In mathematics, as in footwear, order matters. Putting on your socks before your shoes produces a different result to putting on your shoes before your socks. This means that the two operations 'putting on socks' and 'putting on shoes' are noncommutative. Noncommutative structures are widespread in mathematics, appearing in subjects ranging from group theory to analysis and differential equations.
My research focuses on noncommutative algebras. The structure of an algebra is given by two fundamental operations, addition and multiplication. In a noncommutative algebra it is the order of multiplication that matters.
A key difference between commutative and noncommutative algebras is the existence of a dictionary connecting commutative algebras with geometry. There is a direct correspondence between finitely-generated commutative algebras and affine varieties (or schemes), subsets of space defined by polynomial equations. Properties of commutative algebras can be translated into the geometry of affine varieties, and vice versa. Unfortunately, noncommutative algebras have no such dictionary.
However, noncommutative algebras are, in fact, simpler than commutative algebras. One way of measuring this is a number called the Krull dimension, which quantifies complexity.
Let's compare two similar algebras, the polynomial algebra $P_{2n}$ in $2n$ variables, with the Weyl algebra $A_n$. Both algebras are generated by $2n$ elements $X_1, \dots, X_n, Y_1, \dots, Y_n$. In $P_{2n}$ all elements commute. In $A_n$ nearly all of the commutativity is retained, but the relations between elements of the same superscript are noncommutative, $Y_iX_i = X_iY_i+1$. Hermann Weyl introduced this relation to model the noncommutativity at the heart of the Heisenberg uncertainty principle.
Weyl's noncommutative relations simplify the internal structure considerably, by reducing the number of ideals which are possible. The Krull dimension of $A_n$ is $n$, whilst the polynomial algebra $P_{2n}$, by contrast, has Krull dimension $2n$. So, the commutative algebra is twice as complex as its noncommutative relative.
To compute that the Krull dimension of the Weyl algebra is $n$, we use a famous theorem known as Bernstein's inequality. This theorem characterises the representation theory of Weyl algebras - their modules - by describing their canonical dimension. The canonical dimension is a non-negative integer that measures the size of any module (for algebras with the Auslander-Gorenstein condition).
Bernstein's inequality states that the canonical dimension of any module for the $n$th Weyl algebra is always at least $n$ (and at most $2n$). On the other hand, the canonical dimension of a module for the polynomial algebra $P_{2n}$ can be any integer between $0$ and $2n$. This discrepancy explains the difference in Krull dimension between the algebras. Because the representations of the Weyl algebra can only take on around half as many values, its Krull dimension can be at most half that of the polynomial algebra.
During my DPhil (PhD), I attempted to find a Bernstein's inequality for an important type of noncommutative algebra called Iwasawa algebras. Iwasawa algebras encapsulate the representation theory of $p$-adic Lie groups. Whilst Weyl algebras were introduced to study the revolutionary theory of quantum mechanics, Iwasawa algebras play a role in the Langlands programme, an epic project aiming to connect number theory and representation theory on a deep level.
Although a Bernstein's inequality is a simple statement about integers, a barrier to discovering one is that modules are virtually unclassifiable. Given a noncommutative algebra, it's generally impossible to write down a list of all its modules, even if restricted to certain types such as the indecomposables. This means that even if we can calculate the canonical dimension in many cases, this is unlikely to be enough.
In particular, how can we determine when a particular integer won't be realised as a canonical dimension? The answer to this conundrum is to use an invariant that keeps more information than just an integer, but still throws away a lot of obscuring data.
Borrowing a commutative technique, we associate an affine variety to every module. We can do this by making a commutative approximation to our noncommutative algebra, via a process known as taking the 'associated graded'. For both Weyl algebras and Iwasawa algebras, the approximation we obtain is a polynomial algebra. The commutative algebra-geometry dictionary then tells us that any module, via its associated graded, produces an affine variety. We call this the characteristic variety, and its dimension equals the canonical dimension of the module.
This approach introduces geometry into a noncommutative setting! Crucially, a variety contains more information than an integer. This makes it easier to prove that a certain integer $m$ is not the canonical dimension of any module - we can instead prove that each $m$-dimensional variety is not a characteristic variety, 'one at a time'.
I used this idea during my DPhil to prove that a wide class of Iwasawa algebras have no module of canonical dimension equal to 1. In particular, every non-trivial representation of the corresponding $p$-adic Lie groups must have a canonical dimension of at least 2.
This is a proto-Bernstein's inequality: it gives us a previously unknown lower bound on canonical dimension. However, for many Iwasawa algebras we don't know of any modules of canonical dimension two, and suspect that the true lower bound is higher, dependent on the $p$-adic Lie group. Nevertheless, combining the inequality with calculations made by Breuil et al. answers a key finite length conjecture for some of the representations expected to realise a Langlands correspondence. The inequality also implies new bounds on the Krull dimension of Iwasawa algebras, the first progress on this question for nearly twenty years.
James Timmins is a postdoctoral research associate at the University of Edinburgh and formerly a DPhil student in Oxford Mathematics.
Bibliography:
Konstantin Ardakov, Krull dimension of Iwasawa algebras, Journal of Algebra (2004).
Christophe Breuil, Florian Herzig, Yongquan Hu, Stefano Morra, Benjamin Schraen, Gelfand-Kirillov dimension and mod p cohomology for GL2, Inventiones mathematicae (2023)
James Timmins, The canonical dimension of modules for Iwasawa algebras, arXiv (2023)