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| − | An [[Algebra|algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s0841201.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s0841202.png" /> with a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s0841203.png" /> that is at the same time a multiplicative [[Semi-group|semi-group]]. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s0841204.png" /> is a group, one obtains a [[Group algebra|group algebra]]. If the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s0841205.png" /> contains a zero, this zero is usually identified with the zero of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s0841206.png" />. The problem of describing all linear representations of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s0841207.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s0841208.png" /> (cf. [[Linear representation|Linear representation]]; [[Representation of a semi-group|Representation of a semi-group]]) is equivalent to that of describing all representations of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s0841209.png" />. The importance of semi-group algebras in the theory of semi-groups is the possibility they offer of utilizing the richer tools of the theory of algebras to study linear representations of semi-groups. An example of this kind of result is: The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412010.png" /> of a finite semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412011.png" /> is semi-simple if and only if all linear representations of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412012.png" /> over the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412013.png" /> are reducible.
| + | {{MSC|20E35}} |
| | + | {{TEX|done}} |
| | | | |
| − | ====References====
| + | An |
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , '''1''' , Amer. Math. Soc. (1961)</TD></TR></table>
| + | [[Algebra|algebra]] $\Phi(S)$ over a field $\Phi$ with a basis $S$ that is at the same time a multiplicative |
| | + | [[Semi-group|semi-group]]. In particular, if $S$ is a group, one obtains a |
| | + | [[Group algebra|group algebra]]. If the semi-group $S$ contains a [[zero]], this zero is usually identified with the zero of the algebra $\Phi(S)$. The problem of describing all linear representations of a semi-group $S$ over a field $\Phi$ (cf. [[Linear representation]]; [[Representation of a semi-group]]) is equivalent to that of describing all representations of the algebra $\Phi(S)$. The importance of semi-group algebras in the theory of semi-groups is the possibility they offer of utilizing the richer tools of the theory of algebras to study linear representations of semi-groups. An example of this kind of result is: The algebra $\Phi(S)$ of a finite semi-group $S$ is semi-simple if and only if all linear representations of the semi-group $S$ over the algebra $\Phi$ are reducible. |
| | | | |
| | + | ====Comments==== |
| | + | More precisely, let $S$ be a semi-group and $\Phi$ a field. Consider the vector space $V$ of all formal finite sums $V=\{\sum_{s\in S} a_s s\}$, i.e. the vector space over $\Phi$ with basis $S$. The semi-group multiplication $(s,t)\mapsto st$ extends linearly to define an algebra structure on $V$. This is the semi-group algebra $\Phi[S]$. |
| | | | |
| | + | If $S$ is a semi-group with zero $z$, the subspace $\Phi_z$ is an ideal in $\Phi[S]$ and the contracted semi-group algebra $\Phi_0[S]$ is the quotient algebra $\Phi_0[S] = \Phi[S]/\Phi_z$. |
| | | | |
| − | ====Comments====
| + | For an inverse semi-group (cf. [[Inversion semi-group]]) one has the following analogue of Maschke's theorem (cf. [[Group algebra]]). The semi-group algebra $\Phi[S]$ of a finite inverse semi-group $S$ is semi-simple if and only if the characteristic of $\Phi$ is zero or is a prime that does not divided the order of any sub-semi-group of $S$. |
| − | More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412014.png" /> be a semi-group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412015.png" /> a field. Consider the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412016.png" /> of all formal finite sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412017.png" />, i.e. the vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412018.png" /> with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412019.png" />. The semi-group multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412020.png" /> extends linearly to define an algebra structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412021.png" />. This is the semi-group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412022.png" />.
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| − | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412023.png" /> is a semi-group with zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412024.png" />, the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412025.png" /> is an ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412026.png" /> and the contracted semi-group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412027.png" /> is the quotient algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412028.png" />.
| + | ====References==== |
| − | | + | {| |
| − | For an inverse semi-group (cf. [[Inversion semi-group|Inversion semi-group]]) one has the following analogue of Maschke's theorem (cf. [[Group algebra|Group algebra]]). The semi-group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412029.png" /> of a finite inverse semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412030.png" /> is semi-simple if and only if the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412031.png" /> is zero or is a prime that does not divided the order of any sub-semi-group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084120/s08412032.png" />.
| + | |- |
| | + | |valign="top"|{{Ref|ClPr}}||valign="top"| A.H. Clifford, G.B. Preston, "The Algebraic theory of semigroups", '''1''', Amer. Math. Soc. (1961) {{MR|0132791}} {{ZBL|0111.03403}} |
| | + | |- |
| | + | |} |
Latest revision as of 19:41, 8 December 2015
2020 Mathematics Subject Classification: Primary: 20E35 [MSN][ZBL]
An
algebra $\Phi(S)$ over a field $\Phi$ with a basis $S$ that is at the same time a multiplicative
semi-group. In particular, if $S$ is a group, one obtains a
group algebra. If the semi-group $S$ contains a zero, this zero is usually identified with the zero of the algebra $\Phi(S)$. The problem of describing all linear representations of a semi-group $S$ over a field $\Phi$ (cf. Linear representation; Representation of a semi-group) is equivalent to that of describing all representations of the algebra $\Phi(S)$. The importance of semi-group algebras in the theory of semi-groups is the possibility they offer of utilizing the richer tools of the theory of algebras to study linear representations of semi-groups. An example of this kind of result is: The algebra $\Phi(S)$ of a finite semi-group $S$ is semi-simple if and only if all linear representations of the semi-group $S$ over the algebra $\Phi$ are reducible.
More precisely, let $S$ be a semi-group and $\Phi$ a field. Consider the vector space $V$ of all formal finite sums $V=\{\sum_{s\in S} a_s s\}$, i.e. the vector space over $\Phi$ with basis $S$. The semi-group multiplication $(s,t)\mapsto st$ extends linearly to define an algebra structure on $V$. This is the semi-group algebra $\Phi[S]$.
If $S$ is a semi-group with zero $z$, the subspace $\Phi_z$ is an ideal in $\Phi[S]$ and the contracted semi-group algebra $\Phi_0[S]$ is the quotient algebra $\Phi_0[S] = \Phi[S]/\Phi_z$.
For an inverse semi-group (cf. Inversion semi-group) one has the following analogue of Maschke's theorem (cf. Group algebra). The semi-group algebra $\Phi[S]$ of a finite inverse semi-group $S$ is semi-simple if and only if the characteristic of $\Phi$ is zero or is a prime that does not divided the order of any sub-semi-group of $S$.
References
| [ClPr] |
A.H. Clifford, G.B. Preston, "The Algebraic theory of semigroups", 1, Amer. Math. Soc. (1961) MR0132791 Zbl 0111.03403
|
How to Cite This Entry:
Semi-group algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-group_algebra&oldid=11311
This article was adapted from an original article by L.M. Gluskin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article