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href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Over non-<a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative</a> <a class="existingWikiWord" href="/nlab/show/rings">rings</a> <a class="existingWikiWord" href="/nlab/show/determinants">determinants</a> are not useful invariants of <a class="existingWikiWord" href="/nlab/show/matrix">matrices</a> (in fact, various classical formulas for determinants mutually disagree in this case) and other <a class="existingWikiWord" href="/nlab/show/polynomial">polynomial</a> suggestions were not of much success (in some cases the <a class="existingWikiWord" href="/nlab/show/superdeterminant">superdeterminant</a> and <a class="existingWikiWord" href="/nlab/show/Dieudonn%C3%A9+determinant">Dieudonné determinant</a> are of use, but they can be easily expressed in terms of quasideterminants anyway). Quasideterminants will be <a class="existingWikiWord" href="/nlab/show/noncommutative+rational+function">noncommutative rational function</a>s, rather than polynomial, expressions.</p> <p>A quasideterminant generalizes a ratio of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n\times n</annotation></semantics></math>-determinant and a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)\times(n-1)</annotation></semantics></math> minor. Regarding that there are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>n</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">n^2</annotation></semantics></math> such minors – complementary to each entry– there are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>n</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">n^2</annotation></semantics></math> quasideterminants, indexed by labels of the complementary entry. Special cases when useful polynomial determinants are defined like the usual determinant, superdeterminant, quantum determinant and Dieudonné determinant can be obtained as products of quasideterminants.</p> <h2 id="definition">Definition</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">(</mo><msubsup><mi>a</mi> <mi>j</mi> <mi>i</mi></msubsup><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>M</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = (a^i_j)\in M_n(R)</annotation></semantics></math> be an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n\times n</annotation></semantics></math> matrix over an arbitrary noncommutative (but unital and associative) <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. In fact it makes sense to work with many objects (see <a class="existingWikiWord" href="/nlab/show/horizontal+categorification">horizontal categorification</a>): having, say, an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>a</mi> <mi>j</mi> <mi>i</mi></msubsup></mrow><annotation encoding="application/x-tex">a^i_j</annotation></semantics></math> is a morphism from the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> to the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>. Let us choose a row label <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> and a column label <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>. By <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>A</mi> <mover><mi>j</mi><mo stretchy="false">^</mo></mover> <mover><mi>i</mi><mo stretchy="false">^</mo></mover></msubsup></mrow><annotation encoding="application/x-tex">A^{\hat{i}}_{\hat{j}}</annotation></semantics></math> we’ll denote the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)\times(n-1)</annotation></semantics></math> matrix obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> by removing the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>-th row and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>-th column. The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,j)</annotation></semantics></math>-th <strong>quasideterminant</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>A</mi><msub><mo stretchy="false">|</mo> <mi>ij</mi></msub></mrow><annotation encoding="application/x-tex">|A|_{ij}</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>A</mi><msub><mo stretchy="false">|</mo> <mi>ij</mi></msub><mo>=</mo><msubsup><mi>a</mi> <mi>j</mi> <mi>i</mi></msubsup><mo>−</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>≠</mo><mi>i</mi><mo>,</mo><mi>l</mi><mo>≠</mo><mi>j</mi></mrow></munder><msubsup><mi>a</mi> <mi>l</mi> <mi>i</mi></msubsup><mo stretchy="false">(</mo><msubsup><mi>A</mi> <mover><mi>j</mi><mo stretchy="false">^</mo></mover> <mover><mi>i</mi><mo stretchy="false">^</mo></mover></msubsup><msubsup><mo stretchy="false">)</mo> <mi>lk</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msubsup><mi>a</mi> <mi>j</mi> <mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex"> |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 </annotation></semantics></math></div> <p>provided the right-hand side is defined (the corresponding inverses exist).</p> <h2 id="properties">Properties</h2> <p>Up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>n</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">n^2</annotation></semantics></math> quasideterminants of a given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msub><mi>M</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in M_n(R)</annotation></semantics></math> may be defined. If all the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>n</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">n^2</annotation></semantics></math> quasideterminants <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>A</mi><msub><mo stretchy="false">|</mo> <mi>ij</mi></msub></mrow><annotation encoding="application/x-tex">|A|_{ij}</annotation></semantics></math> exist and are invertible then the <a class="existingWikiWord" href="/nlab/show/inverse+matrix">inverse matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">A^{-1}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> exists in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M_n(R)</annotation></semantics></math> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>A</mi><msub><mo stretchy="false">|</mo> <mi>ji</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msubsup><mo stretchy="false">)</mo> <mi>j</mi> <mi>i</mi></msubsup><mo>.</mo></mrow><annotation encoding="application/x-tex"> (|A|_{ji})^{-1} = (A^{-1})^i_j. </annotation></semantics></math></div> <p>Quasideterminants for a matrix with entries in a commutative ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> are, up to a sign, ratios of the determinant of the matrix and the determinant of the appropriate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)\times (n-1)</annotation></semantics></math> 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.</p> <h2 id="references">References</h2> <p>Quasideterminants were introduced by I. Gel’fand and V. Retakh around 1990.</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/I.+M.+Gel%27fand">I. M. Gel'fand</a>, <a class="existingWikiWord" href="/nlab/show/V.+Retakh">V. S. Retakh</a>, <em>Determinants of matrices over noncommutative rings</em>, Funct.Anal.Appl. <strong>25</strong> (1991), no.2, pp. 91–102.</p> <p>engl. transl. <strong>21</strong> (1991), pp. 51–58.</p> </li> <li> <p>I.M. Gel’fand, V.S. Retakh, A theory of noncommutative determinants and characteristic functions of graphs, Funct.Anal.Appl. <strong>26</strong> (1992), no.4, pp. 231–246.</p> </li> <li> <p>I.M. Gel’fand, V.S. Retakh, <em>Quasideterminants I</em>, Selecta Mathematica, N. S. 3 (1997) no.4, pp. 517–546; <a href="https://doi.org/10.1007/s000290050019">doi</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Israel+Gelfand">Israel Gelfand</a>, Sergei Gelfand, <a class="existingWikiWord" href="/nlab/show/Vladimir+Retakh">Vladimir Retakh</a>, Robert Lee Wilson, <em>Quasideterminants</em>, Advances in Mathematics 193 (2005) 56–141 <a href="https://doi.org/10.1016/j.aim.2004.03.018">doi</a></p> </li> <li> <p>D.Krob, <a class="existingWikiWord" href="/nlab/show/Bernard+Leclerc">Bernard Leclerc</a>, <em>Minor identities for quasi-determinants and quantum determinants</em>, Comm. Math. Phys. 169 (1995) 1–23 <a href="https://doi.org/10.1007/BF02101594">doi</a> arXiv:<a href="https://arxiv.org/pdf/hep-th/9411194">hep-th/9411194</a></p> </li> <li> <p>Chapter 16: Quasideterminants and Cohn localization in <a class="existingWikiWord" href="/nlab/show/Z.+%C5%A0koda">Z. Škoda</a>, <em>Noncommutative localization in noncommutative geometry</em>, London Math. Society Lecture Note Series 330, ed. A. Ranicki; pp. 220–313, <a href="http://arxiv.org/abs/math/0403276">arXiv:math.QA/0403276</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/V.+Retakh">V. Retakh</a>, R. Wilson, <em>Advanced course on quasideterminants and universal localization</em> (2007) [<a class="existingWikiWord" href="/nlab/files/RetakhWilson-Quasideterminants.pdf" title="pdf">pdf</a>]</p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/algebra">algebra</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on July 8, 2024 at 15:09:12. 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