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Susan Montgomery: A Journey in Noncommutative Algebra

Yevgenia Kashina

Communicated by Notices Associate Editor Steven Sam

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Susan Montgomery was born in 1943 and grew up in Lansing, Michigan. She did her undergraduate studies at the University of Michigan where her advisor was J. E. McLaughlin, who inspired her interest in algebra. Having obtained her undergraduate degree in Mathematics in 1965, Susan received an NSF Graduate Fellowship and started her graduate studies at the University of Chicago. In 1969, she defended her Ph.D. thesis titled “The Lie Structure of Simple Rings with Involution of Characteristic 2” under the supervision of I. N. Herstein. She then spent one year at DePaul University. In 1970, Susan joined the faculty at the University of Southern California, where she currently is a professor.

Susan received multiple awards and recognition, including a John S. Guggenheim Memorial Foundation Fellowship in 1984, an Albert S. Raubenheimer Distinguished Faculty Award from the Division of Natural Sciences and Mathematics at USC in 1985, and a Gabilan Distinguished Professorship in Science and Engineering at USC in 2017–2020. In 2012, Susan was selected as an Inaugural Fellow of the AMS and elected as a Fellow of the American Association for the Advancement of Science.

Susan served on many professional society committees, including the AMS Board of Trustees in 1986–1996, the Board on Mathematical Sciences of the National Research Council in 1995–1998 and its Executive Committee in 1997–1998, and the AWM Scientific Advisory Committee in 2015–2017. Furthermore, she was elected as vice president of the AMS for the 2014–2017 term. Moreover, she served as the chair of the Mathematics Department at USC in 1996–1999.

While in her early work Susan studied rings with involutions and group actions on rings, her current research is focused on Hopf algebras and their representations. She has published two books and more than 100 research papers; in addition, she was a coeditor for seven collections of papers on various topics in algebra. Her monograph Hopf algebras and their actions on rings became the most cited book on Hopf algebras and quantum groups. Below we discuss Susan’s research accomplishments, focusing on several important topics.

1. Group Algebras and Rings with Involution

The first thesis problem on which Susan worked as a graduate student was to determine whether, in characteristic , left invertible elements of a group algebra are also right invertible. This property was observed by Kaplansky in characteristic ; his proof followed from results concerning von Neumann algebras. Montgomery was able to give a shorter proof of the characteristic  case, based on the properties of -algebras instead (see Mon69), but the characteristic  proof eluded her for almost two years. This question still remains open, even though many mathematicians have tried to solve it in the last fifty years.

Susan’s second thesis problem was on Lie simplicity of simple rings with involution of characteristic . Given an associative ring , one can introduce a Lie structure on  by defining the Lie product via for all , in . This definition immediately raises the question how the ideal structures of  as an associative ring and  as a Lie ring are connected. In particular, if  is simple as an associative ring, what can we say about its Lie simplicity? One of the motivations for such a study, given by Herstein, was to investigate whether the simplicity of four infinite families of simple, finite-dimensional Lie algebras defined as matrices is in fact a consequence of the simplicity of the associative matrix algebra over a field. In the 1950s, there was a series of papers where these questions were considered; in particular, Herstein and Baxter proved several results about the Lie structure of the skew-symmetric elements of a simple ring with involution of characteristic not equal to . The case of characteristic  was still unknown at the time; it then became the main topic of Susan’s thesis. We will now describe her results on Lie simplicity of simple rings with involution of characteristic , which were later published in Mon70. Note that, for many of the results discussed below, the rings were not required to have identity; however, for simplicity, we will always assume that is a ring with identity.

Recall that, if  are additive subgroups of , we say that  is a Lie ideal of  provided that , where is an additive subgroup of  generated by the commutators for all and . Let be a simple ring, that is,  has no proper nontrivial two-sided ideals; then its center  is a field. Assume, in addition, that has an involution , which is, by definition, a self-inverse anti-automorphism of , and that  fixes every element of . Let  and  be respectively the sets of symmetric and skew-symmetric elements of and let be the set of trace elements. It is easy to see that are additive subgroups of  and  is a Lie subring of . Herstein and Baxter obtained the following results on characterization of Lie ideals of  and in the case of characteristic different from :

Theorem 1.

Let  be a simple ring with involution of characteristic not  and assume that . Then

(1)

(Herstein) Every Lie ideal of  either is contained in  or contains .

(2)

(Baxter) Every proper Lie ideal of is contained in .

Note that, since  is fixed by the involution, the intersection of  and  is trivial. Thus every proper Lie ideal of is trivial and therefore the latter part of the theorem implies that is a simple Lie ring. Moreover, since the matrices over a field, with the transposition used for the involution, do not behave well, the condition is necessary.

In her thesis, Montgomery investigated the case of characteristic . It turned out that, in contrast to the case of characteristic not , the Lie ideals were now characterized in terms of , and not . Moreover, Montgomery showed that in this situation and therefore the principal difference is that previously the main object of study was the Lie square of , whereas in characteristic  the main object becomes the Lie cube of . Montgomery’s results can be summarized in the following theorem:

Theorem 2.

Assume that  is a simple ring with involution of characteristic  and that . Then

(1)

Every Lie ideal of  either is contained in  or contains .

(2)

Every Lie ideal of  either is contained in  or contains .

(3)

Every proper Lie ideal of is contained in .

In light of the Lie square of  being equal to the Lie cube of , the latter part of the theorem implies the following Lie simplicity result:

The methods used to prove the theorem involved applications of several results on polynomial identities; the degree of one of these identities was  which caused the condition on the dimension of  over . Then, in joint work with Lanski, Montgomery used similar methods to extend her results to prime rings of characteristic .

Montgomery spent most of the next few years studying rings with involution, in particular when properties of the ring were inherited by the symmetric elements. Note that one can introduce a Jordan structure on  by defining a new product via for all in ; then the set  of symmetric elements becomes its Jordan subring. Using this observation, Montgomery obtained many important results describing the connection between the ideal structure of  as an associative ring and  as a Jordan ring, as well as the ideal structure of the Jordan ring .

Figure 1.

Susan Montgomery at USC in 1971.

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2. Fixed Rings of Automorphism Groups of a Ring

Since the techniques used by Montgomery in her studies of rings with involution relied on polynomial identities, she was led to investigate the question of when having a subring satisfying a polynomial identity implies that the entire ring satisfies a polynomial identity. Recall that a ring satisfies a polynomial identity and is called a PI-ring if there is a polynomial with integral coefficients in noncommuting variables which vanishes under the substitutions from the ring; PI-rings generalize the class of rings which are finitely generated as modules over their center. In Mon74, Montgomery proved several results for the subrings being centralizers and, as an application, gave an affirmative answer to a question raised by Bjork; namely, she showed that under certain conditions the ring itself satisfies a polynomial identity if the fixed subring of an automorphism group satisfies a polynomial identity. A natural direction from this work was studying fixed rings  of automorphism groups  of a ring, in particular, the connection between the structure of the ring and its fixed subring. Another important problem is to investigate the relationship between the fixed ring  and the skew group ring .

Here, we say that a group  acts via automorphisms on a ring  if there exists a homomorphism from  to , the group of automorphisms of . Often this homomorphism is going to be injective, so that  could be considered as a subgroup of . For any , the action of  on  will be denoted by . An automorphism  of  is called inner if there exists a unit in  such that  acts on  via conjugation with this unit; otherwise  is called outer. A subgroup  of is called inner if every element of  is inner and it is called outer if its only inner element is . If  is finite, we say that  has no -torsion provided that implies .

Recall that the fixed subring  under the action of  is

The fixed ring  is guaranteed to be nontrivial under the following sufficient conditions: Either  has no -torsion and  is not nilpotent, as proved by Bergman and Isaacs in 1973, or  has no nilpotent elements, as shown by Kharchenko in 1975. For this reason, most of the following results will, implicitly or explicitly, rely on one of these hypotheses.

Another important ring to study in the situation of a group acting on a ring is the skew group ring , which extends the semidirect product for groups: a free -module with basis with multiplication defined via for in and in . Looking closely at this product formula, we can notice that the action of is responsible for interchanging and via . This construction can be extended to a more general notion of the crossed product with elements of the form , where elements from and commute by the same rule as before, but for a -cocycle , where is a group of units of . If the -cocycle is trivial then , a skew group ring, and if the group action is trivial then , a twisted group ring. Furthermore, if is a field and acts trivially on , then is called a group algebra.

Note that in the original definitions one has a right group action, denoted by , and the skew group multiplication defined via . We chose the current notation so that it matches the (left) Hopf algebra action and smash product defined in Section 3.

This raises an important question: Considering the three rings, , , and , if one of them is prime, semiprime, semisimple Artinian, primitive, semiprimitive, or satisfies polynomial identities, can we say the same about the other two and under what conditions? And what is the relationship between Jacobson radicals of these rings? Montgomery provided answers to many of these questions and her results and techniques motivated the work of other researchers.

All of the notions mentioned above are basic properties of noncommutative rings that one wants to understand to get an initial picture of the ring’s structure. Some of them reduce to well-known properties when the ring is commutative, for example, a commutative ring is primitive if and only if it is a field. Or, when a ring is Artinian (that is, satisfies the descending chain condition on ideals, which generalizes the notion of finite-dimensional algebras), the conditions of being simple, prime, and primitive are equivalent. A ring is called semiprimitive (or Jacobson semisimple) if its Jacobson radical is zero. A ring is semiprimitive if it is semisimple, that is, semisimple as a module over itself; when a ring is Artinian these two notions coincide. A ring is called prime provided that the zero ideal is a prime ideal (that is, if a product of two ideals is zero then one of them is zero) and is called semiprime provided that it has no nontrivial nilpotent ideals (in other words, if a square of an ideal is zero then the ideal itself is zero).

In 1980, Montgomery published a Springer Lecture Notes volume titled Fixed Rings of Finite Automorphism Groups of Associative Rings Mon80, in which she summarized the progress made in the field in the 1970s. We will now discuss some of her most important results in this area.

In 1973, Susan visited Israel, where she started a life-long collaboration with Miriam Cohen. In their first paper, they proved that, assuming that a ring  has no nilpotent ideals and no -torsion, is semisimple Artinian if  is semisimple Artinian. The converse of this result was shown by Levitzki in 1935, under the assumption of being an element of  (see Mon80, Theorem 1.15). In 1976, Montgomery proved a more general result, completely describing the relation between Jacobson radicals of the ring and its fixed subring: Assuming that  is a bijection on  (that is, if and there is no -torsion), , that is, the Jacobson radical of  is the intersection of  with the Jacobson radical of . Note that the assumption of  being a bijection can be replaced by the hypothesis of being an element of , as explained in Mon80, Theorem 1.14.

As mentioned before, the above results relied on  having no -torsion. Montgomery’s next goal was to establish a connection between the ring structures of , , and  with no assumptions about  acting on . It turns out that if  is simple with  or a direct sum of simple rings and if  is outer, the assumption of no -torsion can often be dropped. But if  is semiprime,  being outer is not enough, so further restrictions are necessary. In 1975, Kharchenko used the Martindale ring of quotients to generalize the definition of an inner automorphism of a ring, in a way restricting the definition of an outer automorphism (see Mon80, page 42):

Definition 3.
(1)

An automorphism  is -inner if there exists a nonzero  in such that for all in .

(2)

A subgroup  of is called -inner, if every element of  is -inner, and it is called -outer if its only -inner element is .

Note that if  is -outer then it is always outer, but the converse is not always true.

Kharchenko used this definition to prove that  is prime if  is prime and  is -outer. Montgomery then further developed the method of -inner automorphisms and investigated the connection between , , and  (see Mon80, Theorems 3.17, 5.3, 6.9, Corollary 6.10):

Theorem 4.

Let  be a ring with a finite group  of automorphisms.

(1)

Assuming is semiprime,  is also semiprime. Moreover,  is Goldie if and only if  is Goldie.

(2)

Assuming  is semiprime and  is -outer, both and  and are semiprime. If, in addition,  is Goldie, then  and  have the same PI degree.

(3)

Assuming  is prime and  is -outer, is also prime and  and  have the same PI degree.

The above results were then extended by Montgomery in her joint work with Fisher (see Mon80, Corollary 3.18):

Theorem 5.

Let  be a semiprime ring and let  be -outer.

(1)

If  is simple then  is simple.

(2)

If  is primitive and  is finite then is primitive.

(3)

If  is semiprimitive then is semiprimitive.

Furthermore, Fisher and Montgomery used the method of -inner automorphisms to prove a “Maschke-type” theorem, effectively answering the question of when the skew group ring is semiprime. Recall that, a classical theorem due to Maschke states that, for a field , the group algebra  is semisimple if and only if the characteristic of  does not divide . Generalizations of this fundamental result, often referred to as “Maschke-type” theorems, are extremely important; the Fisher–Montgomery theorem provides such generalization from group algebras to skew group rings (see Mon80, Theorem 7.4):

Theorem 6 (Fisher–Montgomery).

If  is finite and  is semiprime with no -torsion then is semiprime.

The question about the semiprimeness of a skew group ring naturally led to the one about the semiprimeness of a crossed product. In MP78, Montgomery and Passman obtained necessary and sufficient conditions for the crossed product to be prime or semiprime, assuming that the ring itself is prime. As a consequence, they proved that in characteristic , if  is prime, then is semiprime. Because -inner automorphisms played a very important role in describing the above-mentioned conditions, Montgomery and Passman followed this paper with a series of joint works in which they studied -inner automorphisms of various rings including group rings and crossed products.

In related work Mon81, Montgomery studied the connection between the prime ideals of  and , by passing through the skew group ring . In order to formalize this correspondence, she introduced certain equivalence relations on and , the sets of prime ideals of  and , and proved that the sets of equivalence classes are homeomorphic with respect to the quotient Zariski topology.

In a different direction, joint with Small, Montgomery extended Noether’s classical theorem on affine rings of invariants from the commutative to non-commutative case:

Theorem 7.

MS81, Theorems 1 and 2 Let be a Noetherian ring which is affine (that is, finitely generated) over a commutative Noetherian ring  and let  be a finite group of -automorphisms of . Then  is affine over  provided that one of the following conditions holds:

(1)

.

(2)

is a domain satisfying a polynomial identity and  is Noetherian.

The authors also provided examples when these results fail if either  is not a domain and or  is not Noetherian.

3. Hopf Algebras

3.1. Group actions, group gradings, and module algebras

Since group algebras provide the first example of Hopf algebras, in the beginning of the 1980s, Susan Montgomery got interested in these kinds of algebras and started to work on the topic with Miriam Cohen. As for an action of a group on a ring, one can define an action of a group on an algebra over a field  as a homomorphism from  to , the group of -linear automorphisms of . Having such an action of  on  is equivalent to  being a module algebra over the group algebra , and this notion can be generalized to the notion of an -module algebra over any Hopf algebra . In their first paper on the subject CM84, Cohen and Montgomery pointed out that, for a finite group , a grading of  by  is equivalent to  being a -module algebra, where  is the Hopf algebra dual to ; in this case it is possible to define a smash product , which we will discuss later. Similar to how the connection between the structures of , , and were studied in the previous section, Cohen and Montgomery investigated the relationship between , (the identity component of the graded algebra ), and , in particular, they obtained results about the Jacobson radical, prime ideals, and semiprimeness. Furthermore, the authors proved a Maschke-type theorem, analogous to the original Fisher–Montgomery theorem, but for group gradings instead of group actions (see CM84, Theorem 2.9). The most important results of the paper are the following duality theorems relating group actions and group gradings:

Theorem 8.

CM84, Theorems 3.2 and 3.5 Let  be a group of order  and  be an algebra.

(1)

If  acts on , then  is naturally -graded and .

(2)

If  is -graded, then has a natural -action and .

The ideas of duality were inspired partly by results in von Neumann algebras and -algebras. As an application, the authors showed that the graded Jacobson radical  is always contained in the usual Jacobson radical , proving a conjecture of Bergman on radicals of graded rings.

Having proved the duality theorems, Cohen and Montgomery asked the natural question whether the analogs of the duality theorems hold not only for group algebras and their duals, but also for other finite-dimensional Hopf algebras. In a series of papers, joint with Blattner and Cohen, Montgomery extended these ideas in several directions. Since then, Susan has worked almost exclusively on topics related to Hopf algebras. In 1992, Susan Montgomery was the Principle Lecturer at the Conference Board of the Mathematical Sciences conference on Hopf Algebras and, in 1993, she published the CBMS monograph Mon93 Hopf algebras and their actions on rings.

Before describing Montgomery’s results further, we will discuss the motivation behind these generalizations. A group algebra , in addition to being an algebra, has a coalgebra structure; in particular, there is a -linear map called comultiplication and defined via for and extended linearly. This additional structure, used implicitly, allows us to define a -action on a tensor product of two -modules via

that is, we see that a tensor product of two representations is again a representation. This property is not true anymore if the group algebra is replaced by an arbitrary algebra, but it can be extended from group algebras to Hopf algebras , using the comultiplication map . In this case the -module structure is defined on the tensor product of two -modules via

using the so-called Sweedler notation for the coproduct to write for .

For an algebra  over a field , we can treat its multiplication and unit as -linear maps and . When  is finite-dimensional, the notion of an algebra can be dualized in the following way: Consider the dual vector space of -linear maps from  to . This dual vector space has two -linear maps and ; these maps, called comultiplication and counit give  the structure of a coalgebra. We call  a bialgebra if it is both an algebra and a coalgebra and these two structures satisfy a compatibility condition, namely that comultiplication and counit are algebra maps. A bialgebra  becomes a Hopf algebra if in addition it has a map , which is called an antipode and satisfies requirements that generalize the ones for the inverse map in groups.

Historically, the first Hopf algebras studied were cocommutative, that is, the ones where for every element , the coproduct . Over an algebraically closed base field of characteristic , the only finite-dimensional cocommutative Hopf algebras are group algebras. Other examples of cocommutative Hopf algebras include universal enveloping algebras of Lie algebras and restricted Lie algebras. For any Hopf algebra  one can consider all elements  such that and , called group-like elements of . They are linearly independent, form a group, denoted by , and generate a cocommutative Hopf subalgebra of .

When  is both an algebra and an -module over a Hopf algebra , we say that  is an -module algebra, or, equivalently, that  is an algebra in the category of left -modules , if its multiplication and unit are -module maps, that is, and . Similarly to skew group rings arising when groups act on rings as automorphisms, for an -module algebra  the smash product algebra is defined to be as a vector space, but with multiplication

where the elements of are denoted by .

For example, for an algebra , graded by the group , the -action on  is defined via and , where is the dual basis of and for and . Since , the multiplication in is determined by and .

Figure 2.

Susan Montgomery and Bob Blattner visiting Munich in 1994.

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If  is a finite-dimensional Hopf algebra, then the dual vector space  is also a Hopf algebra. In particular, the multiplication in  is the map dual to the comultiplication in , and vice versa, as mentioned before. When  is not finite-dimensional,  is not a Hopf algebra anymore, but one can use the so-called finite dual  instead. In 1985, Susan Montgomery, in collaboration with her husband Bob Blattner, extended the results of CM84 from group algebras to infinite-dimensional Hopf algebras with bijective antipode, where  is replaced by a Hopf subalgebra  of the Hopf algebra , and proved that, under certain conditions,

(see BM85, Theorem 2.1). In order to obtain these results, Bob and Susan combined their perspectives from functional analysis and noncommutative algebra. The authors then discussed several applications of this theorem, in particular, in the case when  is a universal enveloping algebra of a finite-dimensional Lie algebra or a group algebra of a residually -linear FC-group. Furthermore, when  is a finite-dimensional Hopf algebra of dimension , this result yields the generalization of the duality results in Theorem 8 (see BM85, Corollary 2.7):

3.2. Crossed products

In the same way as smash products generalize skew group rings, the notion of a crossed product can be extended to the case when groups are replaced by Hopf algebras; such a crossed product of an algebra  with a Hopf algebra  is denoted by , where is an invertible cocycle. These general crossed products were introduced independently in 1986 by Blattner, Cohen, and Montgomery, and by Doi and Takeuchi (see Mon93, Chapter 7). Next, in a 1989 joint paper with Blattner, Montgomery continued her studies of crossed products and further extended the results of the previous papers. In particular, they generalized the duality theorems from smash to crossed products (see Mon93, Theorem 9.4.17):

Theorem 9.

Let  be a Hopf algebra of dimension  and be a crossed product. Then

Another direction was to extend the Maschke-type results to crossed products for semisimple Hopf algebras . All previous results were obtained either under the assumption that  is a group algebra or its dual or by imposing additional conditions on . Blattner and Montgomery proved a Maschke-type theorem by restricting the action instead (see Mon93, Theorem 7.4.7):

Theorem 10.

Let  be a semisimple Hopf algebra,  be a semiprime algebra, and be a crossed product. Then is semiprime if the action of  is inner.

Many ideas and results discussed in this section, as well as in Section 2, motivated Montgomery, together with Linchenko and Small, to investigate two related questions about -module algebras and smash products. In the first question they asked whether, for a semisimple Hopf algebra  and an -module algebra , the Jacobson radical is stable under the action of . As it was mentioned in the beginning of Subsection 3.1, for , it is equivalent to the question of Bergman about the graded Jacobson radical and it was answered positively in CM84. In 2001, Linchenko showed that it is true when  is finite-dimensional and the base field has either characteristic  or characteristic with additional condition of  being cosemisimple (that is,  being semisimple). In LMS05, Theorem 3.8, Linchenko, Montgomery, and Small proved that the answer is positive for any infinite-dimensional PI-algebra  which is either affine or algebraic over the base field of characteristic . In the case of positive characteristic they showed that the Jacobson radical is -stable under the additional assumptions of  being cosemisimiple and the characteristic being large enough compared to the dimension of  and the degree of the polynomial identity satisfied by . Note that, by the Larson-Radford theorem, in characteristic  semisimplicity and cosemisimplicity are equivalent.

The second question addressed in LMS05 was whether is semiprime provided that is semisimple and is -semiprime. This open question was asked by Cohen and Fischman in 1984, under the stronger hypothesis of being semiprime, and the positive answer would generalize the results of Fisher and Montgomery for and Cohen and Montgomery for . In LMS05, Theorems 2.8 and 2.11, Linchenko, Montgomery, and Small proved that these two questions are connected: First, given a finite-dimensional Hopf algebra , they established two conditions equivalent to the one that the Jacobson radical of every -module algebra is -stable. Then the authors proved that if the first question is answered positively for all -module algebras , then the prime radical of every -module algebra is -stable and is semiprime for all -semiprime -module algebras . In conclusion, they showed that is semiprimitive provided that is semisimple, is an -semiprime -module algebra satisfying a polynomial identity, and the base field has characteristic (in the case of positive characteristic some extra hypotheses on and were needed); in particular, under the above assumption, the second question was answered positively, since semiprimitiveness implies semiprimeness.

Figure 3.

Susan Montgomery with her mathematical siblings Gail Letzter, Daniel Farkas, Lance Small, and Lynne Small in Torrey Pines in 2014.

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3.3. Coalgebras and comodules

In the same fashion how dualizing the multiplication and unit of an algebra , as -linear maps led to the concept of a coalgebra , the notion of a -comodule is dual to the one of an -module. That is, treating the -action on a left -module as a -linear map , one can define a left -comodule via the coaction . The category of left -comodules is denoted by , while the category of right -comodules is denoted by . If  is finite-dimensional, then left -comodules are exactly right -modules; furthermore, for a group , a vector space is a -comodule if and only if it is -graded. Additional background on coalgebras and comodules can be found in Mon93, Chapter 5.

Note that, in the infinite-dimensional case, the dual vector space  of an algebra  is not always a coalgebra, since is larger than , and therefore there is no one-to-one correspondence between the theories of algebras and coalgebras. Nevertheless, by the fundamental theorem of coalgebras, any finite subset of elements of a coalgebra is contained in a finite-dimensional subcoalgebra, and, thus, every simple coalgebra is finite-dimensional. This fact led Montgomery, as well as the other researchers in the area, to working on the extension of the results from the theory of finite-dimensional algebras to general coalgebras.

In Mon95 Montgomery used the classical Brauer theorem about the indecomposable finite-dimensional algebras to prove the decomposition theorem for coalgebras. She considered the quiver  whose vertices are simple subcoalgebras of coalgebra , showed that it is isomorphic to the so-called Ext quiver whose vertices are the isomorphism classes of simple (right) -comodules, and proved that any coalgebra can be decomposed as a direct sum of indecomposable components, each of which corresponds to a connected component of . Montgomery then applied these results about coalgebras to prove that every pointed Hopf algebra, that is, the one for which every simple subcoalgebra is one-dimensional, can be decomposed as a crossed product. It was shown independently by Cartier and Gabriel and by Kostant in the early 1960’s that a pointed cocommutative Hopf algebra is a skew group ring of its group of group-like elements over the irreducible component of the identity element. For an arbitrary pointed Hopf algebra , Montgomery showed that , the indecomposable component containing , is a Hopf subalgebra of , the group of group-like elements acts on  via conjugation, and the group of group-like elements of , , is normal in . She then proved that  is isomorphic to the Hopf algebra , which has a structure of a crossed product with a certain cocycle as an algebra and a structure of a tensor product as a coalgebra.

Then, in CM97, Chin and Montgomery constructed, for a given coalgebra , an associated basic coalgebra  for which every simple subcoalgebra is the dual of some finite-dimensional division algebra and proved that categories of -comodules and -comodules are equivalent, that is, that  and  are Morita–Takeuchi equivalent. In particular, when the base field  is algebraically closed, every finite-dimensional division algebra and, therefore, every simple subcoalgebra of  is one-dimensional, implying that  is pointed. The authors then applied their results to path coalgebras, and showed that, over an algebraically closed base field, any coalgebra is equivalent to a large subcoalgebra of a path algebra of the Ext quiver.

3.4. Extensions

As crossed products, considered in Subsection 3.2, play a fundamental role in the theory of extensions, the next direction of Montgomery’s research was to study certain types of these extensions. In the joint paper with Blattner from 1989, she started working on Hopf Galois extensions, which were first introduced in 1969 by Chase and Sweedler for commutative algebras; the general definition was given by Kreimer and Takeuchi in 1980 (see Mon93, Chapter 8).

Recall that for an -module algebra , the algebra of -invariant elements  is defined via

extending the notion of a fixed subring  from Section 2. Dualizing, if  is a (right) -comodule algebra with coaction (that is, an algebra in the category ), then one can define , its algebra of -coinvariant elements, via

Using this terminology, is called a (right) -extension if  is a right -comodule algebra with and the -extension is called (right) -Galois provided that the canonical map defined by is bijective. This definition extends the notion of the classical Galois extensions as follows: Let be fields,  be a finite group acting as automorphisms of  fixing , and be the set of elements fixed by . Since  is a -module algebra, it becomes a -comodule algebra with . Then  is a classical Galois field extension with Galois group  if and only if is -Galois.

Important examples of -Galois extensions include , where is a right -comodule via coaction , but not every -Galois extension can be written as a crossed product. In fact, combining the results of Doi and Takeuchi and of Blattner and Montgomery, one can show that for an -extension , the algebra  is isomorphic to if and only if is -Galois with the so-called normal basis property (see Mon93, Corollary 8.2.5).

While group crossed products are transitive in the sense that if is a crossed product,  is a normal subgroup of , and , then there exists a cocycle such that , Hopf crossed products are not transitive in general. In order to state the transitivity problem, we first consider an exact sequence of Hopf algebras , where  is a normal Hopf subalgebra of  and is the quotient Hopf algebra for the Hopf ideal of , with . This sequence is called an extension of  by ; note, however, that not every quotient Hopf algebra arises from a normal Hopf subalgebra. Then, by an example of Schneider, it is not true in general that a crossed product , with being an extension of  by , can always be written as