The Lie groupoid analogue of a symplectic Lie group

summary:A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a $t$-symplectic Lie groupoid; the “$t$" is motivated by the fact that each target fiber of a $t$-symplectic Lie groupoid is a symplectic manifold. For a Lie groupoid $\mathcal{G}\rightrightarrows M$, we show that there is a one-to-one correspondence between quasi-Frobenius Lie algebroid structures on $A\mathcal{G}$ (the associated Lie algebroid) and $t$-symplectic Lie groupoid structures on $\mathcal{G}\rightrightarrows M$. In addition, we also introduce the notion of a symplectic Lie group bundle (SLGB) which is a special case of both a $t$-symplectic Lie groupoid and a Lie group bundle. The basic properties of SLGBs are explored

Left-invariant Hermitian connections on Lie groups with almost Hermitian structures

Left-invariant Hermitian and Gauduchon connections are studied on an arbitrary Lie group $G$ equipped with an arbitrary left-invariant almost Hermitian structure $(\langle\cdot,\cdot\rangle,J)$. The space of left-invariant Hermitian connections is shown to be in one-to-one correspondence with the space $\wedge^{(1,1)}\mathfrak{g}^\ast\otimes \mathfrak{g}$ of left-invariant 2-forms of type (1,1) (with respect to $J$) with values in \mathfrak{g}:=\mbox{Lie}(G). Explicit formulas are obtained for the torsion components of every Hermitian and Gauduchon connection with respect to a convenient choice of left-invariant frame on $G$. The curvature of Gauduchon connections is studied for the special case $G=H\times A$, where $H$ is an arbitrary $n$-dimensional Lie group, $A$ is an arbitrary $n$-dimensional abelian Lie group, and the almost complex structure is totally real with respect to \mathfrak{h}:=\mbox{Lie}(H). When $H$ is compact, it is shown that $H\times A$ admits a left-invariant (strictly) almost Hermitian structure such that the Gauduchon connection corresponding to the Strominger (or Bismut) connection in the integrable case is precisely the trivial left-invariant connection and, in addition, has totally skew-symmetric torsion. Moreover, the affine line of Gauduchon connections on $H\times A$ with the aforementioned almost Hermitian structure also contains a (nontrivial) flat connection.Comment: 26 pages, a new reference has been added in regard to Proposition A.

An Invitation to Linear Algebra

This is an OER textbook on linear algebra

Bicovariant differential calculi for finite global quotients

Let ((M,G)) be a finite global quotient, that is, a finite set (M) with an action by a finite group $G$. In this note, we classify all bicovariant first order differential calculi (FODCs) over the weak Hopf algebra (Bbbk(Gltimes M)simeq Bbbk[Gltimes M]^ast), where (Gltimes M) is the action groupoid associated to ((M,G)), and (Bbbk[Gltimes M]) is the groupoid algebra of (Gltimes M). Specifically, we prove a necessary and sufficient condition for a FODC over (Bbbk(Gltimes M)) to be bicovariant and then show that the isomorphism classes of bicovariant FODCs over (Bbbk(Gltimes M)) are in one-to-one correspondence with subsets of a certain quotient space

Loss of Mfn2 results in progressive, retrograde degeneration of dopaminergic neurons in the nigrostriatal circuit

Mitochondria continually undergo fusion and fission, and these dynamic processes play a major role in regulating mitochondrial function. Studies of several genes associated with familial Parkinson's disease (PD) have implicated aberrant mitochondrial dynamics in the disease pathology, but the importance of these processes in dopaminergic neurons remains poorly understood. Because the mitofusins Mfn1 and Mfn2 are essential for mitochondrial fusion, we deleted these genes from a subset of dopaminergic neurons in mice. Loss of Mfn2 results in a movement defect characterized by reduced activity and rearing. In open field tests, Mfn2 mutants show severe, age-dependent motor deficits that can be rescued with L-3,4 dihydroxyphenylalanine. These motor deficits are preceded by the loss of dopaminergic terminals in the striatum. However, the loss of dopaminergic neurons in the midbrain occurs weeks after the onset of these motor and striatal deficits, suggesting a retrograde mode of neurodegeneration. In our conditional knockout strategy, we incorporated a mitochondrially targeted fluorescent reporter to facilitate tracking of mitochondria in the affected neurons. Using an organotypic slice culture system, we detected fragmented mitochondria in the soma and proximal processes of these neurons. In addition, we found markedly reduced mitochondrial mass and transport, which may contribute to the neuronal loss. These effects are specific for Mfn2, as the loss of Mfn1 yielded no corresponding defects in the nigrostriatal circuit. Our findings indicate that perturbations of mitochondrial dynamics can cause nigrostriatal defects and may be a risk factor for the neurodegeneration in PD

A Categorical Approach to Groupoid Frobenius Algebras

In this paper, we show that \C{G}-Frobenius algebras (for \C{G} a finite groupoid) correspond to a particular class of Frobenius objects in the representation category of D(k[\C{G}]), where D(k[\C{G}]) is the Drinfeld double of the quantum groupoid k[\C{G}].Comment: final version; to appear in Applied Categorical Structure