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Oeyvind Solberg
Oeyvind Solberg
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The structure constants $m_{ij}(k)$ ofI realized that I didn't look at the algebra $A$definition careful enough. So I have erased my first answer, since it didn't say anything useful.

The requirements actually says that

$(1)\quad \Delta(ab) = \Delta(a)b$

and

$(2)\quad \Delta(ab) = a\Delta(b).$

In other words, the linear homomorphism $a_i * a_j = \sum_{k} m_{ij}(k)a_k$ for$\Delta\colon A \to A\otimes_k A$ should be a basis $\\{a_k\\}_k$homomorphism of $A$, enters into-$A$-bimodules. Therefore the linear systemFrobenius dimension of equations given by (1) and (2). These constantsan algebra is nothing else than the dimension of the space of homomorphisms

$\operatorname{Hom}_{A\otimes_k A^{\operatorname{op}}}(A,A\otimes_k A).$

This can be easily foundcomputed in QPA by the commandfollowing line of commands:

gap> BAenv := BasisEnvelopingAlgebra( A );;;
gap> strucconstP := ListDirectSumOfQPAModules( B, b -> ListIndecProjectiveModules( B,Aenv c) ->);
M Coefficients:= AlgebraAsModuleOverEnvelopingAlgebra( B,A b);
Length( *HomOverAlgebra( cM, )P ) );

One can also get the same information by usingThe last number is the command StructureConstantsTable on a basis forFrobenius dimension of the algebra $A$.

However, the linear equations defining (1) and (2), I see no way to obtain by using predefined functions in QPA. These needs to be worked out. But in the endSo, it should be possible to compute the Frobenius dimension within QPA/GAP= $\dim_k \operatorname{Hom}_{A\otimes_k A^{\operatorname{op}}}( A, A\otimes_k A )$.

The QPA-team.

The structure constants $m_{ij}(k)$ of the algebra $A$, say $a_i * a_j = \sum_{k} m_{ij}(k)a_k$ for a basis $\\{a_k\\}_k$ of $A$, enters into the linear system of equations given by (1) and (2). These constants can be easily found by the command

gap> B := Basis(A);;
gap> strucconst := List( B, b -> List( B, c -> Coefficients( B, b * c ) ) );

One can also get the same information by using the command StructureConstantsTable on a basis for the algebra $A$.

However, the linear equations defining (1) and (2), I see no way to obtain by using predefined functions in QPA. These needs to be worked out. But in the end, it should be possible to compute the Frobenius dimension within QPA/GAP.

The QPA-team.

I realized that I didn't look at the definition careful enough. So I have erased my first answer, since it didn't say anything useful.

The requirements actually says that

$(1)\quad \Delta(ab) = \Delta(a)b$

and

$(2)\quad \Delta(ab) = a\Delta(b).$

In other words, the linear homomorphism $\Delta\colon A \to A\otimes_k A$ should be a homomorphism of $A$-$A$-bimodules. Therefore the Frobenius dimension of an algebra is nothing else than the dimension of the space of homomorphisms

$\operatorname{Hom}_{A\otimes_k A^{\operatorname{op}}}(A,A\otimes_k A).$

This can be easily computed in QPA by the following line of commands:

Aenv := EnvelopingAlgebra( A );
P := DirectSumOfQPAModules( IndecProjectiveModules( Aenv ) );
M := AlgebraAsModuleOverEnvelopingAlgebra( A );
Length( HomOverAlgebra( M, P ) );

The last number is the Frobenius dimension of the algebra $A$.

So, Frobenius dimension = $\dim_k \operatorname{Hom}_{A\otimes_k A^{\operatorname{op}}}( A, A\otimes_k A )$.

The QPA-team.

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Oeyvind Solberg
Oeyvind Solberg
  • 1.2k
  • 7
  • 8

The structure constants $m_{ij}(k)$ of the algebra $A$, say $a_i * a_j = \sum_{k} m_{ij}(k)a_k$ for a basis $\\{a_k\\}_k$ of $A$, enters into the linear system of equations given by (1) and (2). These constants can be easily found by the command

gap> B := Basis(A);;
gap> strucconst := List( B, b -> List( B, c -> Coefficients( B, b * c ) ) );

One can also get the same information by using the command StructureConstantsTable on a basis for the algebra $A$.

However, the linear equations defining (1) and (2), I see no way to obtain by using predefined functions in QPA. These needs to be worked out. But in the end, it should be possible to compute the Frobenius dimension within QPA/GAP.

The QPA-team.