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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 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 defined as :T(\mathbf_j)=\mathbf_i T^i_j.\, The grading of ''T''''i''''j'' is the sum of the gradings of ''T'', e''i'', e''j'' mod 2. A change of basis to e1', ..., ep', e(''p''+1)', ..., e(''p''+''q'')' is given by the supermatrix :\mathbf_=\mathbf_i A^i_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supermatrix
In mathematics and theoretical physics, a supermatrix is a Z2-graded analog of an ordinary matrix (mathematics), 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 (mathematics), 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 algebra, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace (matrix)
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proved that for any two matrices and . This implies that 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 th row and th column of . The entries of can be real numbers or (more generally) complex numbers. The trace is not def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Berezin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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'' such t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 supe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ''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 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 decomposition (as an abelian group) :E = E_0 \oplus E_1 such that multiplication by el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 added toge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Diagram
In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly modified) graphs in which some edges are labeled by matrices. The simplest trace diagrams represent the 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 dimension ''n'' over a field ''F'' (with ''n''≥2), and let Hom(''V'',''V'') denote the linear transformations on ''V''. An ''n''-trace diagram is a graph \mathcal=(V_1\sqcup V_2\sqcup V_n, E), where the sets ''V''''i'' (''i'' = 1, 2, ''n'') are composed of vertices of degree ''i'', together with the following additional structures: * a ''ciliation'' at each vertex in the graph, which is an explicit ordering of the adjacent edges at that ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
