Projective representations of mapping class groups in combinatorial quantization

Let $\Sigma_{g,n}$ be a compact oriented surface of genus $g$ with $n$ open disks removed. The graph algebra $\mathcal{L}_{g,n}(H)$ was introduced by Alekseev--Grosse--Schomerus and Buffenoir--Roche and is a combinatorial quantization of the moduli space of flat connections on $\Sigma_{g,n}$. We construct a projective representation of the mapping class group of $\Sigma_{g,n}$ using $\mathcal{L}_{g,n}(H)$ and its subalgebra of invariant elements. Here we assume that the gauge Hopf algebra $H$ is finite-dimensional, factorizable and ribbon, but not necessarily semi-simple. We also give explicit formulas for the representation of the Dehn twists generating the mapping class group; in particular, we show that it is equivalent to a representation constructed by V. Lyubashenko using categorical methods.Comment: 32 pages; minor corrections and improvements; new section and new theorem adde

Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras

Let $\Sigma_{g,n}$ be a compact oriented surface of genus $g$ with $n$ open disks removed. The algebra $\mathcal{L}_{g,n}(H)$ was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and is a combinatorial quantization of the moduli space of flat connections on $\Sigma_{g,n}$. Here we focus on the two building blocks $\mathcal{L}_{0,1}(H)$ and $\mathcal{L}_{1,0}(H)$ under the assumption that the gauge Hopf algebra $H$ is finite-dimensional, factorizable and ribbon, but not necessarily semisimple. We construct a projective representation of $\mathrm{SL}_2(\mathbb{Z})$, the mapping class group of the torus, based on $\mathcal{L}_{1,0}(H)$ and we study it explicitly for $H = \overline{U}_q(\mathfrak{sl}(2))$. We also show that it is equivalent to the representation constructed by Lyubashenko and Majid

Unrestricted quantum moduli algebras, III: Surfaces of arbitrary genus and skein algebras

We prove that the quantum moduli algebra associated to a possibly punctured compact oriented surface and a complex semisimple Lie algebra $\mathfrak{g}$ is a Noetherian and finitely generated ring; if the surface has punctures, we prove also that it has no non-trivial zero divisors. Moreover, we show that the quantum moduli algebra is isomorphic to the skein algebra of the surface, defined by means of the Reshetikhin-Turaev functor for the quantum group $U_q(\mathfrak{g})$, and which coincides with the Kauffman bracket skein algebra when $\mathfrak{g}=\mathfrak{sl}_2$.Comment: V1: 60 pages, 26 figures; V2: 75 pages, 37 figures, with typos corrected, section 6. 3 rewritten with simpler arguments and a new result (Corollary 6.12), and section 7 added, with results about the quantum reductio

Mosaic dysfunction of mitophagy in mitochondrial muscle disease

Mitophagy is a quality control mechanism that eliminates damaged mitochondria, yet its significance in mammalian pathophysiology and aging has remained unclear. Here, we report that mitophagy contributes to mitochondrial dysfunction in skeletal muscle of aged mice and human patients. The early disease stage is characterized by muscle fibers with central nuclei, with enhanced mitophagy around these nuclei. However, progressive mitochondrial dysfunction halts mitophagy and disrupts lysosomal homeostasis. Interestingly, activated or halted mitophagy occur in a mosaic manner even in adjacent muscle fibers, indicating cell-autonomous regulation. Rapamycin restores mitochondrial turnover, indicating mTOR-dependence of mitochondrial recycling in advanced disease stage. Our evidence suggests that (1) mitophagy is a hallmark of age-related mitochondrial pathology in mammalian muscle, (2) mosaic halting of mitophagy is a mechanism explaining mosaic respiratory chain deficiency and accumulation of pathogenic mtDNA variants in adult-onset mitochondrial diseases and normal aging, and (3) augmenting mitophagy is a promising therapeutic approach for muscle mitochondrial dysfunction.Peer reviewe

Effects of aging and caloric restriction on fiber type composition, mitochondrial morphology and dynamics in rat oxidative and glycolytic muscles

Aging is associated with a progressive decline in muscle mass and strength, a process known as sarcopenia. Evidence indicates that mitochondrial dysfunction plays a causal role in sarcopenia and suggests that alterations in mitochondrial dynamics/morphology may represent an underlying mechanism. Caloric restriction (CR) is among the most efficient nonpharmacological interventions to attenuate sarcopenia in rodents and is thought to exert its beneficial effects by improving mitochondrial function. However, CR effects on mitochondrial morphology and dynamics, especially in aging muscle, remain unknown. To address this issue, we investigated mitochondrial morphology and dynamics in the oxidative soleus (SOL) and glycolytic white gastrocnemius (WG) muscles of adult (9-month-old) ad libitum-fed (AL; A-AL), old (22-month-old) AL-fed (O-AL), and old CR (O-CR) rats. We show that CR attenuates the aging-related decline in the muscle-to-body-weight ratio, a sarcopenic index. CR also prevented the effects of aging on muscle fiber type composition in both muscles. With aging, the SOL displayed fragmented SubSarcolemmal (SS) and InterMyoFibrillar (IMF) mitochondria, an effect attenuated by CR. Aged WG displayed enlarged SS and more complex/branched IMF mitochondria. CR had marginal anti-aging effects on WG mitochondrial morphology. In the SOL, DRP1 (pro-fission protein) content was higher in O-AL vs YA-AL, and Mfn2 (pro-fusion) content was higher in O-CR vs A-AL. In the gastrocnemius, Mfn2, Drp1, and Fis1 (pro-fission) contents were higher in O-AL vs A-AL. CR reduced this aging-related increase in Mfn2 and Fis1 content. Overall, these results reveal for the first time that aging differentially impacts mitochondrial morphology and dynamics in different muscle fiber types, by increasing fission/fragmentation in oxidative fibers while enhancing mitochondrial size and branching in glycolytic fibers. Our results also indicate that although CR partially attenuates aging-related changes in mitochondrial dynamics in glycolytic fibers, its anti-aging effect on mitochondrial morphology is restricted to oxidative fibers

Effects of Aging and Caloric Restriction on Fiber Type Composition, Mitochondrial Morphology and Dynamics in Rat Oxidative and Glycolytic Muscles

Aging is associated with a progressive decline in muscle mass and strength, a process known as sarcopenia. Evidence indicates that mitochondrial dysfunction plays a causal role in sarcopenia and suggests that alterations in mitochondrial dynamics/morphology may represent an underlying mechanism. Caloric restriction (CR) is among the most efficient nonpharmacological interventions to attenuate sarcopenia in rodents and is thought to exert its beneficial effects by improving mitochondrial function. However, CR effects on mitochondrial morphology and dynamics, especially in aging muscle, remain unknown. To address this issue, we investigated mitochondrial morphology and dynamics in the oxidative soleus (SOL) and glycolytic white gastrocnemius (WG) muscles of adult (9-month-old) ad libitum-fed (AL; A-AL), old (22-month-old) AL-fed (O-AL), and old CR (O-CR) rats. We show that CR attenuates the aging-related decline in the muscle-to-body-weight ratio, a sarcopenic index. CR also prevented the effects of aging on muscle fiber type composition in both muscles. With aging, the SOL displayed fragmented SubSarcolemmal (SS) and InterMyoFibrillar (IMF) mitochondria, an effect attenuated by CR. Aged WG displayed enlarged SS and more complex/branched IMF mitochondria. CR had marginal anti-aging effects on WG mitochondrial morphology. In the SOL, DRP1 (pro-fission protein) content was higher in O-AL vs YA-AL, and Mfn2 (pro-fusion) content was higher in O-CR vs A-AL. In the gastrocnemius, Mfn2, Drp1, and Fis1 (pro-fission) contents were higher in O-AL vs A-AL. CR reduced this aging-related increase in Mfn2 and Fis1 content. Overall, these results reveal for the first time that aging differentially impacts mitochondrial morphology and dynamics in different muscle fiber types, by increasing fission/fragmentation in oxidative fibers while enhancing mitochondrial size and branching in glycolytic fibers. Our results also indicate that although CR partially attenuates aging-related changes in mitochondrial dynamics in glycolytic fibers, its anti-aging effect on mitochondrial morphology is restricted to oxidative fibers