I am trying to learn some basic knowledge on lattices for studying Minkowski's theorems and Dirichlet's unit theorem. My problem is, I could not build the basics of lattice theory and there are some blurred parts. I will give some definitons and then ask my questions.
Let $L \subset \mathbb{R}^n$ be a lattice. It is a set $\{ \sum \alpha_i x_i | \alpha_i \in \mathbb{Z}\}$ where $\{x_1,\dots,x_n\}$ is a linearly independent set in $\mathbb{R}^n$. So we can either say that $L$ is the set above or we can define a matrix $$B=[ (x_1)_{n\times 1} \dots (x_n)_{n \times 1}]_{n \times n}$$- generator matrix of $L$-
and say $L = \{Bv : v \in \mathbb{Z}^{n \times 1} \}$.
I saw that $vol(L)$ is defined as Lebesgue measure $\mu(P)$ where $P=\{ \sum a_i x_i | 0\le a_i <1 \}$, the parallelpipped of $L$ with respect to base $\{x_1,\dots,x_n\}$.
Now,
i)Is $vol(L) = \det(B)$, or, $\det(B)=vol(\mathbb{R}^n / L)$? If the latter is correct, what is $\mathbb{R}^n / L$, exactly?
ii)What is the difference between covolume$(L)$ and volume$(L)$?
Best Answer
Let's think of $\mathbb R^n$ as an abelian topological group, and a lattice $L$ as a discrete subgroup. Then (maybe A. Weil proved such a thing in his book on topological groups; in any case, it is standard) the topological group quotient $\mathbb R^n/L$ has a unique (positive, regular, Borel) measure $d\dot{x}$ such that for every $f\in C^o_c(\mathbb R^n/L)$ $$ \int_{\mathbb R^n} f(x)\;dx \;=\; \int_{\mathbb R^n/L} \sum_{\lambda\in L} f(\dot{x}+\lambda)\;d\dot{x} $$ In the tangible case at hand, for your choice of $\mathbb Z$-basis for $L$, the characteristic function of a "fundamental period parallelogram" for $L$ maps to the identically-$1$ function on the quotient, whose integral on the quotient is the natural volume of it.
(It's not quite the determinant, but, rather, the absolute value thereof...)
EDIT: in response to a comment... one way to see the relationship between (absolute values of) the determinant (of a matrix whose columns (or rows) are a chosen $\mathbb Z$-basis for the given lattice), is to invoke the volume-preserving feature of "shear" matrices, e.g, upper-triangular with $1$'a on the diagonal. One can give a physical plausibility argument that "volume" (perhaps ambiguously defined) is preserved by these. Also, for example, by orthogonal matrices (since they preserve the metric). Every linear transformation is expressible (Iwasawa decomposition...) as $g=nak$, where $n$ is a "shear" map, $k$ is orthogonal, and $a$ is diagonal. The effect of diagonal matrices on "volume" is obviously the absolute value of the product of the diagonal entries. :)