##### Department of Mathematics,

University of California San Diego

****************************

### Algebra Colloquium

## Daniel Goldstein

#### UCSD

## Principal ideals of the exterior algebra

##### Abstract:

Let $F$ be a field and $n>0$ a positive integer. Let $A = A_0 + A_1 + ...+ A_n$ be the exterior algebra of dimension $2^n$ over $F$ with its natural grading. Then a homogenous element w in $A_s$ generates a homogeneous principal ideal $wA$. What is the maximum value of $\dim_F(wA \cap A_r)$ for given $s,r,n$ as $w$ varies in $A_s$? We state a conjecture. The (most interesting?) case $(s,r,n) = (3,6,9)$ (where the max is 84) is directly related to the exceptional Lie algebra $E_8$. (By definition, A is generated by elements $e_1,...,e_n$ that satisfy $(1) e_i^2 =0$ and $(2) e_ie_j +e_je_i=0$ for $1<=i,j<= n$. The generators $e_i$ all lie in $A_1$. By definition, $A_s$ is the $F-span$ of all $s-fold$ products $e_{j_1} ... e_{j_s}.)$

### October 23, 2006

### 3:00 PM

### AP&M 7218

****************************