In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, an Okubo algebra or pseudo-octonion algebra is an 8-dimensional
non-associative algebra
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary operation, binary multiplication operation is not assumed to be associative operation, associative. That is, an algebraic structure ''A'' is a non-ass ...
similar to the one studied by
Susumu Okubo. Okubo algebras are
composition algebra
In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies
:N(xy) = N(x)N(y)
for all and in .
A composition algebra includes an involution ...
s,
flexible algebra In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity:
: a \bullet \left(b \bullet a\right) = \left(a \bullet b\right) \bullet a
for any two elements ''a'' and ''b'' of the s ...
s (''A''(''BA'') = (''AB'')''A''),
Lie admissible algebras, and
power associative, but are not associative, not
alternative algebra In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
*x(xy) = (xx)y
*(yx)x = y(xx)
for all ''x'' and ''y'' in the algebra.
Every associative algebra is o ...
s, and do not have an identity element.
Okubo's example was the algebra of 3-by-3
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
-zero complex matrices, with the product of ''X'' and ''Y'' given by ''aXY'' + ''bYX'' – Tr(''XY'')''I''/3 where ''I'' is the identity matrix and ''a'' and ''b'' satisfy ''a'' + ''b'' = 3''ab'' = 1. The
Hermitian elements form an 8-dimensional real
non-associative division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
. A similar construction works for any cubic alternative separable algebra over a field containing a primitive cube root of unity. An Okubo algebra is an algebra constructed in this way from the trace-zero elements of a degree-3
central simple algebra over a field.
[Max-Albert Knus, ]Alexander Merkurjev
Aleksandr Sergeyevich Merkurjev (russian: Алекса́ндр Сергее́вич Мерку́рьев, born September 25, 1955) is a Russian-American mathematician, who has made major contributions to the field of algebra. Currently Merkurjev ...
, Markus Rost
Markus Rost is a German mathematician who works at the intersection of topology and algebra. He was an invited speaker at the International Congress of Mathematicians in 2002 in Beijing, China. He is a professor at the University of Bielefeld.
He ...
, Jean-Pierre Tignol (1998) "Composition and Triality", chapter 8 in ''The Book of Involutions'', pp 451–511, Colloquium Publications v 44, American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
Construction of Para-Hurwitz algebra
Unital composition algebras are called Hurwitz algebras.
If the ground field is the field of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s and is
positive-definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite fu ...
, then is called a
Euclidean Hurwitz algebra.
Scalar product
If has characteristic not equal to 2, then a
bilinear form is associated with the quadratic form .
Involution in Hurwitz algebras
Assuming has a multiplicative unity, define involution and
right and left multiplication operators by
:
Evidently is an
involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
and preserves the quadratic form. The overline notation stresses the fact that complex and quaternion
conjugation
Conjugation or conjugate may refer to:
Linguistics
* Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the chang ...
are partial cases of it. These operators have the following properties:
* The involution is an antiautomorphism, i.e.
*
* , , where denotes the
adjoint operator
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule
:\langle Ax,y \rangle = \langle x,A^*y \rangle,
where ...
with respect to the form
* where
*
* , , so that is an
alternative algebra In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
*x(xy) = (xx)y
*(yx)x = y(xx)
for all ''x'' and ''y'' in the algebra.
Every associative algebra is o ...
These properties are proved starting from polarized version of the identity :
:
Setting or yields and . Hence . Similarly . Hence . By the polarized identity so . Applied to 1 this gives . Replacing by gives the other identity. Substituting the formula for in gives .
Para-Hurwitz algebra
Another operation may be defined in a Hurwitz algebra as
:
The algebra is a composition algebra not generally unital, known as a para-Hurwitz algebra.
[ In dimensions 4 and 8 these are para-quaternion][The term "para-quaternions" is sometimes applied to unrelated algebras.] and para-octonion algebras.[
A para-Hurwitz algebra satisfies][
:
Conversely, an algebra with a non-degenerate symmetric bilinear form satisfying this equation is either a para-Hurwitz algebra or an eight-dimensional ]pseudo-octonion algebra In abstract algebra, algebra, an Okubo algebra or pseudo-octonion algebra is an 8-dimensional algebra over a field, non-associative algebra similar to the one studied by Susumu Okubo. Okubo algebras are composition algebras, flexible algebras (''A'' ...
.[ Similarly, a ]flexible algebra In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity:
: a \bullet \left(b \bullet a\right) = \left(a \bullet b\right) \bullet a
for any two elements ''a'' and ''b'' of the s ...
satisfying
:
is either a Hurwitz algebra, a para-Hurwitz algebra or an eight-dimensional pseudo-octonion algebra.[
]
References
*
*
* Susumu Okubo & J. Marshall Osborn (1981) "Algebras with nondegenerate associative symmetric bilinear forms permitting composition", Communications in Algebra
''Communications in Algebra'' is a monthly peer-reviewed scientific journal covering algebra, including commutative algebra, ring theory, module theory, non-associative algebra (including Lie algebras and Jordan algebras), group theory, and alg ...
9(12): 1233–61, and 9(20): 2015–73 {{mr, id=0640611.
Composition algebras
Non-associative algebras