Über komplexe Banachverbandsalgebren

It is the aim of this paper to show that complex Banach lattice algebras have a richer spectral theory than complex Banach algebras without lattice structure. Indeed, it turns out that complex Banach lattice algebras having the property that the absolute values of their complex homomorphisms are again multiplicative are interesting with respect to the Gelfand theory. For example the set of complex homomorphisms is “cyclic.” Furthermore among other things it is shown that for central homomorphisms of AL-algebras, a special class of complex Banach lattice algebras, similar results are valid as J. L. Taylor has proved them for convolution measure algebras.     

Classification of graded Hecke algebras for complex reflection groups

The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V). This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction. The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups. Received: July 25, 2001     

On Fomin–Kirillov algebras for complex reflection groups

In this note, we apply classification results for finite-dimensional Nichols algebras to generalizations of Fomin–Kirillov algebras to complex reflection groups. First, we focus on the case of cyclic groups where the corresponding Nichols algebras are only finite-dimensional up to order four, and we include results about the existence of Weyl groupoids and finite-dimensional Nichols subalgebras for this class. Second, recent results by Heckenberger–Vendramin [ArXiv e-prints, 1412.0857 (December 2014)] on the classification of Nichols algebras of semisimple group type can be used to find that these algebras are infinite-dimensional for many non-exceptional complex reflection groups in the Shephard–Todd classification.     

On the rate of change of spectra of operators. II

A bound for the distance between the spectra of two operators lying in a special class, which includes the Lie algebras of the complex orthogonal group and the complex symplectic group, is obtained.     

The Calogero–Moser partition and Rouquier families for complex reflection groups

Let W be a complex reflection group. We formulate a conjecture relating blocks of the corresponding restricted rational Cherednik algebras and Rouquier families for cyclotomic Hecke algebras. We verify part of the conjecture in the case that W is a wreath product of a symmetric group with a cyclic group of order l.     

The algebraic and geometric classification of Zinbiel algebras

This paper is devoted to the complete algebraic and geometric classification of complex 5-dimensional Zinbiel algebras. In particular, we proved that the variety of complex 5-dimensional Zinbiel algebras has dimension 24, it is defined by 16 irreducible components and it has 11 rigid algebras.     

Construction process for simple Lie algebras

We give here a construction process for the complex simple Lie algebras and the non-Hermitian type real forms which intersect the minimal nilpotent complex adjoint orbit, using a finite dimensional irreducible representation of the conformal group, or of some two-fold covering of it, with highest weight vector a semi-invariant of degree four. This process leads to a five-graded simple complex Lie algebra and the underlying semi-invariant is intimately related to the structure of the minimal nilpotent orbit. We also describe a similar construction process for the simple real Lie algebras of Hermitian type.     

The algebraic and geometric classification of nilpotent terminal algebras

We give algebraic and geometric classifications of 4-dimensional complex nilpotent terminal algebras. Specifically, we find that, up to isomorphism, there are 41 one-parameter families of 4-dimensional nilpotent terminal (non-Leibniz) algebras, 18 two-parameter families of 4-dimensional nilpotent terminal (non-Leibniz) algebras, 2 three-parameter families of 4-dimensional nilpotent terminal (non-Leibniz) algebras, complemented by 21 additional isomorphism classes (see Theorem 13). The corresponding geometric variety has dimension 17 and decomposes into 3 irreducible components determined by the Zariski closures of a one-parameter family of algebras, a two-parameter family of algebras and a three-parameter family of algebras (see Theorem 15). In particular, there are no rigid 4-dimensional complex nilpotent terminal algebras.     

ON CLASS PAIRS AND RADICALS

ABSTRACT

We show that, for a class of schurian algebras, which we call schurian almost triangular, the fundamental group of the algebra is isomorphic to the fundamental group of an associated simplicial complex. Moreover, we obtain a simple presentation of this group in terms of generators and relations. Finally we use it to obtain easy short proofs of some known facts on the simple connectedness of incidence algebras.     

Hom-Lie-Rinehart algebras

We introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterization of low dimensional cohomology spaces in terms of the group of automorphisms of certain abelian extension and the equivalence classes of those abelian extensions in the category of hom-Lie-Rinehart algebras, respectively. We also construct a canonical example of hom-Lie-Rinehart algebra associated to a given Poisson algebra and an automorphism.     

Capelli identities for reductive dual pairs

Analogues of the Capelli identity are given for every irreducible reductive dual pair in the complex symplectic group and the complex orthogonal group. They describe a natural correspondence between the “centers” of the two universal enveloping algebras and an algebra of invariant operators.     

REPRESENTATION THEORY IN COMPLEX RANK,I

P. Deligne defined interpolations of the tensor category of representations of the symmetric group S _n to complex values of n. Namely, he defined tensor categories Rep(S _t ) for any complex t. This construction was generalized by F. Knop to the case of wreath products of S _n with a finite group. Generalizing these results, we propose a method of interpolating representation categories of various algebras containing S _n (such as degenerate affine Hecke algebras, symplectic reflection algebras, rational Cherednik algebras, etc.) to complex values of n. We also define the group algebra of S _n for complex n, study its properties, and propose a Schur-Weyl duality for Rep(S _t ).     

Finite quantum groups over abelian groups of prime exponent

We classify pointed finite-dimensional complex Hopf algebras whose group of group-like elements is abelian of prime exponent p, p>17. The Hopf algebras we find are members of a general family of pointed Hopf algebras we construct from Dynkin diagrams. As special cases of our construction we obtain all the Frobenius-Lusztig kernels of semisimple Lie algebras and their parabolic subalgebras. An important step in the classification result is to show that all these Hopf algebras are generated by group-like and skew-primitive elements.     

Braided doubles and rational Cherednik algebras

We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of quasi-Yetter-Drinfeld (QYD) modules over Hopf algebras which turn out to be a generalisation of the ordinary Yetter-Drinfeld modules. To each braiding (a solution to the braid equation) we associate a QYD-module and the corresponding braided Heisenberg double—this is a quantum deformation of the Weyl algebra where the role of polynomial algebras is played by Nichols-Woronowicz algebras. Our main result is that any rational Cherednik algebra canonically embeds in the braided Heisenberg double attached to the corresponding complex reflection group.     

Abelian Hermitian geometry

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a Euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.     

Classification of complex simply connected homogeneous spaces of dimension not greater than 2

The paper is devoted to the classification of finite-dimensional complex Lie algebras of analytic vector fields on the complex plane and the corresponding actions of Lie groups on complex two-dimensional manifolds. These Lie algebras were specified by Sophus Lie. He specified vector fields which form bases of the Lie algebras. However the structure of the Lie algebras was not clarified, and isomorphic Lie algebras among listed were not established. Thus, the classification was far from complete, and the situation has not been essentially changed until now. This paper is devoted to the completion of the above mentioned classification. We consider the part of this classification which concerns transitive actions of Lie groups.     

On the denseness of the invertible group in Banach algebras

We examine the condition that a complex Banach algebra has dense invertible group. We show that, for commutative algebras, this property is preserved by integral extensions. We also investigate the connections with an old problem in the theory of uniform algebras.

     

Modules for Yokonuma-type Hecke Algebras

This paper describes the module categories for a family of generic Hecke algebras, called Yokonuma-type Hecke algebras. Yokonuma-type Hecke algebras specialize both to the group algebras of the complex reflection groups G(r,1,n) and to the convolution algebras of (B \(^{\prime }\),B \(^{\prime }\))-double cosets in the group algebras of finite general linear groups, for certain subgroups B \(^{\prime }\) consisting of upper triangular matrices. In particular, complete sets of inequivalent, irreducible modules for semisimple specializations of Yokonuma-type Hecke algebras are constructed.     

Cyclic Homology of Hopf Algebras

A cyclic cohomology theory adapted to Hopf algebras has been introduced recently by Connes and Moscovici. In this paper, we consider this object in the homological framework, in the spirit of Loday and Quillen and Karoubi's work on the cyclic homology of associative algebras. In the case of group algebras, we interpret the decomposition of the classical cyclic homology of a group algebra in terms of this homology. We also compute both cyclic homologies for truncated quiver algebras.