arXiv subjects

Krishnendu Gongopadhyay

Publications and source records attributed to Krishnendu Gongopadhyay.

77 records · Page 5Linked to original sources

arXiv2010

On the existence of an invariant non-degenerate bilinear form under a linear map

Let $\V$ be a vector space over a field $\F$. Assume that the characteristic of $\F$ is \emph{large}, i.e. $char(\F)>\dim \V$. Let $T: \V \to \V$ be an invertible linear map. We answer the following question in this paper: When does $\V$ admit a $T$-invariant non-degenerate symmetric (resp. skew-symmetric) bilinear form? We also answer the infinitesimal version of this question. Following Feit-Zuckerman \cite{fz}, an element $g$ in a group $G$ is called real if it is conjugate in $G$ to its own inverse. So it is important to characterize real elements in $\G(\V, \F)$. As a consequence of the answers to the above question, we offer a characterization of the real elements in $\G(V, \F)$. Suppose $\V$ is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form $B$. Let $S$ be an element in the isometry group $I(\V, B)$. A non-degenerate $S$-invariant subspace $\W$ of $(\V, B)$ is called orthogonally indecomposable with respect to $S$ if it is not an orthogonal sum of proper $S$-invariant subspaces. We classify the orthogonally indecomposable subspaces. This problem is nontrivial for the unipotent elements in $I(\V, B)$. The level of a unipotent $T$ is the least integer $k$ such that $(T-I)^k=0$. We also classify the levels of unipotents in $I(\V, B)$.

math.AC↗

arXiv2010

Conjugacy classes in Möbius groups

Let $\H^{n+1}$ denote the $n + 1$-dimensional (real) hyperbolic space. Let $\s^{n}$ denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of $\s^n$ is denoted by $M (n)$. Let $M_o (n)$ be its identity component which consists of all orientation-preserving elements in $M (n)$. The conjugacy classification of isometries in $M_o (n)$ depends on the conjugacy of $T$ and $T^{-1}$ in $M_o (n)$. For an element $T$ in $M (n)$, $T$ and $T^{-1}$ are conjugate in $M (n)$, but they may not be conjugate in $M_o (n)$. In the literature, $T$ is called real if $T$ is conjugate in $M_o (n)$ to $T^{-1}$. In this paper we classify real elements in $M_o (n)$. Let $T$ be an element in $M_o(n)$. Corresponding to $T$ there is an associated element $T_o$ in $SO(n+1)$. If the complex conjugate eigenvalues of $T_o$ are given by $\{e^{iθ_j}, e^{-iθ_j}\}$, $0 < θ_j \leq π$, $j=1,...,k$, then $\{θ_1,...,θ_k\}$ are called the \emph{rotation angles} of $T$. If the rotation angles of $T$ are distinct from each-other, then $T$ is called a \emph{regular} element. After classifying the real elements in $M_o (n)$ we have parametrized the conjugacy classes of regular elements in $M_o (n)$. In the parametrization, when $T$ is not conjugate to $T^{-1}$, we have enlarged the group and have considered the conjugacy class of $T$ in $M (n)$. We prove that each such conjugacy class can be induced with a fibration structure.

math.GT↗

arXiv2009

On the Conjugacy Classes in the orthogonal and symplectic groups over algebraically closed fields

Let $\F$ be an algebraically closed field. Let $\V$ be a vector space equipped with a non-degenerate symmetric or symplectic bilinear form $B$ over $\F$. Suppose the characteristic of $\F$ is \emph{large}, i.e. either zero or greater than the dimension of $\V$. Let $I(\V, B)$ denote the group of isometries. Using the Jacobson-Morozov lemma we give a new and simple proof of the fact that two elements in $I(\V,B)$ are conjugate if and only if they have the same elementary divisors.

math.GR↗

arXiv2009

Algebraic characterization of the isometries of the hyperbolic 5-space

Using the representation of the isometries as 2x2 invertible matrices over the division algebra $\H$ of quaternions, we give an algebraic characterization of the dynamical types of the orientation-preserving isometries of the hyperbolic 5-space. We also determine the conjugacy classes and the conjugacy classes of centralizers or the z-classes in $GL(2, \H)$.

math.GT↗

arXiv2009

$z$-Classes of Isometries of The Hyperbolic Space

Let $G$ be a group. Two elements $x, y$ are said to be {\it $z$-equivalent} if their centralizers are conjugate in $G$. The class equation of $G$ is the partition of $G$ into conjugacy classes. Further decomposition of conjugacy classes into $z$-classes provides an important information about the internal structure of the group. Let $I(\H^n)$ denote the group of isometries of the hyperbolic $n$-space. We show that the number of $z$-classes in $I(\H^n)$ is finite. We actually compute their number, cf. theorem 1.3. We interpret the finiteness of $z$-classes as accounting for the finiteness of "dynamical types" in $I(\H^n)$. Along the way we also parametrize conjugacy classes. We mainly use the linear model for the hyperbolic space for this purpose. This description of parametrizing conjugacy classes appears to be new.

math.GT↗