n such that (H/J(H))^?n contains every projective indecomposable H/J _w (H)-module as a direct summand. This also equals the minimal n such that the tensor product of n suitable simple H-modules contains the projective cover of the trivial H/J _w (H)-module as a direct summand. We define projective homomorphisms between H-modules, which are used to obtain various reciprocity laws for tensor products of simple H-modules and their projective indecomposable direct summands. We also discuss some consequences of our general results in case H = kG is a group algebra of a finite group G and k is a field of characteristic p.     

Quantum differential operators on the quantum plane

The universal enveloping algebra of a Lie algebra acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or quantum group) is a deformation of a universal enveloping algebra and acts not through the differential operators of its representation ring but through the quantised differential operators of its representation ring. We present this situation for the quantum group of sl₂.