Graded Algebras and PI theory – an introduction

Graded Algebras:

Let A be an algebra over a field F, and let G be some group. A decomposition  A=\bigoplus_{g\in G} A_g as an F vector space is called a G-grading if it satisfies A_g \cdot A_h \subseteq A_{gh} for all g,h \in G.

For example:

  • The polynomial algebra F[x] has a \mathbb{Z}-grading with (F[x])_n = F\cdot x^n.
  • The complex field \mathbb{C} is C_2 graded as an \mathbb{R} algebra with the decomposition \mathbb{R}\oplus i\mathbb{R}.
  • The algbera M_2(F) of 2\times 2 has a C_2 grading
    \left(M_{2}(F)\right)_{1}=\left\{ \left(\begin{array}{cc} a & 0\\ 0 & b \end{array}\right):a,b\in F\right\} \;;\;\left(M_{2}(F)\right)_{-1}=\left\{ \left(\begin{array}{cc} 0 & a\\ b & 0 \end{array}\right):a,b\in F\right\}
  • (Twisted group algebra) Other examples for such gradings can be constructed as follows. Let \alpha\in Z^2(G,F^\times) be a 2-cocycle, namely \alpha is a function \alpha:G\times G\to F^\times which satisfies \alpha(g_1,g_2)\alpha(g_1 g_2,g_3)=\alpha(g_1,g_2 g_3)\alpha(g_2,g_3). The twisted group algebra F^\alpha G is the F vector space with basis \{U_g \mid g\in G\} with the product defined by (aU_g)(bU_h)=ab\alpha(g,h)U_{gh}. Note that if \alpha \equiv 1, then F^1 G is the usual group algebra. The 2-cocycle condition makes this an associative algebra with a G-grading (F^\alpha G)_g=F\cdot U_g.
  • (Grassmann\Exterior Algebra) Define the Grassmann algebra E to be the free noncommutative algebra F<e_i \mid i\in \mathbb{N}> modulo the relations e_i ^2=0 and e_i e_j = - e_j e_i for all i \neq j. Setting E_1 = span \{e_{i_1}\cdots e_{i_{2n}} \mid \; i_1 <\cdots <i_{2n} \} and E_{-1} = span \{e_{i_1}\cdots e_{i_{2n+1}} \mid \; i_1 <\cdots <i_{2n+1} \}, one easily checks that E=E_1 \oplus E_2 is a C_2-grading of the Grassmann algebra.

The elements of A_g are called homogeneous of degree g.

A homomorphism \varphi:A\to B between two G-graded algebras is called graded if it preserves the G-grading, namely \varphi(A_g)\subseteq B_g. A subalgebra\one sided\two sided ideal I in A is called graded (or homogeneous) ideal if it is generated by homogeneous elements, or equivalently I=\sum I\cap A_g . It is easy to see the the kernel of a graded homomorphism is a graded ideal of A and its image is a graded subalgebra of B.

Polynomial Identities:

Let F be a field, X=\{x_n \mid n\in \mathbb {N} \} be  infinitely many noncommuting indeterminates over F, and denote by F the free noncommutative algebra over X, namely all the noncommutative polynomial in X.

A polynomial f(x_1,...,x_n)\in F is called a polynomial identity of an F algebra A, if f(a_1,...,a_n)=0 for all a_i \in A. Equivalently, f is an identity if it is in the kernel of every homomorphism from F to A. We denote the set of all polynomial identities of A by Id(A). This is an ideal in F which in addition is closed under endomorphisms of F, or in other words, if f(x_1,...,x_n) is an identity of A, then so is f(g_1(x_1,...,x_m),...,g_n(x_1,...,x_m)). This property is called the T-property and Id(A) is called a T-ideal.

For example, the polynomial xy-yx is an identity of A if and only if A is commutative. Another less trivial example is the Hall polynomial identity [[x,y]^2,z] for M_2(k). This follows from the fact that [x,y] is a traceless matrix, hence it either have two distinct eigenvalues \lambda, -\lambda, or has only zero as an eigenvalue. In any way [x,y]^2 is a scalar matrix and therefore [[x,y]^2,z]=0.

An algebra that satisfies a nontrivial polynomial identity is called a PI-algebra. As mentioned above, all the commutative algebras are PI. Another important family of PI algebras are the finite dimensional algebras. Thus, the class of PI-algebras are a generalization of the class of commutative algebras. This generalization is more profound than just a mere definition – the main objects of study for noncommutative algebraic geometry are the PI-algebras.

A polynomial of the form f(x_1,...,x_n)=\sum_{\sigma \in S_n} c_\sigma x_{\sigma(1)}\cdots x_{\sigma(n)} where c_\sigma \in F is called a multilinear polynomial. A standard linearization technique shows that if an algebra satisfies a nonzero identity, then it satisfies a nonzero multilinear identity. On the other hand, if the field F is of characteristic zero, then the T-ideal generated by the multilinear identities of an algebra A is exactly Id(A), thus in order to study the identities of A it is enough to consider only the multilinear identities.

Two algebras are said to be PI-equivalent if they have the same identities. As we mentioned above, if the field’s characteristic is zero, then this is equivalent to saying that the algebras have the same multilinear identities.

Graded Polynomial Identities:

A similar process can be done for G-graded algebras for a given group G. Letting X_G=\{x_g,i \mid g\in G\; i\in \mathbb{N}\}, we set F<x_g> to be the free G-graded noncommutative algebra where the grading is given by \deg(x_{g_1,i_1} \cdots x_{g_n,i_n})= g_1 \cdots g_n.

A polynomial f(x_{g_1,i_1},...,x_{g_n,i_n}) is called a graded identity of a G-graded algebra A, if f(a_1,...,a_n) =0 for all graded assignments, namely a_i \in A_{g_i}. The set Id_G(A) of all graded polynomial identities of A is a graded T-ideal in F<X_G>.

For example, consider the C_2 grading of M_2(k) given above. The algebra M_2(k)  is not commutative, and therefore xy-yx is not a polynomial identity, while as a C_2 graded algebra it does satisfy the graded identity x_e y_e -y_e x_e, since its e component, which is all the scalar matrices, is commutative.

Consider now the algebra F^\alpha G for some 2-cocycle \alpha. If g_1,...,g_n \in G then there is some scalar \alpha(g_1,...,g_n) such that U_{g_1} \cdots U_{g_n} = \alpha(g_1 ,...,g_n) U_{g_1 \cdots g_n}. If \sigma \in S_n is any permutation such that g_{\sigma(1)} \cdots g_{\sigma(n)}=g_1 \cdots g_n, then we have

U_{g_1} \cdots U_{g_n} = \alpha(g_1 ,...,g_n) U_{g_1 \cdots g_n} = \frac{\alpha(g_1 ,...,g_n)}{\alpha(g_{\sigma(1)} ,...,g_{\sigma(n)})}U_{g_{\sigma(1)}} \cdots U_{g_{\sigma(n)}}

Since the scalars in the field commute with the U_g we get that

x_{g_1,i_1} \cdots x_{g_n,i_n} - \frac{\alpha(g_1 ,...,g_n)}{\alpha(g_{\sigma(1)} ,...,g_{\sigma(n)})}x_{g_{\sigma(1)},i_{\sigma(1)}} \cdots x_{g_{\sigma(n)},i_{\sigma(n)}}

is a multilinear polynomial identity of F^\alpha G. It is not hard to show that vector space of multilinear identities of F^\alpha G is spanned by these binomial identities, hence these binomial identities generate Id_G(F^\alpha G) as a T-ideal. If in addition G is abelian, it is enough to consider only degree two binomial identities, namely x_g y_h - \frac{\alpha(g,h)}{\alpha(h,g)}y_h x_g.