## Return to Answer

Post Made Community Wiki by S. Carnahan♦
1

IN addition to what has been said, I would like to mention the following aspect: a group action usually does not come alone. When $G$ acts on a set $X$ you will almost for sure be interested also in the induced action of $G$ on certain kind of functions on $X$: think of a smooth manifold $M$ and of smooth functions, or of smooth tensor fields on $M$, or what not. Then the induced group action can be either viewed as a right action when $G$ acts from the left on $X$ or you have to plug in a $^{-1}$ to turn things again into a left action. In various situations (say equivariant maps in the dual of a Lie algebra etc) it might become a notational desaster if you want to specify all kind of actions with a separate symbol. However, it might be important to keep track of whether you have a left or a right action.

So my habit is to denote the action of a group element $g$ on some object $x \in X$ by $g \triangleright x$ if it is a left action and by $g \triangleleft x$ or better $x \triangleleft g$ if it is a right action. Then you don't have to bother whether you have already included the $^{-1}$ in the coadjoint action to make it a left action or just stay with a right action :)
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 15, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9458324909210205, "perplexity_flag": "head"}