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# Susan Montgomery: A Journey in Noncommutative Algebra

Communicated by *Notices* Associate Editor Steven Sam

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 instead (see -algebrasMon69), 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 :

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: .

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 .

## 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 provided that -torsion 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 and -torsion 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 , with basis -module 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

Note that in the original definitions one has a right group action, denoted by

This raises an important question: Considering the three rings,

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

As mentioned before, the above results relied on

Note that if

Kharchenko used this definition to prove that

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

Furthermore, Fisher and Montgomery used the method of

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

In related work Mon81, Montgomery studied the connection between the prime ideals of

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:

The authors also provided examples when these results fail if either

## 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

The ideas of duality were inspired partly by results in von Neumann algebras and

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

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 so-called Sweedler notation for the coproduct to write

For an algebra

Historically, the first Hopf algebras studied were cocommutative, that is, the ones where for every element

When

where the elements of

For example, for an algebra

If

(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

### 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

Another direction was to extend the Maschke-type results to crossed products

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

The second question addressed in LMS05 was whether

### 3.3. Coalgebras and comodules

In the same fashion how dualizing the multiplication and unit of an algebra

Note that, in the infinite-dimensional case, the dual vector space

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

Then, in CM97, Chin and Montgomery constructed, for a given coalgebra

### 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

extending the notion of a fixed subring

Using this terminology,

Important examples of

While group crossed products are transitive in the sense that if

One of the advantages of studying Hopf Galois extensions rather than crossed products is that, unlike Hopf crossed products, faithfully flat Hopf Galois extensions are transitive, which enables the use of inductive arguments. This was proved by Montgomery and Schneider in their joint paper MS99: