PI-algebra

An algebra over a field for which certain polynomial identities are true.

Let $ A $ be an associative algebra (cf. Associative rings and algebras) over a field $ F $, let

$$ F [ X ] = F [ x _ {1} \dots x _ {n} , . . . ] $$

be the free associative algebra (the algebra of non-commutative polynomials) on a countable set of generators $ x = ( x _ {1} \dots x _ {n} , . . . ) $ over $ F $, and let $ f ( x _ {1} \dots x _ {n} ) $ be a non-zero element of $ F [ X] $. Then

$$ f ( x _ {1} \dots x _ {n} ) = 0 $$

is said to be a polynomial identity of the algebra $ A $ if $ f( a _ {1} \dots a _ {n} ) = 0 $ for every choice of elements $ a _ {1} \dots a _ {n} \in A $.

Examples of PI-algebras and of identities. The following identity is true in a commutative algebra:

$$ [ x _ {1} , x _ {2} ] = x _ {1} x _ {2} - x _ {2} x _ {1} = 0 $$

(identity of commutativity); in the exterior algebra of a linear space the metAbelian identity $ [[ x _ {1} , x _ {2} ] , x _ {3} ] = 0 $ is satisfied; an algebra $ A $ of finite dimension $ n - 1 $ over a field $ F $ satisfies the so-called standard identity of $ n $- th degree

$$ S _ {n} ( x _ {1} \dots x _ {n} ) = \sum _ {\sigma \in S _ {n} } ( - 1 ) ^ \sigma x _ {\sigma ( 1) } \dots x _ {\sigma ( n) } = 0 , $$

where $ S _ {n} $ is the group of permutations of the set consisting of the first $ n $ positive integers, $ (- 1) ^ \sigma = \mathop{\rm sgn} \sigma $; it also satisfies the more general Capelli identity

$$ K _ {n} ( x _ {1} \dots x _ {n} , y _ {1} \dots y _ {n+ 1 } ) = $$

$$ = \ \sum _ {\sigma \in S _ {n} } ( - 1 ) ^ \sigma y _ {1} x _ {\sigma ( 1) } \dots y _ {n} x _ {\sigma ( n) } y _ {n+ 1 } = 0. $$

In the algebra $ F _ {n} $ of square matrices of order $ n $ over a field $ F $ the standard identity of degree $ 2n $ is satisfied (cf. Amitsur–Levitzki theorem). A tensor product of PI-algebras is a PI-algebra.

For any PI-algebra $ A $ over a field $ F $ of characteristic zero it is possible to find a positive integer $ n $ such that the identities of $ A $ are implied by the powers of the identities of the matrix algebra $ F _ {n} $; moreover, some power of any identity of $ F _ {n} $ is an identity of the algebra $ A $. Thus, in any PI-algebra over a field of characteristic zero some power of the standard identity is satisfied.

The totality of all left-hand sides of the identities which are satisfied in a given algebra $ A $ forms a fully characteristic ideal ( $ T $- ideal for short) of the free algebra $ F [ x] $; conversely, for any $ T $- ideal there exists an algebra whose set of identities coincides with this $ T $- ideal (for example, the quotient algebra $ F [ x] /T $). If $ F $ is of characteristic zero, the identities can be differentiated, and the $ T $- ideals of $ F [ x] $ are precisely the differentially closed one-sided ideals. For instance, repeated differentiation of the nil identity $ x ^ {n} = 0 $ yields the identity

$$ \frac \partial {\partial x } ( x _ {n} ) \dots \frac \partial {\partial x } ( x _ {1} ) x ^ {n\ } = $$

$$ = \ \sum _ {\sigma \in S _ {n} } x _ {\sigma ( 1) } \dots x _ {\sigma ( n) } = 0, $$

which is multi-linear (or, more exactly, $ n $- linear), i.e. linear with respect to each one of its constituent variables. Conversely, setting $ x _ {1} = \dots = x _ {n} = 0 $ in the last identity one obtains the identity $ n ! x ^ {n} = 0 $, or $ x ^ {n} = 0 $. This process of linearization of identities makes it possible to state (for fields of characteristic zero) that all the identities of the algebra are consequences of its multi-linear identities. For an algebra with unit element, moreover, all its identities result from those of its multi-linear identities which are representable by linear combinations of products of right-normalized commutators (cf. Commutator) of different degrees in the generators $ x _ {i} $. The Specht problem deals with the question of whether all associative algebras have a finite basis for the identities.

The totality of all algebras which satisfy a given system of identities is called a variety. A variety may also be defined as a class of algebras closed with respect to taking subalgebras, homomorphic images and subdirect products (cf. also Algebraic systems, variety of). A number of varieties of algebras have been demonstrated to be finitely based (i.e. Specht's problem has a positive solution in such varieties). Such varieties include those (again over a field of characteristic zero) of nilpotent algebras of a given index $ n $, algebras in which the additive commutators of length $ n $ are zero (Lie-nilpotent algebras), and the variety of algebras defined by the $ T $- ideal of semi-identities of $ M _ {2} $( the algebra of $ 2 \times 2 $- matrices). However, the problem remains open for the variety defined by an ideal of identities of $ M _ {n} $, i.e. for the matrix algebras of order $ n > 2 $.

The existence of a polynomial identity rigidly determines the structure of an associative algebra. A primitive algebra $ A $( cf. Primitive ring) which satisfies a polynomial identity of degree $ d $ is isomorphic to a matrix algebra $ D _ {n} $ over a skew-field $ D $ with centre $ Z $, and

$$ \mathop{\rm dim} _ {Z} A \leq \left ( \frac{1}{2} d \right ) ^ {2} . $$

Accordingly, a semi-simple (in the sense of the Jacobson radical) PI-algebra can be expanded into a subdirect sum of complete matrix algebras over skew-fields, the orders of the matrix algebras and the dimensions of the skew-fields over the centres being bounded in the set, and the $ T $- ideal of identities of the semi-simple algebra coinciding with some "matrix" $ T $- ideal of $ M _ {n} $. An ordered PI-algebra is commutative. A primary PI-algebra $ A $( cf. Primary ring) has a two-sided classical quotient ring $ Q( A) $, which is isomorphic to a matrix algebra $ D _ {m} $ over a skew-field $ D $, the latter being finite-dimensional over its centre $ Z $. The ring $ Q( A) $ is a central extension of the algebra $ A $ in the sense that $ Q( A) = AZ $. The ideals of identities of the algebras $ A $ and $ Q( A) $ are the same. PI-algebras satisfy a number of conditions of Burnside type (cf. Burnside problem). For instance, an algebraic (nil) PI-algebra is locally finite (locally nilpotent). An associative nil algebra of bounded index $ n $ is nilpotent if the characteristic of the ground field is zero or larger than $ n $.

A PI-algebra without non-zero nil ideals is representable by matrices over a commutative ring. However, not all PI-algebras are representable in this way. For example, the exterior algebra of a countably-dimensional space is not so representable, since it does not satisfy any standard identity. The internal characterization of the representability of an algebra by matrices over a commutative ring is an independent branch of study in the theory of PI-algebras.

The Jacobson radical of a finitely-generated PI-algebra over a field of characteristic zero is a nil ideal. At the time of writing (1977) the question of its nilpotency is still open. If the Jacobson radical of a PI-algebra is nilpotent, this algebra satisfies all the identities of a matrix algebra of order $ n $ for some value of $ n $. The converse proposition has been demonstrated for finitely-generated algebras. Moreover, for a finitely-generated algebra over a field of characteristic zero, nilpotency of the Jacobson radical is equivalent to the validity of some standard identity in this algebra.

If an identity is satisfied for "a part" of the elements of an algebra, it follows in many cases that some identity is satisfied for all elements of the algebra. For instance, if the symmetric elements in an algebra with involution (cf. Involution algebra) satisfy an identity, the algebra is a PI-algebra; if a finite group of automorphisms acts on an algebra over a field of characteristic zero and if an invariant subalgebra satisfies a given identity, the initial algebra will be a PI-algebra.

It is interesting to inquire into the conditions under which given special algebras satisfy a polynomial identity.

For the group algebra $ F( G) $ of a group $ G $ over a field of characteristic zero to satisfy some polynomial identity it is necessary and sufficient that $ G $ have an Abelian subgroup of finite index. If the characteristic of $ F $ is finite and equal to $ p $, $ F [ G] $ is a PI-algebra if and only if $ G $ has a $ p $- Abelian subgroup of finite index (a group is said to be $ p $- Abelian if its commutator is a finite $ p $- group).

The universal enveloping algebra $ U _ {L} $ of a Lie algebra $ L $ over a field $ F $ of characteristic zero is a PI-algebra if and only if $ L $ is Abelian (i.e. $ U _ {L} $ is commutative). If $ F $ is a field of a finite characteristic, $ U _ {L} $ will be a PI-algebra if and only if $ L $ has an Abelian ideal of finite codimension while the adjoint representation of the algebra $ L $ is of bounded algebraic degree.

All PI-subalgebras of a free associative algebra are commutative.

The theory of PI-algebras is a natural extension of commutative algebra. It contains deep and decisive analogues of theorems in commutative algebra, so that one may speak of the construction of non-commutative algebraic geometry.

Any finitely-generated PI-algebra with generators $ a _ {1} \dots a _ {k} $ over a field $ F $ satisfies the condition of boundedness of heights, i.e. there exists a finite number of words $ v _ {1} \dots v _ {m} $ in the generators $ a _ {i} $ and a positive integer $ h $ such that any word $ u $ in the generators $ a _ {i} $ is representable in $ A $ by a linear combination of the words

$$ v _ {i _ {1} } ^ {S _ {i _ {1} } } \dots v _ {i _ {d} } ^ {S _ {i _ {d} } } , $$

where $ d \leq h $, the composition of which with respect to $ a _ {i} $ coincides with the word $ u $. In the commutative case the generators $ a _ {i} $ themselves may be used as the words $ v _ {i} $. A free non-commutative affine ring is a quotient algebra

$$ \mathfrak A ( F , k , n ) = F [ x _ {1} \dots x _ {k} ] / M _ {n} , $$

where $ F [ x _ {1} \dots x _ {k} ] $ is a free algebra with a finite number of generators $ x _ {i} $ over a field $ F $ of characteristic zero while $ M _ {n} $ is a $ T $- ideal of identities of the matrix algebra $ F _ {n} $ as defined above. The algebra $ \mathfrak A ( F, k, n) $ is a PI-algebra without divisors of zero, and has classical skew-field of fractions $ D ( F, k, n) $ which is finite-dimensional over its centre $ Z $. Further, let $ {( F _ {n} ) } _ {k} $ be a space whose elements are columns of length $ k $, constituted by matrices of the algebra $ F _ {n} $. One may speak of the zeros of the elements of $ \mathfrak A ( F, k, n) $ that are located in $ {( F _ {n} ) } _ {k} $, of algebraic varieties in $ {( F _ {n} ) } _ {k} $, etc., the fundamental assumptions of classical algebraic geometry being satisfied. Thus, the non-commutative analogue of Hilbert's Nullstellensatz (cf. Hilbert theorem) is satisfied. Primary ideals of the algebra which meet the condition of being Noetherian correspond to irreducible algebraic varieties. Krull's theorem describing the coincidence of the maximum chain length of primary ideals of the algebra $ \mathfrak A ( F, k, n) $ with the degree of transcendence of $ Z $ over $ F $, which, in the present case, is

$$ kn ^ {2} - ( n ^ {2} - 1 ), $$

is satisfied.

By analogy with associative algebras it is possible to define, using the elements of free algebras, PI-algebras in other classes of algebras comprising free algebras (Lie algebras, alternative algebras, etc.).

A Lie algebra over a field of characteristic zero which satisfies the $ n $- th Engel identity

$$ [ x , y \dots y ] = 0 \ ( n \textrm{ factors } y ), $$

is locally nilpotent. Higgins' problem (namely, does Engel's identity imply nilpotency?) has been positively solved for $ n = 4 $ only. Its solution is negative for fields of positive characteristic.

References

[1]	C. Procesi, "Rings with polynomial identities" , M. Dekker (1973) MR0366968 Zbl 0262.16018
[2]	N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) MR0081264 Zbl 0073.02002
[3]	I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) MR1535024 MR0227205 Zbl 0177.05801
[4]	P.M. Cohn, "Universal algebra" , Reidel (1981) MR0620952 Zbl 0461.08001
[5]	A.I. Shirshov, "On rings with identity relations" Mat. Sb. , 43 (85) : 2 (1957) pp. 277–283 (In Russian)
[6]	A.I. Kostrikin, "On Burnside's problem" Izv. Akad. Nauk SSSR Ser. Mat. , 3 : 1 (1959) pp. 3–34 (In Russian) MR132100
[7]	G. Higman, "On a conjecture of Nagata" Proc. Cambridge Philos. Soc. (1) , 52 (1956) pp. 1–4 MR0073581 Zbl 0072.02502
[8]	P.J. Higgins, "Lie rings satisfying the Engel condition" Proc. Cambridge Philos. Soc. (1) , 50 (1954) pp. 8–15 MR0059890 Zbl 0055.02601

Comments

A.R. Kemer has shown that every variety of associative algebras over a field of characteristic zero is finitely based [a1].

References

[a1]	A.R. Kemer, "Solution of the problem as to whether associative algebras have a finite basis of identities" Soviet Math. Dokl. , 37 (1988) pp. 60–64 Dokl. Akad. Nauk SSSR , 298 (1988) pp. 273–277 MR0937115
[a2]	L.H. Rowen, "Polynomial identities in ring theory" , Acad. Press (1980) pp. Chapt. 7 MR0576061 Zbl 0461.16001