Noncommutative Case: Gelfand-Kirillov dimension

Let $R=k<x_1,...,x_n>$, the polynomial ring over $k$ in $n$ noncommuting variables. The standard filtration on $R$ is the sequence of vector subspaces of $R$, $R_t,$ consisting of all polynomials of degree $t$ or less. If $I$ is an ideal of $R$, set $T=R/I.$ Then the standard filtration $T_t$ on $T$ is the sequence $\{R_t/(I\cap R_t)\}.$ If we define $H_T(t)=\dim_k(T_t),$ the Hilbert series of $T$ corresponding to this filtration is

$$\sum_{t\geq 0}H_T(t)z^t$$

where z is an indeterminate. The standard gradation on $R$ is the sequence $R^{(s)}$ of vector subspaces of $R$ consisting of polynomials that are homogeneous of degree $s.$ This has the nice property that

$$R=\bigoplus_{s \geq 0} R^{(s)}.$$

Now $T$ may not have a gradation, but in the case that $I$ is a homogeneous ideal then $T^{(s)}$ is defined similarly as in the filtration case. Note that $T_t=\bigcup_{0\leq s \leq t} T^{(s)}.$ Setting $H^T(s)=\dim_k T^{(s)},$ the Hilbert series corresponding to this gradation is

$$\sum_{s\geq 0} H^T(s)z^s.$$

By the note above we have that

$$H_T(t)=\sum_{0 \leq s \leq t} H^T(s).$$

Usually when we refer to the Hilbert series of an algebra we must be specific about the gradation or filtration we are using as there are many besides the standard ones. However, here we take the convention that if an algebra is graded we take the Hilbert series to mean the one with respect to the standard gradation, otherwise we are using the standard filtration. This should cause no confusion, as we are dealing strictly with finitely generated polynomial algebras.

The following dimension has been around since the 1960s and has become a main tool in noncommutative algebra. It was first introduced by I.M. Gelfand and A.A. Kirillov in a paper in 1966 describing work on enveloping algebras of Lie algebras. References for background and proofs of theorems are at the end of this chapter.

Definition 0   The Gelfand-Kirillov dimension of $T$ is defined to be

$$\limsup_{t\to\infty} \frac {\log(H_T(t))}{\log(t)}.$$

We need a little bit more notation. Let $A$ be an affine $k$-algebra, that is, an algebra finitely generated over a field $k.$ Then $A=k[V]$ where $V=Span_k\{1,x_1,\ldots,x_m\}$ for some $\{x_j\} \subset A.$ Define

$$V^n=Span_k\{x_{j_1}x_{j_2}\cdots x_{j_s} : 1\leq j_i \leq m, \sum_{i}j_i \leq n\}$$

so that $k,V,V^2,\ldots,V^n,\ldots$ gives the standard filtration of $A$. Note also that $H_A(t)=\dim_k V^t.$

A quick fact about GK dimension: It is known that there exist algebras of GK dimension $r$ for any real number $r \in \{0\} \cup \{1\} \cup [2,\infty),$ and moreover no other number is attained.

There are several reasons that the Gelfand-Kirillov (hereafter abreviated $GK$) dimension is an apropriate measure of degrees of freedom. The following four standard theorems tell us that the GK dimension behaves much like the transcendance degree from commutative algebra. They follow easily from the definition, and a little combinitorics. Proofs can be found in the references at the end of the chapter.

Theorem 0   If $A$ is a commutative algebra, then

$$GK(A)=$$Krull$$(A).$$

Theorem 0   If $B \subset A$ is a subalgebra, then

$$GK(B) \leq GK(A).$$

Theorem 0   If $C$ is a homomorphic image of $A$, then

$$GK(C) \leq GK(A).$$

Theorem 0   Let $A$ be an affine algebra, with $GK$ dimension $r$. Then $GK(A[x])=r+1$, where $x$ is a new central indeterminate.

The following theorem shows that the $GK$ dimension is well-behaved with respect to quotient algebras.

Theorem 0   Let $A$ be an affine algebra and $I$ an ideal of $A$ that contains a left (or right) non-zero-divisor, $c \in A.$ Then

$$GK(A/I) + 1 \leq GK(A).$$

Proof. Using the notation above, suppose $c$ is a degree $l$ non-zero-divisor. Let $C_n$ be the vector space complement of
$V^n \cap I$ in $V^n,$ So $C_n \simeq V^n/(V^n \cap I).$ Note that the sequence $\{V^n/(V^n \cap I)\}_{n=1}^{\infty}$ gives a filtration of $A/I.$ Now $C_n \cap \langle c \rangle = \{0\},$ as $C_n \cap I = \{0\}.$ So $C_n \oplus C_n c \oplus \cdots \oplus C_n c^n \subset V^{n(l+1)}$ and the $\oplus$ are valid as $a c^i = b c^j, i < j$ implies $c^i ( a - b c^{j-i}) = 0$ and $c$ a left non-zero-divisor then gives $a - b c^{j-i}=0$ so $a \in \langle c \rangle \cap C_n$ which implies $a=0.$ So

$$\dim_k(V^{n(l+1)}) \geq n \cdot \dim_k(C_n) = \dim_k(V^n/(V^n \cap I)).$$

Thus

$$GK(A/I) + 1 = \limsup_{n \to \infty} \frac {\log(\dim_k(C_n))}{\log(n)} +1 =$$

$$\limsup_{n \to \infty} \frac {\log(\dim_k(C_n))}{\log(n)} + \frac... ...log(n)} = \limsup_{n \to \infty} \frac {\log(n\cdot \dim_k(C_n))}{\log(n)} \leq$$

$$\limsup_{n \to \infty} \frac {\log(\dim_k(V^{n(l+1)}))}{\log(n)} \leq \limsup_{n \to \infty} \frac {\log(\dim_k(V^n))}{\log(n)} = GK(A).$$

$\qedsymbol$

The computation of the $GK$ dimension is in general difficult. Since the sequence whose limit is the $GK$ dimension converges very slowly, the only hope is to compute some of the coefficients of the Hilbert series and then guess a generating formula for them. Then the $GK$ dimension can be computed by taking the limit of this sequence. This may sound a bit ad hoc, but it has been implemented for several interesting algebras, an example of which follows.

The algebra we consider is the algebra on two variables generated by the relations that say that any two degree 2 monomials commute with each other. The commands used here are explained in the section on Commands.

We use the input file as follows:

 
<<NCHilbertCoefficient.m;

SNC[x,y];

SetMonomialOrder[{x,y}];

rels = {x**x**y**y - y**y**x**x, x**x**x**y - x**y**x**x,
    -x**y**y**y + y**y**x**y, x**x**y**x - y**x**x**x,
    -y**x**y**y + y**y**y**x, x**y**y**x - y**x**x**y};

NCHilbertCoefficient[18,rels,3,ExpressionForm->Homogeneous];

The call to NCMakeGB finishes after only 2 iterations, so we know that the coefficients are being computed using a full Groebner basis, hence they are exact. The output is the following:

{2, 4, 8, 10, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72,
81, 90, 100}

After staring at this sequence for a while, one sees that every other term is a square and the intermediate terms are products of sucessive integers after the 6th term. A formula then is $[n][n+1]$ where $[a]$ is the greatest integer less than or equal to $a$. This is then clearly asymtotic to a quadratic polynomial. This sequence came from a gradation, so to get the sequence coming from the filtration we simply take the partial sums. This in turn will be asymtotic to a cubic polynomial in $n$, which we will call $P(n)$. So the $GK$ dimension of the quotient of the free polynomial algebra on two variables by the ideal generated by rels is:

$$\limsup_{n \to \infty}\frac{\log(P(n))}{\log(n)}=3.$$

For further information on the function NCHilbertCoefficient used above, see the documentation section.