Algebra group structures for elementary abelian group:E8

This article gives specific information, namely, algebra group structures, about a particular group, namely: elementary abelian group:E8.
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There exist (at least) two isomorphism classes of nilpotent associative algebras $N_{1},N_{2}$ over field:F2 such that both the algebras have algebra group isomorphic to direct product of Z4 and Z2.

Of these, $N_{1}$ is a direct product of algebra groups, whereas $N_{2}$ is not. Further, $N_{1}$ can be viewed as an algebra over field:F8, but $N_{2}$ cannot.

Summary

Algebra	Smallest $n$ for which it can be embedded in $UT(n,2)$	Maximum of element nilpotencies	Nilpotency index of whole algebra
direct product of algebras each corresponding to cyclic group:Z2; equivalently, all products are zero (we call this $N_{1}$) -- note that this can also be given the structure of an algebra over field:F8	4	2	2
has some nonzero products (we call this $N_{2}$)	4	2	3

Algebra that splits as a direct product, all products are zero

Multiplication table (structure constants)

We choose basis letters $a,b,c$ for $N_{1}$, and give it the following multiplication table. The row element is multiplied on the left and the column element on the right (though this is irrelevant since multiplication is commutative anyway).

	$a$	$b$	$c$
$a$	0	0	0
$b$	0	0	0
$c$	0	0	0

Verification of properties

• $N_{1}$ is associative: By the linearity, it suffices to check associativity on basis triples. It's easy to see from the multiplication table that all products for basis triples are zero, so associativity holds.
• $N_{1}$ is nilpotent: All products of length two or more are zero, so the algebra is nilpotent.
• The algebra group of $N_{1}$ is isomorphic to elementary abelian group:E8: It's easy to see that in the corresponding group, every element has order two, because every element of the algebra squares to zero.

Description as subalgebra of niltriangular matrix Lie algebra

The algebra can be realized explicitly as a subalgebra of niltriangular matrix Lie ring:NT(4,2) as follows:

$a={\begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\\\end{pmatrix}},\qquad b={\begin{pmatrix}0&0&1&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\\\end{pmatrix}},\qquad c={\begin{pmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\0&0&0&0\\\end{pmatrix}}$

Algebra with nonzero products

Multiplication table (structure constants)

We choose basis letters $x,y,z$ for $N_{2}$, and give it the following multiplication table. The row element is multiplied on the left and the column element on the right (though this is irrelevant since multiplication is commutative anyway).

	$x$	$y$	$z$
$x$	0	$z$	0
$y$	$z$	0	0
$z$	0	0	0

Verification of properties

• $N_{2}$ is associative: By the linearity, it suffices to check associativity on basis triples. All products of length three are zero.
• $N_{2}$ is nilpotent: All products of length three or more are zero, so the algebra is nilpotent.
• The algebra group of $N_{2}$ is isomorphic to elementary abelian group:E8: We can verify that all the elements of the group have order two, because all elements of the algebra square to zero. This needs some checking, but is not hard

Description as subalgebra of niltriangular matrix Lie algebra

The algebra can be realized explicitly as a subalgebra of niltriangular matrix Lie ring:NT(4,2) as follows:

$x={\begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0\\\end{pmatrix}},\qquad y={\begin{pmatrix}0&0&1&0\\0&0&0&1\\0&0&0&0\\0&0&0&0\\\end{pmatrix}},\qquad z={\begin{pmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\0&0&0&0\\\end{pmatrix}}$