nLab quasideterminant

Idea

Over ground rings which are not commutative, the notion of determinants does not provide useful invariants of matrices (in fact, various classical formulas for determinants mutually disagree in this case).

The notion of quasideterminants is meant to fix this: these are noncommutative rational functions of the given matrix entries rather than polynomials.

Given an $n \times n$ matrix, its quasideterminants may be thought of as generalized ratios of an $n\times n$-determinant by an $(n-1)\times (n-1)$ minor. Since there are $n^2$ such minors – one complementary to each entry – there are $n^2$ quasideterminants.

Ordinary determinants, but also superdeterminants, quantum determinants and Dieudonné determinants can be expressed as products of quasideterminants.

Definition

Let $A = (a^i_j)\in M_n(R)$ be an $n\times n$ matrix over an arbitrary noncommutative (but unital and associative) ring $R$. In fact it makes sense to work with many objects (see horizontal categorification): having, say, an abelian category where $a^i_j$ is a morphism from the object $i$ to the object $j$. Let us choose a row label $i$ and a column label $j$. By $A^{\hat{i}}_{\hat{j}}$ we’ll denote the $(n-1)\times(n-1)$ matrix obtained from $A$ by removing the $i$-th row and the $j$-th column. The $(i,j)$-th quasideterminant $|A|_{ij}$ is

$$|A|_{ij} = a^i_j - \sum_{k \neq i, l\neq j} a^i_l (A^{\hat{i}}_{\hat{j}})^{-1}_{lk} a^k_j$$

provided the right-hand side is defined (the corresponding inverses exist).

Properties

Up to $n^2$ quasideterminants of a given $A \in M_n(R)$ may be defined. If all the $n^2$ quasideterminants $|A|_{ij}$ exist and are invertible then the inverse matrix $A^{-1}$ of $A$ exists in $M_n(R)$ and

$$\big( {|A|}_{ji}\big)^{-1} = (A^{-1})^i_j.$$

Quasideterminants for a matrix with entries in a commutative ring $R$ are, up to a sign, ratios of the determinant of the matrix and the determinant of the appropriate $(n-1)\times (n-1)$ submatrix, as one can see by remembering the formula for matrix inverse in terms of the cofactor matrices. For systems of linear equations with coefficients (from one side) in a noncommutative ring, there are solution formulas involving quasideterminants and generalizing Cramer’s rule (hence left and right Cramer’s rule). Quasideterminants with generic entries (entries in a free skewfield) satisfy generalizations of many classical identities: Muir’s law of extensionality, Silvester’s law, etc., and a new “heredity” principle and so-called “homological identities”. Furthermore, for polynomial equations over noncommutative rings, noncommutative Viete’s formulas have been found and used in applications.

References

The original articles:

• Israel M. Gel'fand, Vladimir S. Retakh: Determinants of matrices over noncommutative rings, Funct. Anal. Appl. 25 2 (1991) 91–102 [doi:10.1007/BF01079588]

• Israel M. Gel'fand, Vladimir S. Retakh: A theory of noncommutative determinants and characteristic functions of graphs, Funct. Anal. Appl. 26 4 (1992) 231–246 [doi:10.1007/BF01075044]

• Israel M. Gel'fand, Vladimir S. Retakh: Quasideterminants I, Selecta Mathematica, N. S. 3 4 (1997) 517–546 [doi:10.1007/s000290050019]

Review:

• Israel M. Gelfand, Sergei Gelfand, Vladimir S. Retakh, Robert Lee Wilson: Quasideterminants, Advances in Mathematics 193 (2005) 56–141 [doi:10.1016/j.aim.2004.03.018]

• Vladimir S. Retakh, Robert Lee Wilson: Advanced course on quasideterminants and universal localization, CRM Barcelona, (2007) [pdf]

See also:

• Wikipedia: Quasideterminant

Further discussion:

• Israel Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir S. Retakh, J.-Y. Thibon: Noncommutative symmetric functions, Adv. in Math. 112 2 (1995) 218–348 [arXiv:hep-th/9407124, doi:10.1006/aima.1995.1032]

• Daniel Krob, Bernard Leclerc: Minor identities for quasi-determinants and quantum determinants, Comm. Math. Phys. 169 (1995) 1-23 [doi:10.1007/BF02101594, arXiv:hep-th/9411194]

• Zoran Škoda: Quasideterminants and Cohn localization, Chapter 16 in: Noncommutative localization in noncommutative geometry, London Math. Society Lecture Note Series 330 (2006) 220–310 [arXiv:math.QA/0403276, doi:10.1017/CBO9780511526381.015]

category: algebra