Celebratio Mathematica

A. A. Albert: Algebras and their radicals and division algebras. Ph.D. thesis, University of Chicago, 1928. Advised by L. E. Dickson. MR 2611246 phdthesis

A. A. Albert: “A determination of all normal division algebras in sixteen units,” Trans. Am. Math. Soc. 31 : 2 (1929), pp. 253–260. An abstract was published in Bull. Am. Math. Soc. 34:4 (1928). MR 1501481 JFM 55.0090.04 article

The chief outstanding problem in the theory of linear associative algebras over an infinite field \( F \) is the determination of all division algebras. This problem is equivalent to that of the determination of all normal division algebras, or algebras \( A \) in which the only elements of \( A \) commutative with every element of \( A \) are the quantities of its reference field \( F \).

The order of a normal division algebra is the square of an integer. All normal division algebras in \( 1^2 \), \( 2^2 \) [Dickson 1927, p. 46], and \( 3^2 \) [Wedderburn 1921, p. 132] units have been determined. In this paper all normal division algebras in \( 4^2 \) units, the next case, are determined and shown to be the algebras of Cecioni [1923].

A. A. Albert and H. Hasse: “A determination of all normal division algebras over an algebraic number field,” Trans. Am. Math. Soc. 34 : 3 (1932), pp. 722–726. An abstract was published in Bull. Am. Math. Soc. 38:3 (1932). MR 1501659 Zbl 0005.05003 article

A. A. Albert: “On a certain algebra of quantum mechanics,” Ann. Math. (2) 35 : 1 (January 1934), pp. 65–73. MR 1503142 JFM 60.0902.03 Zbl 0008.42104 article

P. Jordan, J. von Neumann, and E. Wigner [1934] have discussed certain linear real non-associative algebras of importance in quantum mechanics. Their algebras \( \mathfrak{M} \) satisfy the ordinary postulates for addition, the commutative law for multiplication, and the distributive law, but they are non-associative.

In the paper quoted above it is shown that, with a single exception, every algebra satisfying the above postulates is equivalent to an algebra \( \mathfrak{M} \) whose elements are ordinary real matrices \( x, y,\dots \) with products \( xy \) in \( \mathfrak{M} \) defined by quasi-multiplication, \[ xy = \tfrac{1}{2}(x\cdot y + y\cdot x), \] where \( x\cdot y \) is the ordinary matrix product. This single exception is the algebra \( \mathfrak{M}_3^8 \) of all three rowed Hermitian matrices with elements in the real non-associative algebra \( C \) of Cayley numbers.

The algebras obtained by quasi-multiplication of real matrices were considered in earlier papers so that, as stated by the above authors, this seemingly exceptional case, if proved really exceptional, is the only algebra of the above type which could lead to any new form of quantum mechanics.

In the present paper I shall prove that \( \mathfrak{M}_3^8 \) is a new algebra and that it is not equivalent to any algebra obtained by quasi-multiplication of real matrices. Moreover, I shall show that the relation \[ x(yx^2) = (xy)x^2 \] is satisfied by \( \mathfrak{M}_3^8 \).

A. A. Albert: Modern higher algebra. University of Chicago Press, 1937. JFM 63.0868.01 Zbl 0017.29201 book

A. A. Albert: Structure of algebras. AMS Colloquium Publications 24. American Mathematical Society (Providence, RI), 1939. MR 0000595 Zbl 0023.19901 book

A. A. Albert: “Non-associative algebras. II: New simple algebras,” Ann. Math. (2) 43 : 4 (October 1942), pp. 708–723. MR 0007748 Zbl 0061.04901 article

A. A. Albert: “Non-associative algebras. I: Fundamental concepts and isotopy,” Ann. Math. (2) 43 : 4 (October 1942), pp. 685–707. MR 0007747 Zbl 0061.04807 article

A. A. Albert: “A structure theory for Jordan algebras,” Ann. Math. (2) 48 : 3 (July 1947), pp. 546–567. MR 0021546 Zbl 0029.01003 article

A. A. Albert: “Absolute-valued algebraic algebras,” Bull. Am. Math. Soc. 55 : 8 (1949), pp. 763–768. MR 0030941 Zbl 0033.34901 article

An algebra \( \mathfrak{A} \) over a field \( \mathfrak{F} \) is a vector space over \( \mathfrak{F} \) which is closed with respect to a product \( xy \) which is linear in both \( x \) and \( y \). The product is not necessarily associative. Every element \( x \) of \( \mathfrak{A} \) generates a subalgebra \( \mathfrak{F}[x] \) of \( \mathfrak{A} \) and we call \( \mathfrak{A} \) an algebraic algebra if every \( \mathfrak{F}[x] \) is a finite-dimensional vector space over \( \mathfrak{F} \).

We have shown elsewhere [1947] that every absolute-valued real finite-dimensional algebra has dimension 1, 2, 4, or 8 and is either the field \( \mathfrak{R} \) of all real numbers, the complex field \( \mathfrak{C} \), the real quaternion algebra \( \mathfrak{Q} \), the real Cayley algebra \( \mathfrak{D} \), or certain isotopes without unity quantities of \( \mathfrak{Q} \) and \( \mathfrak{D} \). In the present paper we shall extend these results to algebraic algebras over \( \mathfrak{R} \) showing that every algebraic algebra over \( \mathfrak{R} \) with a unity quantity is finite-dimensional and so is one of the algebras listed above. The results are extended immediately to absolute-valued algebraic division algebras, that is, to algebras without unity quantities whose nonzero quantities form a quasigroup.

A. A. Albert: “A theory of power-associative commutative algebras,” Trans. Am. Math. Soc. 69 : 3 (November 1950), pp. 503–527. MR 0038959 Zbl 0039.26501 article

A. A. Albert: “On simple alternative rings,” Can. J. Math. 4 : 2 (1952), pp. 129–135. MR 0048420 Zbl 0046.25403 article

The only known simple alternative rings which are not associative are the Cayley algebras. Every such algebra has a scalar extension which is isomorphic over its center \( \mathsf{F} \) to the algebra \[ \mathsf{C} = e_{11}\mathsf{F} + e_{00}\mathsf{F} + \mathsf{C}_{10} + \mathsf{C}_{01} ,\] where \[ \mathsf{C}_{ij} = e_{ij}\mathsf{F} + f_{ij}\mathsf{F} + g_{ij}\mathsf{F} \qquad (i,j = 0,1;\ i \neq j) .\] The elements \( e_{11} \) and \( e_{00} \) are orthogonal idempotents and \[ e_{ii}x_{ij} = x_{ij}e_{jj} = x_{ij},\quad e_{jj}x_{ij} = x_{ij}e_{ii} = 0,\quad x_{ij}^2 = 0 \] for every \( x_{ij} \) of \( \mathsf{C}_{ij} \). The multiplication table of \( \mathsf{C} \) is then completed by the relations \begin{gather*} f_{10}g_{10} = e_{01},\quad g_{10}e_{10} = f_{01},\quad e_{10}f_{10} = g_{01},\\ g_{01}f_{01} = e_{10},\quad e_{01}g_{01} = f_{10},\quad f_{01}e_{01} = g_{10},\\ e_{ij}e_{ji} = f_{ij}f_{ji} = g_{ij}e_{ji} = e_{ii},\\ e_{ij}f_{ji} = e_{ij}g_{ji} = f_{ij}e_{ji} = f_{ij}g_{ji} = g_{ij}e_{ij} = g_{ij}f_{ij} = 0. \end{gather*}

Every simple alternative ring which contains an idempotent not its unity quantity is either associative or is the Cayley algebra \( \mathsf{C} \).

A. A. Albert: “On nonassociative division algebras,” Trans. Am. Math. Soc. 72 (1952), pp. 296–309. MR 0047027 Zbl 0046.03601 article

A. A. Albert: “On associative division algebras,” Bull. Am. Math. Soc. 74 (1968), pp. 438–454. Retiring Presidential Address delivered at the 74th annual meeting of the AMS on January 23, 1967, in San Francisco. MR 0222114 Zbl 0157.08001 article

We shall study the structure of a central division algebra \( \mathfrak{D} \), of odd prime degree \( p \) over any field \( \mathfrak{F} \) of characteristic \( p \), which has the property that there exists a quadratic extension field \( \mathfrak{K} \) of \( \mathfrak{F} \) such that the algebra \( \mathfrak{D}_0 = \mathfrak{D}\times\mathfrak{K} \) is cyclic over \( \mathfrak{K} \). We shall obtain a simplified version of the condition that a cyclic algebra \( \mathfrak{D}_0 \), of degree \( p \) over \( \mathfrak{K} \), shall possess the factorization property \( \mathfrak{D}_0 = \mathfrak{D}\times\mathfrak{K} \). We shall also derive a new sufficient condition that such a \( \mathfrak{D} \) shall be cyclic over \( \mathfrak{F} \), and shall present a large class of our algebras \( \mathfrak{D}_0 \) which satisfy this condition.

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A. Adrian Albert

Abraham Adrian Albert: 1905–1972

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