#### by Daniel Zelinsky

Abraham Adrian Albert died on June 6, 1972. The world lost a renowned mathematician, a vigorous force for the advancement of mathematics, and a very warm and understanding human being. From his birth to his death, he was associated with Chicago. As an inveterate traveler, he left that city often, for far parts of the world, but he always returned. He was born in Chicago on November 9, 1905, he went to school in Chicago (except for two years when his family moved to Iron Mountain, Michigan), he did all his undergraduate and graduate work at the University of Chicago. After receiving his Ph.D., he left for three years at Princeton and at Columbia Universities, then returned to the University of Chicago where he was a faculty member until the end of his life. With this as his base, he worked in many mathematical centers at various times in his career: The Institute for Advanced Study in Princeton (1933-34), Universities of Brazil and Buenos Aires (1947), University of Southern California (1950), Yale University (1956-1957), University of California at Los Angeles (1958). He operated in Washington in many capacities, and in the International Mathematical Union. His most recent official trip was a visit to the USSR in 1971 as a guest of the Soviet Academy.

To his friends Professor Albert was known as Adrian. Many
mathematicians referred to him affectionately as A__\( ^3 \)__. He was the son
of a Jewish family that came to America from England. His father
insisted on a Jewish but not very religious training. Albert
distinguished himself early in his schools (Herzl and Marshall) on the
West Side of Chicago, where the intellectual competition from the
other budding scholars was keen. He spent four years earning his
Bachelor’s degree at the University of Chicago, but one year later he
had his Master’s degree, and a year after that, his Ph.D. In 1928, at
age 22, his Ph.D. dissertation already stamped him as one of the
outstanding algebraists of his day.

Those were the days when the mathematical leaders at the University of Chicago were L. E. Dickson in algebra and E. H. Moore in general topology. Dickson was Albert’s thesis advisor and is the one mainly responsible for steering Albert into the subject of algebras over fields, which is the subject that primarily concerned him throughout his career.

He was one of the early National Research Council Fellows (1928–29). This fellowship was the forerunner of the modern NSF Postdoctoral Fellowships (which unfortunately were discontinued recently) and has been held by some of the most famous American mathematicians.

The precocity continued. At the age 35, Albert was promoted to a full professorship at the University of Chicago (at that time it was virtually unheard of to hold such a position before the age of 40). Two years later he was elected to membership in the National Academy of Sciences, a 37-year old academician.

The list of other honors heaped on him, and of honorific duties he was asked to perform would run to more pages than this article. We mention just a sample: chairmanship (1958–1962) and deanship (1962–1971) at the University of Chicago, presidency of the American Mathematical Society (1965–66), trusteeship of the Institute for Advanced Study (1969–72), chairmanship of the International Mathematical Union’s organizing committee for the 1970 Congress in Nice, membership in the Brazilian and Argentine Academies of Sciences, several editorships, the Cole Prize in Algebra (1939), and three honorary degrees. He seemed to collect these honors with enthusiasm, and executed the duties with vigor.

Although Albeit worked on matrix theory, on quadratic forms, and other
aspects of algebra, there is no question that his central interest
was always the study of finite dimensional algebras over a field. In
the old days, they were called hypercomplex systems. They are finite
dimensional vector spaces with a multiplication that associates to
every two vectors in the space another vector, the product. A
suggestive example is the four-dimensional algebra of quaternions over
the field of real numbers. The classical Wedderburn theorems
essentially reduce the study of associative algebras over a field to
the classification of the division algebras (like the algebra of
quaternions, for example). Over any field __\( F \)__, a four-dimensional
division algebra with center __\( F \)__ must be an algebra of “generalized
quaternions” whose multiplication rules are much like the ordinary
quaternions: a basis 1, __\( i, j, ij \)__ with __\( ij = ji \)__ and __\( i^2 = a \)__
and __\( j^2 = \beta \)__ elements of __\( F \)__, which have no square roots in __\( F \)__
(but are not necessarily __\( - 1 \)__). If one wants to generalize to
dimensions higher than 4 there are two candidates: the cyclic algebras
and the still more general crossed product algebras. (A theorem
asserts that, in any case, the dimension of any central division
algebra is a perfect square.)
Wedderburn had already proved that
central division algebras of dimension 9 ate all cyclic algebras. In
Albert’s dissertation (1928) he proved that central division algebras
of dimension 16 are not necessarily cyclic algebras, but are always
crossed products. Although Albert’s theorem raised the obvious
question about algebras of dimension 25, 36, etc., his result has
stood without essential improvement or embellishment (though not for
lack of trying) until some nice, complementary, but still not
definitive results of
Amitsur and others in 1971.

This study put the young Albert in the center of what was to be one of
the major breakthroughs in the theory of algebras: the determination
of all central division algebras over the special field of rational
numbers, or more generally over any algebraic number field. In this
case, it turns out that they are all cyclic algebras — this is the
famous Hasse–Brauer–Noether Theorem (1931). An interesting article by
Hasse and
Albert
in the *Transactions of the American Mathematical
Society* (1932) traces the history of this theorem and relates the
story of Albert’s near miss. On the basis of his results on algebras
and some results announced by Hasse, Albert published some theorems
that nearly proved the big theorem, and he wrote Hasse about it.
Somehow the communication was bad, and the Brauer–Hasse–Noether
manuscript was submitted for publication without mention of Albert’s
independent contributions. The 1932 *Transactions* article shows that
in fact the big theorem follows from Albert’s results in just a few
lines.

Albert was hurt and disappointed by this incident. But the depth of that hurt could not compare with his feelings about the subsequent Nazi scourge which caused some important German mathematicians to begin distinguishing between “Aryan” and “Semitic” mathematics, and which resulted in the exodus of so many German scientists, Jews and non-Jews alike, including both Richard Brauer and Emmy Noether. Albert was invited to be a member of the Institute for Advanced Study in Princeton during its opening year in 1933–34. (Another distinguished member, who arrived that year and remained on a permanent basis was Albert Einstein.) This contact with Princeton was profitable for Albert. His associations with Lefschetz in particular resulted in one of Albert’s mathematical accomplishments that he always regarded with greatest pleasure, and for which he later won the American Mathematical Society’s Cole Prize in algebra. Already in 1929, Lefschetz had interested Albert in a major unsolved problem in the theory of algebraic functions, Riemann surfaces and Abelian varieties. In a series of papers (1929–1934) Albert produced a definitive solution. What was required was a classification of the algebraic correspondences of a Riemann surface (automorphisms of a complex curve). This had been reduced to the problem of finding the matrices that commute with a certain “Riemann matrix” of periods of basic Abelian integrals on the Riemann surface. These commuting matrices form an algebra, and in the basic cases, a central simple algebra over the rational number field. This version of the problem was right in the center of Albert’s special expertise, and he demolished it.

Later, he attacked the problem of general nonassociative algebras that are finite dimensional over a field. Almost single-handed he influenced a large number of young mathematicians to break this seemingly unpromising ground. Special algebras had been studied that were not associative but which obeyed axioms substituting for the associative law (Lie algebras, Jordan algebras, alternative algebras). Results like the Wedderburn theorems had been proved for some of them; in fact, the results for Lie algebras over the complex number field were proved by E. Cartan before Wedderburn obtained his corresponding theorems for the associative algebras. But Albert had the idea of using associative algebra theory to prove analogs of the Wedderburn theorems for quite arbitrary nonassociative algebras (even a nonassociative algebra has an associative “regular representation” algebra). It is a sign of his genius that he was actually able to develop a reasonable theory and also significantly influence the theory and applications of the special algebras we mentioned.

Albert’s style of algebra was almost inimitable. He had a diabolical facility with manipulation of identities — an enterprise in which most mathematicians founder, never being able to see the forest for the trees. Somehow, Albert could see through mazes of symbols to the inner workings of all those polynomials in several variables or multiplication tables of complicated algebras.

Mathematics was Albert’s great enthusiasm. It was impossible to associate with him for any length of time without feeling the vigor with which he pursued his theorems. He was always willing to talk about his latest mathematical exploits. When his son, Alan, was still very young, Albert insisted on explaining even to him his latest theorems, patiently describing the necessary ingredients to the intrigued schoolboy who had not yet formally seen any real mathematics.

Albert was a prolific author of textbooks, research treatises, and more than a hundred and thirty research papers, the last of which is due to appear soon.

A minor theme running through Albert’s life was his fascination with cameras, radios and other gadgets. I have always thought that this streak was responsible for his activity in the Applied Mathematics Group at Northwestern University during World War II and his association with the Rand Corporation (1951 and 1952), Southern California Applied Mathematics Project (1953–55; he was its chairman 1959–60) and the Institute for Defense Analysis (director, Communication Research Division 1961–62, trustee 1969–72). His principal contribution in these activities was to cryptanalysis and coding. He was also a lifelong aficionado of detective stories, which he devoured at an enormous rate.

To his friends, Professor Albert is remembered as a devoted family man. He married Frieda Davis in his second year of graduate study, and they shared a close relationship for all the subsequent forty-four years. They had two sons and a daughter and five grandchildren. (Tragically one son died of an illness at the age of 25.) Perhaps one should also count his 29 Ph.D. students whom he treated almost as members of his family.

He was always pleased to use his influence in Washington to improve the status of mathematicians in general, and he was willing to do the same for individual mathematicians whom he considered worthy. One of the more homey causes to which he lent the weight of his reputation was retaining an apartment building at the University of Chicago for visiting mathematics faculty and their families. There are families throughout the world that remember this little mathematical microcosm with pleasure.

Everyone who knew him will remember his vigorous but round, medium build, curly hair, and often boyish demeanor; but especially one must remember his great, pleased grin that he flashed to welcome news of new successes for any of his extended family anywhere in the world of mathematics.