Non-associative Frobenius algebras for simply laced Chevalley groups

Abstract“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius (QcF) coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or right) rational dual Rat(C*) in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, QcF, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra A that is isomorphic to its complete topological dual Aν. We show that A is a (quasi)Frobenius algebra if and only if A is the dual C* of a (quasi)co-Frobenius coalgebra C. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer q-homological algebra, topology or graph theory, showing the importance of the concept.

Download Full-text

Elements of Order Coxeter Number +1 in Chevalley Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1982-067-4 ◽

1982 ◽

Vol 34 (4) ◽

pp. 945-951 ◽

Cited By ~ 1

Author(s):

Bomshik Chang

Keyword(s):

Weyl Group ◽

Chevalley Group ◽

Trivial Solution ◽

Affirmative Answer ◽

Chevalley Groups ◽

General Solutions ◽

Coxeter Number ◽

Image Position

Following the notation and the definitions in [1], let L(K) be the Chevalley group of type L over a field K, W the Weyl group of L and h the Coxeter number, i.e., the order of Coxeter elements of W. In a letter to the author, John McKay asked the following question: If h + 1 is a prime, is there an element of order h + 1 in L(C)? In this note we give an affirmative answer to this question by constructing an element of order h + 1 (prime or otherwise) in the subgroup Lz = 〈xτ(1)|r ∈ Φ〉 of L(K), for any K.Our problem has an immediate solution when L = An. In this case h = n + 1 and the (n + l) × (n + l) matrixhas order 2(h + 1) in SLn+1(K). This seemingly trivial solution turns out to be a prototype of general solutions in the following sense.

Download Full-text

Frobenius algebra with relaxed associativity constraint

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518410080 ◽

2018 ◽

Vol 27 (07) ◽

pp. 1841008

Author(s):

Zbigniew Oziewicz ◽

William Stewart Page

Keyword(s):

Associative Algebra ◽

Monoidal Category ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Frobenius Algebra ◽

Frobenius Algebras ◽

Frobenius Structures ◽

Necessary And Sufficient

Frobenius algebra is formulated within the Abelian monoidal category of operad of graphs. A not necessarily associative algebra [Formula: see text] is said to be a Frobenius algebra if there exists a [Formula: see text]-module isomorphism. A new concept of a solvable Frobenius algebra is introduced: an algebra [Formula: see text] is said to be a solvable Frobenius algebra if it possesses a nonzero one-sided [Formula: see text]-module morphism with nontrivial radical. In the category of operad of graphs, we can express the necessary and sufficient conditions for an algebra to be a solvable Frobenius algebra. The notion of a solvable Frobenius algebra makes it possible to find all commutative nonassociative Frobenius algebras (Conjecture 10.1), and to find all Frobenius structures for commutative associative Frobenius algebras. Frobenius algebra allows [Formula: see text]-permuted opposite algebra to be extended to [Formula: see text]-permuted algebras.

Download Full-text