Planar spider theorem and asymmetric Frobenius algebras

Abstract

The `spider theorem' for a general Frobenius algebra $A$, classifies all maps $A^{\otimes m}\to A^{\otimes n}$ that are built from the operations and, in a graphical representation, represented by a {\it connected} diagram. Here the algebra can be noncommutative and the Frobenius form can be asymmetric. We view this theorem as reducing any connected diagram to a standard form with $j$ beads $B$, where $j$ is the number of bounded connected components of the original diagram. We study the associated F-dimension Hilbert series ${\dim }_x={\sum }_{j=0}^{\infty }{x}^j{\dim }_j$, where ${\dim }_j=\epsilon \circ B^j\circ 1$ are invariants of the Frobenius structure. We also study moduli of asymmetric quasispecial and `weakly symmetric' Frobenius structures and their F-dimensions. Examples include general Frobenius structures on matrix algebras $A=M_d(k)$ and on group algebras $kG$ as well as on $u_q(s{l}_2)$ at low roots of unity.

20 pages ams latex, several pdf graphics; significantly extended to include u_q(sl_2) and higher F-dimensions

References in corpus (1)

• Quantum and braided ZX calculus