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Martin Brandenburg
Martin Brandenburg
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A $R$-algebra $A$ is a group algebra over $R$ if and only if there exists a $R$-module basis of $A$, which is also central and a subgroup of $R^*$$A^*$. Of course, this is trivial, but I don't think that there is a nice characterization.

Observe that some $R$-algebras $A$ don't have any $R$-homomorphism $A \to R$ at all. However, group algebras admit the augmentation map $R[G] \to R$. Besides, the diagonal map $G \to G \times G$ induces $R[G] \to R[G] \otimes R[G]$ and endow $R[G]$ with the structure of a coalgebra.

A $R$-algebra $A$ is a group algebra over $R$ if and only if there exists a $R$-module basis of $A$, which is also central and a subgroup of $R^*$. Of course, this is trivial, but I don't think that there is a nice characterization.

Observe that some $R$-algebras $A$ don't have any $R$-homomorphism $A \to R$ at all. However, group algebras admit the augmentation map $R[G] \to R$. Besides, the diagonal map $G \to G \times G$ induces $R[G] \to R[G] \otimes R[G]$ and endow $R[G]$ with the structure of a coalgebra.

A $R$-algebra $A$ is a group algebra over $R$ if and only if there exists a $R$-module basis of $A$, which is also central and a subgroup of $A^*$. Of course, this is trivial, but I don't think that there is a nice characterization.

Observe that some $R$-algebras $A$ don't have any $R$-homomorphism $A \to R$ at all. However, group algebras admit the augmentation map $R[G] \to R$. Besides, the diagonal map $G \to G \times G$ induces $R[G] \to R[G] \otimes R[G]$ and endow $R[G]$ with the structure of a coalgebra.

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Martin Brandenburg
Martin Brandenburg
  • 63.1k
  • 11
  • 207
  • 424

A $R$-algebra $A$ is a group algebra over $R$ if and only if there exists a $R$-module basis of $A$, which is also a central submonoidand a subgroup of $A$$R^*$. Of course, this is trivial, but I don't think that there is a nice characterization.

Observe that some $R$-algebras $A$ don't have any $R$-homomorphism $A \to R$ at all. However, group algebras admit the augmentation map $R[G] \to R$. Besides, the diagonal map $G \to G \times G$ induces $R[G] \to R[G] \otimes R[G]$ and endow $R[G]$ with the structure of a coalgebra.

A $R$-algebra $A$ is a group algebra over $R$ if and only if there exists a $R$-module basis of $A$, which is also a central submonoid of $A$. Of course, this is trivial, but I don't think that there is a nice characterization.

Observe that some $R$-algebras $A$ don't have any $R$-homomorphism $A \to R$ at all. However, group algebras admit the augmentation map $R[G] \to R$. Besides, the diagonal map $G \to G \times G$ induces $R[G] \to R[G] \otimes R[G]$ and endow $R[G]$ with the structure of a coalgebra.

A $R$-algebra $A$ is a group algebra over $R$ if and only if there exists a $R$-module basis of $A$, which is also central and a subgroup of $R^*$. Of course, this is trivial, but I don't think that there is a nice characterization.

Observe that some $R$-algebras $A$ don't have any $R$-homomorphism $A \to R$ at all. However, group algebras admit the augmentation map $R[G] \to R$. Besides, the diagonal map $G \to G \times G$ induces $R[G] \to R[G] \otimes R[G]$ and endow $R[G]$ with the structure of a coalgebra.

Source Link
Martin Brandenburg
Martin Brandenburg
  • 63.1k
  • 11
  • 207
  • 424

A $R$-algebra $A$ is a group algebra over $R$ if and only if there exists a $R$-module basis of $A$, which is also a central submonoid of $A$. Of course, this is trivial, but I don't think that there is a nice characterization.

Observe that some $R$-algebras $A$ don't have any $R$-homomorphism $A \to R$ at all. However, group algebras admit the augmentation map $R[G] \to R$. Besides, the diagonal map $G \to G \times G$ induces $R[G] \to R[G] \otimes R[G]$ and endow $R[G]$ with the structure of a coalgebra.