Skein construction of idempotents in Birman-Murakami-Wenzl algebras

We give skein theoretic formulas for minimal idempotents in the Birman-Murakami-Wenzl algebras. These formulas are then applied to derive various known results needed in the construction of quantum invariants and modular categories. In particular, an elementary proof of the Wenzl formula for quantum dimensions is given. This proof does not use the representation theory of quantum groups and the character formulas. Received: 26 September 2000 / Published online: 17 August 2001     

Representation type of endomorphism algebras of exceptional sequences of typeA n

LetF be an algebraically closed field, be a quiver of typeA _n . In this paper we prove that the endomorphism algebras of exceptional sequences over are sums of finitely many tilted algebras of typeA _m wherem≤n by using perpendicular categories, and thus the endomorphism algebras of exceptional sequences of typeA _n are representation-finite. Supported by Chinese Postdoctorate Fund and Beijing Youth Fund     

The J-invariant and Tits algebras for groups of inner type E 6

A connection between the indices of the Tits algebras of a split linear algebraic group G and the degree one parameters of its motivic J-invariant was introduced by Quéguiner-Mathieu, Semenov and Zainoulline through use of the second Chern class map in the Riemann-Roch theorem without denominators. In this paper we extend their result to higher Chern class maps and provide applications to groups of inner type E ₆.     

Classification of bernstein algebras of type (3, n - 3)

A classification of Bernstein algebras in dimensions n ? 4 has been made by Holgate in [2], however that article contains no classification up to isomorphism, the problem is solved by Lyubich in [4] when K = R or C, and by Cortes [1] in the general case. Also Lyubich has given in [5] a classification of the regular nonexceptional Bernstein algebra of type (3,n?3) and a classification but not up to isomorphism of nonregular nonexceptional Bernstein algebras of type (3,n ? 3) when K = C. The aim of this paper it to characterize, up to isomorphism, Bernstein algebras of type(2, n ? 2) and nonexceptional of type(3, n ?3) over a infinite commutative field K whose characteristic is different from 2.     

I n -symmetrical Heyting algebras

In [11, p. 210] A. Monteiro suggested the possibility of generalizing the results of L. Iturrioz [4] to theI _n -symmetrical Heyting algebras which he namedI _n -algebras. We prove that these algebras are semi-simple. We characterize simple algebras and their subalgebras. Finally, we determine the structure of theI _n -algebra with a finite set of free generators and we give an answer to one of the problems posed by A. Monteiro.Presented by W. Taylor.Some of the results of this paper were presented at the Annual Meeting of the Union Matemática Argentina (September, 1987) ([14]).     

The melikyan algebras as lie algebras of the type G2

In this paper, we find an exact sequence involving the automorphism group of a self-dual type-A complex lattice (including the Niemeier lattice A¹² ₂ and the complex Leech lattice) and the automorphism of a certain group central extension of the lattice. Such an exact sequence is important for the study of finite groups which have a moonshine representation analogous to that of the Monster     

Lie algebras and algebras of associative type

In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type A = ⊕ _αεG A _α graded by some group G, over a field of characteristic zero, has a nonzero component A ₁ (where 1 stands for the identity element of G), and A ₁ is a semisimple associative algebra. Let B = ⊕ _αεG B _α be a finite-dimensional semisimple Lie algebra over a prime field F _p , and let B be graded by a commutative group G. If B = F _p ? _? A _L , where A _L is the commutator algebra of a ?-algebra A = ⊕ _αεG A _α ; if ? ? _? A is an algebra of associative type, then the 1-component of the algebra K ? _? B, where K stands for the algebraic closure of the field F _p , is the sum of some algebras of the form gl(n _i ,K).     

The representation type of Ariki-Koike algebras and cyclotomic q-Schur algebras

We give a necessary and sufficient condition on parameters for Ariki-Koike algebras (resp. cyclotomic q-Schur algebras) to be of finite representation type.     

Construction of representation-finite selfinjective artin algebras of classB n andC n

In this paper, we discuss the representation-finite selfinjective artin algebras of classB _n andC _n and obtain the following main results: For any fieldk, let Λ be a representation-finite selfinjective artin algebras of classB _n orC _n overk.

Calderón algebras of smoothing operators

Summary Certain classes of integral operators between generalized Sobolev spaces are shown to form algebras, en lowed with an approximate functional calculus, having properties similar to those of pseudo-differential operators. This research was supported under National Science Foundation grant GP-12295 and was completed while the author was Visiting Professor at the University of Genoa under a grant from the Comitato Nazionale per la Matematica, Consiglio Nazionale Ricerche (C.N.R.), Italy. Entrata in Redazione il 1 giugno 1970.     

The representation type of Hecke algebras of type B

This paper determines the representation type of the Iwahori-Hecke algebras of type B when q≠±1. In particular, we show that a single parameter non-semisimple Iwahori-Hecke algebra of type B has finite representation type if and only if q is a simple root of the Poincaré polynomial, confirming a conjecture of Uno's (J. Algebra 149 (1992) 287).     

On minimal representations of Chevalley groups of type D n , E n and G 2

Let G be a simply connected Chevalley group of type D _n , E _n or G₂. In this paper, we show that the minimal representation of G is unique for types D _n and E _n and it does not exist for the type G₂.     

Cellular algebras arising from Hecke algebras of type H_n

We study a finite-dimensional quotient of the Hecke algebra of type for general n, using a calculus of diagrams. This provides a basis of monomials in a certain set of generators. Using this, we prove a conjecture of C.K. Fan about the semisimplicity of the quotient algebra. We also discuss the cellular structure of the algebra, with certain restrictions on the ground ring. Received February 24, 1997; in final form May 9, 1997     

Representation type of Jordan algebras

The problem of classification of Jordan bimodules over (non-semisimple) finite dimensional Jordan algebras with respect to their representation type is considered. The notions of diagram of a Jordan algebra and of Jordan tensor algebra of a bimodule are introduced and a mapping Qui is constructed which associates to the diagram of a Jordan algebra J the quiver of its universal associative enveloping algebra S(J). The main results are concerned with Jordan algebras of semi-matrix type, that is, algebras whose semi-simple component is a direct sum of Jordan matrix algebras. In this case, criterion of finiteness and tameness for one-sided representations are obtained, in terms of diagram and mapping Qui, for Jordan tensor algebras and for algebras with radical square equals to 0.     

Representation type of Schur algebras

We give a complete classification of the classical Schur algebras and the infinitesimal Schur algebras which have tame representation type. In combination with earlier work of some of the authors on semisimplicity and finiteness, this completes the classification of representation type of all classical and infinitesimal Schur algebras in all characteristics. Received October 17, 1997; in final form March 5, 1998     

Simple algebras of Weyl type

Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceA[D] =A?F[D] from any pair of a commutative associative algebra,A with an identity element and the polynomial algebraF[D] of a commutative derivation subalgebraD ofA We prove thatA[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only ifA isD-simple andA[D] acts faithfully onA. Thus we obtain a lot of simple algebras.