Let be an algebra over a field , and let be some group. A decomposition as an vector space is called a -grading if it satisfies for all .
- The polynomial algebra has a -grading with .
- The complex field is graded as an algebra with the decomposition .
- The algbera of has a grading
- (Twisted group algebra) Other examples for such gradings can be constructed as follows. Let be a 2-cocycle, namely is a function which satisfies . The twisted group algebra is the F vector space with basis with the product defined by . Note that if , then is the usual group algebra. The 2-cocycle condition makes this an associative algebra with a G-grading .
- (Grassmann\Exterior Algebra) Define the Grassmann algebra to be the free noncommutative algebra modulo the relations and for all . Setting and , one easily checks that is a -grading of the Grassmann algebra.
The elements of are called homogeneous of degree .
A homomorphism between two G-graded algebras is called graded if it preserves the G-grading, namely . A subalgebra\one sided\two sided ideal in is called graded (or homogeneous) ideal if it is generated by homogeneous elements, or equivalently . It is easy to see the the kernel of a graded homomorphism is a graded ideal of and its image is a graded subalgebra of .
Let be a field, be infinitely many noncommuting indeterminates over , and denote by the free noncommutative algebra over , namely all the noncommutative polynomial in .
A polynomial is called a polynomial identity of an algebra , if for all . Equivalently, is an identity if it is in the kernel of every homomorphism from to . We denote the set of all polynomial identities of by . This is an ideal in which in addition is closed under endomorphisms of , or in other words, if is an identity of , then so is . This property is called the T-property and is called a T-ideal.
For example, the polynomial is an identity of if and only if is commutative. Another less trivial example is the Hall polynomial identity for . This follows from the fact that is a traceless matrix, hence it either have two distinct eigenvalues , or has only zero as an eigenvalue. In any way is a scalar matrix and therefore .
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 where 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 is of characteristic zero, then the T-ideal generated by the multilinear identities of an algebra is exactly , thus in order to study the identities of 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 , we set to be the free G-graded noncommutative algebra where the grading is given by .
A polynomial is called a graded identity of a G-graded algebra , if for all graded assignments, namely . The set of all graded polynomial identities of is a graded T-ideal in .
For example, consider the grading of given above. The algebra is not commutative, and therefore is not a polynomial identity, while as a graded algebra it does satisfy the graded identity , since its component, which is all the scalar matrices, is commutative.
Consider now the algebra for some 2-cocycle . If then there is some scalar such that . If is any permutation such that , then we have
Since the scalars in the field commute with the we get that
is a multilinear polynomial identity of . It is not hard to show that vector space of multilinear identities of is spanned by these binomial identities, hence these binomial identities generate as a T-ideal. If in addition G is abelian, it is enough to consider only degree two binomial identities, namely .