The Birman–Murakami–Wenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free over Z[δ±1, l±1]/(m(1− δ)− (l− l−1)) of rank (2n + 1)n!! − (2n−1 + 1)n!, where n!! = 1 · 3 · · · (2n − 1). We also show it is a cellular algebra over suitable rings. The Brauer algebra of type Dn is a homomorphic ring image and is also semisimple and free of the same rank, but over the ring Z[δ±1]. A r...

The invariants of 3-manifolds defined by Kuperberg for involutory Hopf algebras and those defined by the authors for spherical Hopf algebras are the same for Hopf algebras on which they are both defined. Introduction The purpose of this paper is to compare two previously defined invariants of 3-manifolds. Let A be a finite-dimensional Hopf algebra over a field F with antipode S. Then if S = 1 t...

Let H be a semisimple Hopf algebra over an algebraically closed field k. It is assumed that either char k = 0 or char k > dimH . Semisimplicity of H means that H is a semisimple left H-module. In that case H has finite dimension. Throughout the paper we shall keep to notations from [M]. For example H stands for the dual Hopf algebra with the natural pairing 〈−,−〉 : H ⊗ H → k. The algebra H is a...

1. A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Bull. Amer. Math. Soc. 67 (1961), 579-583. 2. , Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485-535. 3. Harish-Chandra, On the characters of a semisimple Lie group, Bull. Amer. Math. Soc. 61 (1955), 389-396. 4. , Differential operators on a semisimple Lie algebra, Amer. J. Math. 79 (1957), 87-12...

We prove an explicit formula for the total Chern character of the Verlinde bundle over Mg,n in terms of tautological classes. The Chern characters of the Verlinde bundles define a semisimple CohFT (the ranks, given by the Verlinde formula, determine a semisimple fusion algebra). According to Teleman’s classification of semisimple CohFTs, there exists an element of Givental’s group transforming ...

The Lie algebra of an algebraic group is the (first) linear approximation to the group. The study of Lie algebras is much more elementary than that of algebraic groups. For example, most of the results on Lie algebras that we shall need are proved already in the undergraduate text Erdmann and Wildon 2006. After many preliminaries, in 7 we describe the structure and classification of the semisi...

We extend the basic fact that every ideal of a finite dimensional semisimple Lie algebra has a unique complement to the case of closed ideals of prosemisimple Lie algebras. We prove that if A is a closed ideal of a prosemisimple Lie algebra L = lim ←−−Ln (n ∈ N), where the Ln are finite dimensional semisimple Lie algebras, then there exists a unique ideal B of L such that L = A ⊕ B. Mathematics...

In this paper, we outline the rudiments of the representation theory of semisimple Lie algebras. We build the necessary theory in order to analyze the representations of sl2, which includes proving that representations of semisimple Lie algebras are completely reducible and preserve the Jordan decomposition. We only assume the reader has a working knowledge of linear algebra and a little famili...

Let $X$ be a $BCH$-algebra and $I$ be an ideal of $X$. In this paper, we introduce the concept of $sqrt{I}$. We show that it is an ideal of $X$, when $I$ is closed ideal of $X$. Then we verify some useful properties of it. We prove that it is the ::::union:::: of all $k-$nil ideals of $I$. Moreover, if $I$ is a closed ideal of $X$, then $sqrt{I}$ is a closed translation ideal and so we can cons...