Bilinear forms on Frobenius algebras

Abstract

We analyze the homothety types of associative bilinear forms that can occur on a Hopf algebra or on a local Frobenius k-algebra R with residue field k. If R is symmetric, then there exists a unique form on R up to homothety iff R is commutative. If R is Frobenius, then we introduce a norm based on the Nakayama automorphism of R. We show that if two forms on R are homothetic, then the norm of the unit separating them is central, and we conjecture the converse. We show that if the dimension of R is even, then the determinant of a form on R, taken ink/k 2 , is an invariant for R.  2005 Elsevier Inc. All rights reserved.

Metabelian Associative Algebras

“…Two bilinear forms θ and θ ∈ Bil (V ×V, k) are called homothetic [15], and we denote this by θ ≡ θ , if there exists a pair (u, ψ) ∈ k * × Aut k (P) such that u θ( p, q) = θ ψ( p), ψ(q) , for all p, q ∈ P. Any isometric bilinear forms are homothetic and if k = k 2 := {x 2 | x ∈ k}, then any homothetic bilinear forms are isometric. By an algebra we always mean an associative algebra, i.e.…”

“…Then θ ≈ θ ′ if and only if there exists an invertible matrix C ∈ gl(n, k), such that θ = C T θ ′ C -where C T is the transposed of C. For future references to the classification problem of bilinear forms up to an isometry see [11,16] and the references therein. Two bilinear forms θ and θ ′ ∈ Bil (V × V, k) are called homothetic [15], and we denote this by θ ≡ θ ′ , if there exists a pair (u, ψ) ∈ k * × Aut k (P ) such that u θ(p, q) = θ ′ ψ(p), ψ(q) , for all p, q ∈ P . Any isometric bilinear forms are homothetic and if k = k 2 := {x 2 | x ∈ k} then any homothetic bilinear forms are isometric.…”

“…In Theorem 3.8, all metabelian algebras having the derived algebra of dimension 1 are explicitly described, classified and the automorphism groups of these algebras are determined. The algebras of this family are classified by bilinear forms on a vector space P up to the following equivalence relation: two bilinear forms θ and θ ∈ Bil (P × P, k) are called homothetic [15] if there exists a pair (u, ψ) ∈ k * × Aut k (P) such that u θ( p, q) = θ ψ( p), ψ(q) , for all p, q ∈ P. If k = k 2 , this relation is equivalent to the classical classification of bilinear forms [17]. Examples are given in Corollary 3.10.…”

Some graphical aspects of Frobenius structures

“…We want to emphasize especially the difference between the multiplications in endormorphisms rings End R (M) over an R -module M and the left (right) action induced by an algebra multiplications m A : A ⊗ A → A , where we look at the right (left) factor A as a left A -module A A (right A -module A A ). Our main sources are (Yamagata 1996, Kadison 1999, Caenepeel et al 2002, Murray 2005, Lorenz 2011, Lorenz & Fitzgerald Tokoly 2010. General texts on Hopf algebras and modules are (Sweedler 1969, Abe 1980, Kasch 1982, Caenepeel 1998, Street 2007).…”

“…This is essentially using the parastrophic matrix from (1.1). Definition 3.11: (Murray 2005) Let k be a residual field. A regular associative bilinear form is a k -linear map β ∈ Bil r ass (A, k) :…”