For two algebras $A$ and $B$, a linear map $T:A longrightarrow B$ is called separating, if $xcdot y=0$ implies $Txcdot Ty=0$ for all $x,yin A$. The general form and the automatic continuity of separating maps between various Banach algebras have been studied extensively. In this paper, we first extend the notion of separating map for module case and then we give a description of a linear se...

Let H be a finite-dimensional Hopf algebra over an algebraically closed field of characteristic 0. If H is not semisimple and dim(H) = 2n for some odd integers n, then H or H * is not unimodular. Using this result, we prove that if dim(H) = 2p for some odd primes p, then H is semisimple. This completes the classification of Hopf algebras of dimension 2p. In recent years, there has been some pro...

In this lecture I will explain the classification of finite dimensional semisimple Lie algebras over C. Semisimple Lie algebras are defined similarly to semisimple finite dimensional associative algebras but are far more interesting and rich. The classification reduces to that of simple Lie algebras (i.e., Lie algebras with non-zero bracket and no proper ideals). The classification (initially d...

We study the complexity of multiplication in noncommutative group algebras which is closely related to the complexity of matrix multiplication. We characterize such semisimple group algebras of the minimal bilinear complexity and show nontrivial lower bounds for the rest of the group algebras. These lower bounds are built on the top of Bläser’s results for semisimple algebras and algebras with ...

Let H be a finite-dimensional Hopf algebra over an algebraically closed field of characteristic 0. If H is not semisimple and dim(H) = 2n for some odd integer n, then H or H * is not unimodular. Using this result, we prove that if dim(H) = 2p for some odd prime p, then H is semisimple. This completes the classification of Hopf algebras of dimension 2p. In recent years, there has been some progr...

We study forms of coalgebras and Hopf algebras (i.e. coalgebras and Hopf algebras which are isomorphic after a suitable extension of the base field). We classify all forms of grouplike coalgebras according to the structure of their simple subcoalgebras. For Hopf algebras, given a W ∗-Galois field extension K ⊆ L for W a finite-dimensional semisimple Hopf algebra and a K-Hopf algebra H, we show ...

One way to obtain Quantized Universal Enveloping Algebras (QUEAs) of non-semisimple Lie algebras is by contracting QUEAs of semisimple Lie algebras. We prove that every contracted QUEA in a certain class is a cochain twist of the corresponding undeformed universal envelope. Consequently, these contracted QUEAs possess a triangular quasi-Hopf algebra structure. As examples, we consider κ-Poincar...

We define the class of rigid Frobenius algebras in a (non-semisimple) modular category and prove that their categories local modules are, again, modular. This generalizes previous work Kirillov Ostrik (Adv Math 171(2):183–227, 2002) semisimple setup. Examples non-semisimple via modules, as well connections to authors’ prior on relative monoidal centers, are provided. In particular, we classify ...