On Isotypic Decompositions for Non-Semisimple Hopf Algebras

Abstract

In this paper we study the isotypic decomposition of the regular module of a finite-dimensional Hopf algebra over an algebraically closed field of characteristic zero. For a semisimple Hopf algebra, the idempotents realizing the isotypic decomposition can be explicitly expressed in terms of characters and the Haar integral. In this paper we investigate Hopf algebras with the Chevalley property, which are not necessarily semisimple. We find explicit expressions for idempotents in terms of Hopf-algebraic data, where the Haar integral is replaced by the regular character of the dual Hopf algebra. For a large class of Hopf algebras, these are shown to form a complete set of orthogonal idempotents. We give an example which illustrates that the Chevalley property is crucial.

Appendix A: Calculations with Magma

We describe here a Magma code for calculating explicitly the products of the different conjectured idempotents pi for the Hopf algebra discussed in Section 4.2. We begin with the remark that, when considering the grading we have for H^∗,

$$H^{*}=\bigoplus\limits_{k=0}^{4} H^{*}_{k},$$

the only direct summand on which the product pipj (where i ≠ j) might not vanish is $H^{*}_{4}$. This follows from the fact that all pi’s vanish on H_k for k ≠ 0, 4, and on the coalgebra grading. The calculation with Magma will be done in the following way: for different values of λ_a, λ_b, λ_c we will define the algebra A = H^∗ in Magma. Then we will present it in a matrix form, and calculate the trace of the regular representation, as well as the translations of this trace by multiples of irreducible characters. For the calculations of the product we will calculate (pi ⊗ 1)Δ(abace_g) where g ∈ G and pi ∈{p, p_V} by hand, and apply the relevant functionals pj ∈{p, p_sgn, p_V} to them. Finally, since all the relevant values are polynomials of degree at most 3 in λ_abλ_ac it will be enough to show that they vanish on four different values of λ_abλ_ac.

The code is enclosed here. We ran it on http://magma.maths.usyd.edu.au/calc/, the online version of Magma.

/* Values of lambdas */ lam_a:=0; lam_b:=23; lam_c:=11; K:=RationalField(); A<e1,e2,e3,e4,e5,e6,a,b,c>:= FPAlgebra<K, e1,e2,e3,e4,e5,e6,a,b, c| e1*e1-e1, e2*e1, e3*e1, e4*e1, e5*e1, e6*e1, e1*e2, e2*e2-e2, e3*e2, e4*e2, e5*e2, e6*e2, e1*e3, e2*e3, e3*e3-e3, e4*e3, e5*e3, e6*e3, e1*e4, e2*e4, e3*e4, e4*e4-e4, e5*e4, e6*e4, e1*e5, e2*e5, e3*e5, e4*e5, e5*e5-e5, e6*e5, e1*e6, e2*e6, e3*e6, e4*e6, e5*e6, e6*e6-e6,e1+e2+e3+e4+e5+e6-1, a*e1-e2*a,a*e2-e1*a, a*e3-e5*a,e5*a-a*e3,a*e4-e6*a,a*e6-e4*a, b*e1-e3*b,b*e3-e1*b, b*e4-e5*b,e5*b-b*e4,b*e2-e6*b,b*e6-e2*b, c*e1-e4*c,c*e4-e1*c, c*e2-e5*c,e5*c-c*e2,c*e3-e6*c,c*e6-e3*c, a*b+b*c+c*a, a*c+c*b+b*a, a^2 - (lam_a-lam_b)*(e4+e6) - (lam_a-lam_c)*(e3+e5), b^2 - (lam_b-lam_c)*(e2+e6) - (lam_b-lam_a)*(e4+e5), c^2 - (lam_c-lam_a)*(e3+e6) - (lam_c-lam_b)*(e2+e5)>; /* Defining A=H^* by generators and relations */ D:=Dimension(A); S,f:= Algebra(A); /* S is now the algebra A considered as a subalgebra of the 72 x 72 matrix algebra. f:A\to S is the natural isomorphism */ Y:=AssociativeArray(); B,h:=ChangeBasis(S,[f(e1),f(e2),f(e3),f(e4),f(e5),f(e6), f(a*e1),f(a*e2),f(a*e3),f(a*e4),f(a*e5),f(a*e6), f(b*e1),f(b*e2),f(b*e3),f(b*e4),f(b*e5),f(b*e6), f(c*e1),f(c*e2),f(c*e3),f(c*e4),f(c*e5),f(c*e6), f(a*b*e1),f(a*b*e2),f(a*b*e3),f(a*b*e4),f(a*b*e5),f(a*b*e6), f(b*c*e1),f(b*c*e2),f(b*c*e3),f(b*c*e4),f(b*c*e5),f(b*c*e6), f(a*c*e1),f(a*c*e2),f(a*c*e3),f(a*c*e4),f(a*c*e5),f(a*c*e6), f(c*b*e1),f(c*b*e2),f(c*b*e3),f(c*b*e4),f(c*b*e5),f(c*b*e6), f(a*b*a*e1),f(a*b*a*e2),f(a*b*a*e3),f(a*b*a*e4),f(a*b*a*e5), f(a*b*a*e6), f(a*b*c*e1),f(a*b*c*e2),f(a*b*c*e3),f(a*b*c*e4), f(a*b*c*e5),f(a*b*c*e6), f(b*a*c*e1),f(b*a*c*e2),f(b*a*c*e3), f(b*a*c*e4),f(b*a*c*e5),f(b*a*c*e6), f(a*b*a*c*e1), f(a*b*a*c*e2), f(a*b*a*c*e3), f(a*b*a*c*e4),f(a*b*a*c*e5),f(a*b*a*c*e6)]); /* we now fix for S the basis described in the paper. This is given by the algebra B. The map h:S\to B is then the isomorphism */ for i:=1 to D do Y[i]:=0; for j:=1 to D do Y[i]:= Y[i] + BasisProduct(B,i,j)[j]/72; end for; end for; /* We calculate p as the trace of the regular representation divided by the dimension. Notice that we think of p as an element in H=A^*. */ "print p"; for i:=1 to D do Y[i]; end for; "end p"; ""; chi:= e1-e2-e3-e4+e5+e6; chiV:= 2*e1 - e5-e6; /* the characters of the two non-trivial representations of A^*. Both are elements of A */ Z:=AssociativeArray(); for i:=1 to D do Z[i]:=0; for j:=1 to D do Z[i]:= Z[i] + (h(f(chi))*BasisProduct (B,i,j))[j]/72; end for; end for; /* The array Z contains now the translation of p by the sign representation. In other words, it is p_{sign}, considered as an element of A^*. */ W:=AssociativeArray(); for i:=1 to D do W[i]:=0; for j:=1 to D do W[i]:= W[i] + 2*(h(f(chiV))* BasisProduct(B,i,j))[j]/72; end for; end for; /* Similarly, we calculate p_V for the 2-dimensional irreducible representation of A. */ E2:=AssociativeArray(); for i:=1 to D do E2[i]:=Y[i]+Z[i] + 2*W[i]; end for; "print epsilon"; for i:=1 to D do E2[i]; end for; /* We calculate and print the sum p + p_{sign} + p_V. If it is the counit, then we are on the right path. */ "print p_sign"; for i:=1 to D do Z[i]; end for; "end p_sign"; ""; "print p_V"; for i:=1 to D do W[i]; end for; "end p_V"; ""; /* Next, we calculated manually the elements v_i:=(p \otimes 1) {\Delta}(a*b*a*c*ei). */ v1:= h(f(1/6*(lam_a-lam_b)*(lam_a-lam_c)* (e3 + e5) + 1/6*((lam_c-lam_a)*b*b*e6 - (lam_a-lam_b)*c*c*e4- (lam_a-lam_b)*a*a*e5 - (lam_a-lam_c)*a*a*e3)+ 1/6*(a*b*a*c*e1 + a*c*a*b*e2 + c*b*c*a*e3 + b*a*b*c*e4 + c*a*c* b*e6 + b*c*b*a*e5))); v2:= h(f(1/6*(lam_a-lam_b)*(lam_a-lam_c)* (e6 + e4) + 1/6*((lam_c-lam_a)*b*b*e3 - (lam_a-lam_b)*c*c*e5- (lam_a-lam_b)*a*a*e4 - (lam_a-lam_c)*a*a*e6)+ 1/6*(a*b*a*c*e2 + a*c*a*b*e1 + c*b*c*a*e6 + b*a*b*c*e5 + c*a*c*b*e3 + b*c*b*a*e4))); v3:= h(f(1/6*(lam_a-lam_b)* (lam_a-lam_c)*(e1 + e2) + 1/6*((lam_c-lam_a)*b*b*e4 - (lam_a-lam_b)*c*c*e6- (lam_a-lam_b)*a*a*e2 - (lam_a-lam_c)*a*a*e1)+ 1/6*(a*b*a*c*e3 + a*c*a*b*e5 + c*b*c*a*e1 + b*a*b*c*e6 + c*a*c*b*e4 + b*c*b*a*e2))); v4:= h(f(1/6*(lam_a-lam_b)* (lam_a-lam_c)*(e5 + e3) + 1/6*((lam_c-lam_a)*b*b*e2 - (lam_a-lam_b)*c*c*e1- (lam_a-lam_b)*a*a*e3 - (lam_a-lam_c)*a*a*e5)+ 1/6*(a*b*a*c*e4 + a*c*a*b*e6 + c*b*c*a*e5 + b*a*b*c*e1 + c*a*c*b*e2 + b*c*b*a*e3))); v5:= h(f(1/6*(lam_a-lam_b)* (lam_a-lam_c)*(e4 + e6) + 1/6*((lam_c-lam_a)*b*b*e1 - (lam_a-lam_b)*c*c*e2- (lam_a-lam_b)*a*a*e6 - (lam_a-lam_c)*a*a*e4)+ 1/6*(a*b*a*c*e5 + a*c*a*b*e3 + c*b*c*a*e4 + b*a*b*c*e2 + c*a*c*b*e1 + b*c*b*a*e6))); v6:=h(f(1/6*(lam_a-lam_b)* (lam_a-lam_c)*(e2 + e1) + 1/6*((lam_c-lam_a)*b*b*e5 - (lam_a-lam_b)*c*c*e3- (lam_a-lam_b)*a*a*e1 - (lam_a-lam_c)*a*a*e2)+ 1/6*(a*b*a*c*e6 + a*c*a*b*e4 + c*b*c*a*e2 + b*a*b*c*e3 + c*a*c*b*e5 + b*c*b*a*e1))); E:= AssociativeArray(); for i:=1 to 6 do E[i]:=0; end for; for i:= 1 to D do E[1]:= E[1] + Y[i]* v1[i]; E[2]:= E[2] + Y[i]*v2[i]; E[3]:= E[3] + Y[i]*v3[i]; E[4] := E[4] + Y[i]*v4[i]; E[5]:= E[5] + Y[i]*v5[i]; E[6]:= E[6] + Y[i]*v6[i]; end for; /* This calculates p (p \otimes 1){\Delta}(a*b*a*c*ei) = p p (a*b*a*c*ei). Since a*b*a*c*ei are the only elements on which p^2 might be non-zero, it is enough to consider them. After that we do a similar calculation for p*p_V and p*p_{sign}.*/ "results of p*p - p"; for i:=1 to 6 do E[i]-Y[D-6+i]; end for; for i:=1 to 6 do E[i]:=0; end for; for i:= 1 to D do E[1]:= E[1] + Z[i]*v1[i]; E[2]:= E[2] + Z[i]*v2[i]; E[3]:= E[3] + Z[i] *v3[i]; E[4]:= E[4] + Z[i]*v4[i]; E[5]:= E[5] + Z[i]*v5[i]; E[6]:= E[6] + Z[i]*v6[i]; end for; "Results of p*p_{sign}"; for i:=1 to 6 do E[i]; end for; for i:=1 to 6 do E[i]:=0; end for; for i:= 1 to D do E[1]:= E[1] + W[i] *v1 [i]; E[2]:= E[2] + W[i]*v2[i]; E[3]:= E[3] + W[i]*v3[i]; E[4]:= E[4] + W[i]*v4[i]; E[5]:= E[5] + W[i]*v5[i]; E[6]:= E[6] + W[i]* v6[i]; end for; "results of p*p_V"; for i:=1 to 6 do E[i]; end for; /* Similarly to the elements vi from the previous part, we define wi= (p_V \otimes 1){\Delta}(a*b*a*c*ei) and similarly calculate the products.*/ w1:= 2*h(f(1/6*((lam_a-lam_c)*b*b*e6 + (lam_a-lam_b)* a*a*e5) - 1/6*((lam_a-lam_b)*(lam_a-lam_c)*e5) + 1/3*a*b*a*c*e1 - 1/6*(c*a*c*b*e6 + b*c*b*a*e5))); w2:= 2*h(f(1/6*((lam_a-lam_c)*b*b*e3 + (lam_a-lam_b)*a*a*e4) - 1/6*((lam_a-lam_b)*(lam_a-lam_c)*e4) + 1/3*a*b*a*c*e2 - 1/6*(c*a*c*b*e3 + b*c*b*a*e4))); w3:= 2*h(f(1/6*((lam_a-lam_c)*b*b*e4 + (lam_a-lam_b)*a*a*e2) - 1/6*((lam_a-lam_b)*(lam_a-lam_c)*e2) + 1/3*a*b*a*c*e3 - 1/6*(c*a*c*b*e4 + b*c*b*a*e2))); w4:= 2*h(f(1/6*((lam_a-lam_c)*b*b*e2 + (lam_a-lam_b)*a*a*e3) - 1/6*((lam_a-lam_b)*(lam_a-lam_c)*e3) + 1/3*a*b*a*c*e4 - 1/6*(c*a*c*b*e2 + b*c*b*a*e3))); w5:= 2*h(f(1/6*((lam_a-lam_c)*b*b*e1 + (lam_a-lam_b)*a*a*e6) - 1/6*((lam_a-lam_b)*(lam_a-lam_c)*e6) + 1/3*a*b*a*c*e5 - 1/6*(c*a*c*b*e1 + b*c*b*a*e6))); w6:= 2*h(f(1/6*((lam_a-lam_c)*b*b*e5 + (lam_a-lam_b)*a*a*e1) - 1/6*((lam_a-lam_b)*(lam_a-lam_c)*e1) + 1/3*a*b*a*c*e6 - 1/6*(c*a*c*b*e5 + b*c*b*a*e1))); /* "print ws"; for i:= 1 to D do w1[i], w2[i], w3[i], w4[i], w5[i], w6[i]; end for; */ for i:=1 to 6 do E[i]:=0; end for; for i:= 1 to D do E[1]:= E[1] + Y[i]*w1[i]; E[2]:= E[2] + Y[i]*w2[i]; E[3]:= E[3] + Y[i]* w3[i]; E[4]:= E[4] + Y[i]*w4[i]; E[5]:= E[5] + Y[i]*w5[i]; E[6] := E[6] + Y[i]*w6[i]; end for; " "; "results of p_V*p"; for i:=1 to 6 do E[i]; end for; for i:=1 to 6 do E[i]:=0; end for; for i:= 1 to D do E[1]:= E[1] + Z[i]*w1 [i]; E[2]:= E[2] + Z[i]*w2[i]; E[3]:= E[3] + Z[i]*w3[i]; E[4]:= E[4] + Z[i]*w4[i]; E[5]:= E[5] + Z[i]*w5[i]; E[6]:= E[6] + Z[i]*w6[i]; end for; "Results of p_V*p_{sign}"; for i:=1 to 6 do E[i]; end for; for i:=1 to 6 do E[i]:=0; end for; for i:= 1 to D do E[1]:= E[1] + W[i]*w1[i]; E[2]:= E[2] + W[i]*w2[i]; E[3]:= E[3] + W[i]*w3[i]; E[4]:= E[4] + W[i]*w4[i]; E[5]:= E[5] + W[i]*w5[i]; E[6]:= E[6] + W[i]*w6[i]; end for; "results of p_V*p_V - p_V"; for i:=1 to 6 do E[i]-W[D-6+i]; end for;