We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n graded commutative associative algebra. The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, w…
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16 results for “Berezinian”
We study power expansions of the characteristic function of a linear operator in a -dimensional superspace . We show that traces of exterior powers of satisfy universal recurrence relations of period . `Underlying' recurrence relations hold in the Grothendieck ring of representations of $\GL(V)$. The…
We develop the theory of linear algebra over a (Z_2)^n-commutative algebra (n in N), which includes the well-known super linear algebra as a special case (n=1). Examples of such graded-commutative algebras are the Clifford algebras, in particular the quaternion algebra H. Following a cohomological approach, we introduc…
The paper addresses the expansion of Berezinian and super exterior powers, revealing new insights into supertraces.
problem The breakdown of classical volume elements in supergeometric settings and the need for generalized forms.
method Introduces and analyzes -forms, demonstrates the expansion of , and identifies supertraces.
result Intermediate expansions in annular regions encode supertraces of representations on vector spaces.
Let G be a Lie supergroup and H a closed subsupergroup. We study the unimodularity of the homogeneous supermanifold G/H, i.e. the existence of G-invariant sections of its Berezinian line bundle. To that end, we express this line bundle as a G-equivariant associated bundle of the principal H-bundle G over G/H. We also s…
We consider an example of tubes of hypersurfaces in Euclidean space and generalise the tube formula to supercase. By this we assign to a point of the hypersurface in superspace a rational characteristic function. Does this rational function appear when we calculate the zeta-function of an arithmetic variety?
Constructs Dirac generating operators for split Courant algebroids.
problem Defines Dirac generating operators for split Courant algebroids.
method Explicit construction of Dirac generating operators.
result Square of Dirac generating operator yields invariant.
Study super cluster algebras from super Plücker and Ptolemy relations.
problem Developing super cluster algebra structure in super Grassmannians.
method Analyzing super Plücker and Ptolemy relations, developing super cluster structure.
result New simple form of super Plücker relations for $\Gr_{r|1}(n|1)$.
An intrinsic description of the Hamilton-Cartan formalism for first-order Berezinian variational problems determined by a submersion of supermanifolds is given. This is achieved by studying the associated higher-order graded variational problem through the Poincaré-Cartan form. Noether theorem and examples from superfi…
Basic elements of integral calculus over algebras of iterated differential forms, are presented. In particular, defining complexes for modules of integral forms are described and the corresponding berezinians and complexes of integral forms are computed. Various applications and the integral calculus over the algebra $…
We consider differential operators on a supermanifold of dimension . We define non-degenerate operators as those with an invertible top coefficient in the expansion in the "superderivative" (which is the square root of the shift generator, the partial derivative in an even variable, with the help of an odd ind…
Various semigroups of noninvertible supermatrices of the special (antitriangle) shape having nilpotent Berezinian which appear in supersymmetric theories are defined and investigated. A subset of them continuously represents left and right zero semigroups and rectangular bands. The ideal properties of higher order rect…
We recall the main facts about the odd Laplacian acting on half-densities on an odd symplectic manifold and discuss a homological interpretation for it suggested recently by P. {Š}evera. We study the relationship of odd symplectic geometry with classical objects. We show that the Berezinian of a canonical transformatio…
The paper details local forms of morphisms in colored supermanifolds.
problem Understanding local forms of morphisms in colored supermanifolds.
method Detailed account of -differential calculus and local theorems.
result Detailed insights into local forms of morphisms in colored supermanifolds.
This paper extends foliation concepts to singular foliations using Lie -algebroids.
problem Cohomological obstruction to volume forms in singular foliations.
method Replacing singular foliations with universal Lie -algebroids to define modular class.
result Geometric meaning of modular class as an obstruction to universal Lie -algebroids.
Quantization of (-1)-shifted derived Poisson manifolds via BV-infinity operators.
problem Quantizing -shifted derived Poisson manifolds.
method Using BV-infinity operators on the space of Berezinian half-densities, proving quantization via lifting of Maurer-Cartan elements.
result Quantization of -shifted derived Poisson manifolds is equivalent to the vanishing of the second Poisson cohomology group.