Supertrace
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Supertrace
In the theory of superalgebras, if ''A'' is a commutative superalgebra, ''V'' is a free right ''A''- supermodule and ''T'' is an endomorphism from ''V'' to itself, then the supertrace of ''T'', str(''T'') is defined by the following trace diagram: : More concretely, if we write out ''T'' in block matrix In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix w ... form after the decomposition into even and odd subspaces as follows, :T=\beginT_&T_\\T_&T_\end then the supertrace :str(''T'') = the ordinary trace of ''T''00 − the ordinary trace of ''T''11. Let us show that the supertrace does not depend on a basis. Suppose e1, ..., ep are the even basis vectors and e''p''+1, ..., e''p''+''q'' are the odd basis vectors. Then, the components of ''T'', which are elements of ''A'', are de ...
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Supermatrix
In mathematics and theoretical physics, a supermatrix is a Z2-graded analog of an ordinary matrix. Specifically, a supermatrix is a 2×2 block matrix with entries in a superalgebra (or superring). The most important examples are those with entries in a commutative superalgebra (such as a Grassmann algebra) or an ordinary field (thought of as a purely even commutative superalgebra). Supermatrices arise in the study of super linear algebra where they appear as the coordinate representations of a linear transformations between finite-dimensional super vector spaces or free supermodules. They have important applications in the field of supersymmetry. Definitions and notation Let ''R'' be a fixed superalgebra (assumed to be unital and associative). Often one requires ''R'' be supercommutative as well (for essentially the same reasons as in the ungraded case). Let ''p'', ''q'', ''r'', and ''s'' be nonnegative integers. A supermatrix of dimension (''r'', ''s'')&time ...
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Trace (matrix)
In linear algebra, the trace of a square matrix , denoted , is the sum of the elements on its main diagonal, a_ + a_ + \dots + a_. It is only defined for a square matrix (). The trace of a matrix is the sum of its eigenvalues (counted with multiplicities). Also, for any matrices and of the same size. Thus, similar matrices have the same trace. As a consequence, one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the row and column of . The entries of can be real numbers, complex numbers, or more generally elements of a field . The trace is not defined for non-square matrices. Example Let be a matrix, with \mathb ...
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Berezinian
In mathematics and theoretical physics, the Berezinian or superdeterminant is a generalization of the determinant to the case of supermatrices. The name is for Felix Berezin. The Berezinian plays a role analogous to the determinant when considering coordinate changes for integration on a supermanifold. Definition The Berezinian is uniquely determined by two defining properties: *\operatorname(XY) = \operatorname(X)\operatorname(Y) *\operatorname(e^X) = e^\, where str(''X'') denotes the supertrace of ''X''. Unlike the classical determinant, the Berezinian is defined only for invertible supermatrices. The simplest case to consider is the Berezinian of a supermatrix with entries in a field ''K''. Such supermatrices represent linear transformations of a super vector space over ''K''. A particular even supermatrix is a block matrix of the form :X = \beginA & 0 \\ 0 & D\end Such a matrix is invertible if and only if both ''A'' and ''D'' are invertible matrices over ''K''. The Berez ...
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Superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. The prefix ''super-'' comes from the theory of supersymmetry in theoretical physics. Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, supermanifolds and superschemes. Formal definition Let ''K'' be a commutative ring. In most applications, ''K'' is a field of characteristic 0, such as R or C. A superalgebra over ''K'' is a ''K''-module ''A'' with a direct sum decomposition :A = A_0\oplus A_1 together with a bilinear multiplication ''A'' × ''A'' → ''A'' ...
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Commutative Superalgebra
In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements ''x'', ''y'' we have :yx = (-1)^xy , where , ''x'', denotes the grade of the element and is 0 or 1 (in Z) according to whether the grade is even or odd, respectively. Equivalently, it is a superalgebra where the supercommutator : ,y= xy - (-1)^yx always vanishes. Algebraic structures which supercommute in the above sense are sometimes referred to as skew-commutative associative algebras to emphasize the anti-commutation, or, to emphasize the grading, graded-commutative or, if the supercommutativity is understood, simply commutative. Any commutative algebra is a supercommutative algebra if given the trivial gradation (i.e. all elements are even). Grassmann algebras (also known as exterior algebras) are the most common examples of nontrivial supercommutative algebras. The supercenter of any superalgebra is the set of elements that sup ...
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Supermodule
In mathematics, a supermodule is a Z2-graded module over a superring or superalgebra. Supermodules arise in super linear algebra which is a mathematical framework for studying the concept supersymmetry in theoretical physics. Supermodules over a commutative superalgebra can be viewed as generalizations of super vector spaces over a (purely even) field (mathematics), field ''K''. Supermodules often play a more prominent role in super linear algebra than do super vector spaces. These reason is that it is often necessary or useful to extend the field of scalars to include odd variables. In doing so one moves from fields to commutative superalgebras and from vector spaces to modules. :''In this article, all superalgebras are assumed be associative and unital algebra, unital unless stated otherwise.'' Formal definition Let ''A'' be a fixed superalgebra. A right supermodule over ''A'' is a right module ''E'' over ''A'' with a direct sum of modules, direct sum decomposition (as an abelia ...
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Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group is a group homomorphism . In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set ''S'' to itself. In any category, the composition of any two endomorphisms of is again an endomorphism of . It follows that the set of all endomorphisms of forms a monoid, the full transformation monoid, and denoted (or to emphasize the category ). Automorphisms An invertible endomorphism of is called an automorphism. The set of all automorphisms is a subset of with a group structure, called the automorphism group of and denoted . In the following diagram, the arrows denote implication: Endomorphism rings Any two endomorphisms of an abelian group, , can be ad ...
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Trace Diagram
In mathematics, trace diagrams are a graphical means of performing computations in linear algebra, linear and multilinear algebra. They can be represented as (slightly modified) graph theory, graphs in which some edges are labeled by matrix (mathematics), matrices. The simplest trace diagrams represent the trace (linear algebra), trace and determinant of a matrix. Several results in linear algebra, such as Cramer's Rule and the Cayley–Hamilton theorem, have simple diagrammatic proofs. They are closely related to Penrose's graphical notation. Formal definition Let ''V'' be a vector space of vector space, dimension ''n'' over a field (mathematics), field ''F'' (with ''n''≥2), and let Hom(''V'',''V'') denote the linear transformations on ''V''. An ''n''-trace diagram is a Graph (discrete mathematics), graph \mathcal=(V_1\sqcup V_2\sqcup V_n, E), where the sets ''V''''i'' (''i'' = 1, 2, ''n'') are composed of Vertex (graph theory), vertices of Degree (graph ...
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