HDMI cables are backward compatible, so an HDMI cable with 4K capabilities will transmit 1080p video, which is the resolution of your 1080p TV.
Does 4K HDMI work on 1080p monitor?
Can you use 4K HDMI cable on non 4K monitor?
HDMI 1.4 – If you want your HDMI cables to support 4K resolution, you need to make sure that they are High-Speed HDMI cables. They are tested to transmit video resolutions from 1080p to 4K with a richer color palette. With or without HDR, you need High-Speed HDMI cables.