Quasideterminant Explained
In mathematics, the quasideterminant is a replacement for the determinant for matrices with noncommutative entries. Example 2 × 2 quasideterminants are as follows:
\left|\begin{array}{cc}a11&a12\\
a21&a22\end{array}
\right|11=a11-a12{a22
}^a_\qquad \left|\begin a_ & a_ \\ a_ & a_ \end \right|_ = a_ - a_^a_.
In general, there are n2 quasideterminants defined for an n × n matrix (one for each position in the matrix), but the presence of the inverted terms above should give the reader pause: they are not always defined, and even when they are defined, they do not reduce to determinants when the entries commute. Rather,
\left|A\right|ij=(-1)i+j
,
where
means delete the
ith row and
jth column from
A.
The
examples above were introduced between 1926 and 1928 by
Richardson[1] [2] and Heyting,
[3] but they were marginalized at the time because they were not polynomials in the entries of
. These examples were rediscovered and given new life in 1991 by
Israel Gelfand and
Vladimir Retakh.
[4] [5] There, they develop quasideterminantal versions of many familiar determinantal properties. For example, if
is built from
by rescaling its
-th row (on the left) by
, then
\left|B\right|ij=\rho\left|A\right|ij
. Similarly, if
is built from
by adding a (left) multiple of the
-th row to another row, then
\left|B\right|ij=\left|A\right|ij(\forallj;\forallk ≠ i)
. They even develop a quasideterminantal version of
Cramer's rule.
Definition
Let
be an
matrix over a (not necessarily commutative) ring
and fix
. Let
denote the (
)-entry of
, let
denote the
-th row of
with column
deleted, and let
denote the
-th column of
with row
deleted. The (
)-quasideterminant of
is defined if the submatrix
is invertible over
. In this case,
\left|A\right|ij=aij-
l(Aijr)-1
.
Recall the formula (for commutative rings) relating
to the determinant, namely
. The above definition is a generalization in that (even for noncommutative rings) one has
whenever the two sides makes sense.
Identities
One of the most important properties of the quasideterminant is what Gelfand and Retakh call the "heredity principle". It allows one to take a quasideterminant in stages (and has no commutative counterpart). To illustrate, suppose
\left(\begin{array}{cc}A11&A12\\
A21&A22\end{array}
\right)
is a
block matrix decomposition of an
matrix
with
a
matrix. If the (
)-entry of
lies within
, it says that
\left|A\right|ij=\left|A11-A12{A22
}^\,A_\right|_.That is, the quasideterminant of a quasideterminant is a quasideterminant. To put it less succinctly: UNLIKE determinants, quasideterminants treat matrices with block-matrix entries no differently than ordinary matrices (something determinants cannot do since block-matrices generally don't commute with one another). That is, while the precise form of the above identity is quite surprising, the existence of
some such identity is less so. Other identities from the papers
[4] [5] are (i) the so-called "homological relations", stating that two quasideterminants in a common row or column are closely related to one another, and (ii) the
Sylvester formula.
(i) Two quasideterminants sharing a common row or column satisfy
\left|A\right|ij|Ail
=-\left|A\right|il|Aij
or
|Akj
\left|A\right|ij=-|Aij
\left|A\right|kj,
respectively, for all choices
,
so that the quasideterminants involved are defined.
(ii) Like the heredity principle, the Sylvester identity is a way to recursively compute a quasideterminant. To ease notation, we display a special case. Let
be the upper-left
submatrix of an
matrix
and fix a coordinate (
) in
. Let
be the
matrix, with
defined as the (
)-quasideterminant of the
matrix formed by adjoining to
the first
columns of row
, the first
rows of column
, and the entry
. Then one has
\left|B\right|ij=\left|A\right|ij.
Many more identities have appeared since the first articles of Gelfand and Retakh on the subject, most of them being analogs of classical determinantal identities. An important source is Krob and Leclerc's 1995 article.[6] To highlight one, we consider the row/column expansion identities. Fix a row
to expand along. Recall the determinantal formula
\detA=\suml(-1)i+lail ⋅ \detAil
. Well, it happens that quasideterminants satisfy
\left|A\right|ij=aij-\suml ≠ ail ⋅ |Aij
|Ail|kj
(expansion along column
), and
\left|A\right|ij=aij-\sumk ≠ |Akj|il|Aij
⋅ akj
(expansion along row
).
Connections to other determinants
The quasideterminant is certainly not the only existing determinant analog for noncommutative settings - perhaps the most famous examples are the Dieudonné determinant and quantum determinant. However, these are related to the quasideterminant in some way. For example,
{\det}qA=l|Ar|11\left|A11\right|22\left|A12,12\right|33 … |ann|nn,
with the factors on the right-hand side commuting with each other. Other famous examples, such as
Berezinians,
Moore and Study determinants,
Capelli determinants, and Cartier-Foata-type determinants are also expressible in terms of quasideterminants. Gelfand has been known to define a (noncommutative) determinant as "good" if it may be expressed as products of quasiminors.
Applications
Paraphrasing their 2005 survey article with Sergei Gelfand and Robert Wilson,[7] Israel Gelfand and Vladimir Retakh advocate for the adoption of quasideterminants as "a main organizing tool in noncommutative algebra, giving them the same role determinants play in commutative algebra." Substantive use has been made of the quasideterminant in such fields of mathematics as integrable systems,[8] [9] representation theory,[10] [11] algebraic combinatorics,[12] the theory of noncommutative symmetric functions,[13] the theory of polynomials over division rings,[14] and noncommutative geometry.[15] [16] [17]
Several of the applications above make use of quasi-Plücker coordinates, which parametrize noncommutative Grassmannians and flags in much the same way as Plücker coordinates do Grassmannians and flags over commutative fields. More information on these can be found in the survey article.[7]
See also
Notes and References
- Archibald Read . Richardson . Archibald Read Richardson. Hypercomplex determinants. Messenger of Mathematics. 55. 1926. 145–152.
- Archibald Read . Richardson . Archibald Read Richardson. Simultaneous linear equations over a division algebra. Proceedings of the London Mathematical Society. 28. 1928. 1 . 395–420. 10.1112/plms/s2-28.1.395.
- Arend . Heyting. Die theorie der linearen gleichungen in einer zahlenspezies mit nichtkommutativer multiplikation. Mathematische Annalen. 98. 1928. 1. 465–490. 10.1007/BF01451604 . free.
- Israel . Gelfand . Israel Gelfand. Vladimir . Retakh . Vladimir Retakh. Determinants of matrices over noncommutative rings. Functional Analysis and Its Applications. 25. 1991. 2 . 91–102. 10.1007/BF01079588 . free.
- Israel . Gelfand . Israel Gelfand. Vladimir . Retakh . Vladimir Retakh. A theory of noncommutative determinants, and characteristic functions of graphs. Functional Analysis and Its Applications. 26. 1992. 4 . 231–246. 10.1007/BF01075044 . free.
- Daniel . Krob. Bernard . Leclerc. Minor identities for quasi-determinants and quantum determinants. Communications in Mathematical Physics. 169. 1995. 1. 1–23. 10.1007/BF02101594. hep-th/9411194 . 1995CMaPh.169....1K. free.
- Israel . Gelfand . Israel Gelfand. Sergei . Gelfand. Vladimir . Retakh . Vladimir Retakh. Robert Lee . Wilson. Quasideterminants. Advances in Mathematics. 193. 2005. 1 . 56–141. 10.1016/j.aim.2004.03.018 . free. math/0208146.
- Pavel . Etingof. Israel . Gelfand. Vladimir . Retakh. Nonabelian integrable systems, quasideterminants, and Marchenko lemma. Mathematical Research Letters. 5. 1998. 1. 1–12. 10.4310/MRL.1998.v5.n1.a1 . free. q-alg/9707017.
- Claire R. . Gilson. Jonathan J.C. . Nimmo. C.M. . Sooman. On a direct approach to quasideterminant solutions of a noncommutative modified KP equation. Journal of Physics A: Mathematical and Theoretical. 41. 2008. 8. 085202. 0711.3733. 10.1088/1751-8113/41/8/085202. 2008JPhA...41h5202G. 14109958.
- A. Molev, Yangians and their applications, in Handbook of algebra, Vol. 3, North-Holland, Amsterdam, 2003. (eprint)
- Jonathan . Brundan. Alexander . Kleshchev. Parabolic presentations of the Yangian
. Communications in Mathematical Physics. 254. 2005. 1. 191–220. math/0407011. 10.1007/s00220-004-1249-6 . 2005CMaPh.254..191B. free.
- Matjaž . Konvalinka. Igor . Pak . Igor Pak. Non-commutative extensions of the MacMahon Master Theorem. Advances in Mathematics. 216. 2007. 1. 29–61. math/0607737. 10.1016/j.aim.2007.05.020 . free.
- Israel . Gelfand . Israel Gelfand. Daniel . Krob. Alain . Lascoux. Bernard . Leclerc. Vladimir . Retakh . Vladimir Retakh. Jean-Yves . Thibon. Noncommutative Symmetric Functions. Advances in Mathematics. 112. 1995. 2 . 218–348. 10.1006/aima.1995.1032 . free. hep-th/9407124.
- Israel Gelfand, Vladimir Retakh, Noncommutative Vieta theorem and symmetric functions. The Gelfand Mathematical Seminars, 1993–1995.
- Zoran Škoda, Noncommutative localization in noncommutative geometry, in "Non-commutative localization in algebra and topology", London Math. Soc. Lecture Note Ser., 330, Cambridge Univ. Press, Cambridge, 2006. (eprint)
- Aaron . Lauve. Quantum and quasi-Plücker coordinates. Journal of Algebra. 296. 2006. 2. 440–461. math/0406062. 10.1016/j.jalgebra.2005.12.004 . free.
- Arkady . Berenstein. Vladimir . Retakh . Vladimir Retakh. Noncommutative double Bruhat cells and their factorizations. International Mathematics Research Notices. 2005. 477–516. 2005. 8. math/0407010. 10.1155/IMRN.2005.477. . 15154129.